Introduction
A Kakeya set in R n \mathbb{R}^{n} R n n n n
Theorem 1.1. Let K ⊂ R 4 K \subset\mathbb{R}^{4} K ⊂ R 4 Suppose that for every e ∈ S 3 e \in S^{3} e ∈ S 3 there is a point a e ∈ R 4 a_{e} \in\mathbb{R}^{4} a e ∈ R 4 such that
{ a e + t e : 0 ≤ t ≤ 1 } ⊂ K . \{a_{e}+te:0\le t\le1\}\subset K. { a e + t e : 0 ≤ t ≤ 1 } ⊂ K . Then dim H K = 4 \dim_{H}K=4 dim H K = 4
Theorem 1.1 resolves the Hausdorff-dimension Kakeya conjecture positively in four dimensions. Neither K K K
For arbitrary sets, we use the covering definition
H δ s ( K ) = inf { ∑ i ( diam U i ) s : K ⊂ ⋃ i U i , diam U i ≤ δ } , H s ( K ) = lim δ ↓ 0 H δ s ( K ) , \mathcal{H}^{s}_{\delta}(K)=\inf\left\{\sum_{i}(\operatorname{diam}U_{i})^{s}:K\subset\bigcup_{i}U_{i},\ \operatorname{diam}U_{i}\le\delta\right\},\qquad\mathcal{H}^{s}(K)=\lim_{\delta\downarrow0}\mathcal{H}^{s}_{\delta}(K), H δ s ( K ) = inf { i ∑ ( diam U i ) s : K ⊂ i ⋃ U i , diam U i ≤ δ } , H s ( K ) = δ ↓ 0 lim H δ s ( K ) , where the covers are countable, and dim H K = inf { s : H s ( K ) = 0 } \dim_{H}K=\inf\{s:\mathcal{H}^{s}(K)=0\} dim H K = inf { s : H s ( K ) = 0 }
Related dimension statements and geometric consequences are developed in Section 11. Besides the packing- and bounded-set Minkowski-dimension implications, these include a lower bound in higher dimensions by orthogonal projection, arbitrary direction sets and extension of segments to lines, and Nikodym and curved-Kakeya applications. The latter implications use the transfer results of Keleti and Máthé [20 ] , Gao et al. [11 ] , and Nadjimzadah [23 ] , with their distinct hypotheses stated there.
Context and input estimates
The Kakeya problem connects the geometry of line segments with estimates for oscillatory operators. The classical needle problem asks how small a planar region can be if a unit segment is to turn continuously within it and return with its endpoints reversed; the area can be arbitrarily small [3 , 10 ] . A central difficulty in the dimension problem is that many segments can overlap substantially even when their directions are well distributed. Besicovitch’s constructions show that a Kakeya set can have Lebesgue measure zero, whereas Davies proved full Hausdorff dimension in the plane [3 , 10 ] .
For bounded Kakeya sets, the upper Minkowski formulation asks for dimension n n n n ≥ 2 n \ge2 n ≥ 2 T δ ( a , e ) T_\delta(a,e) T δ ( a , e ) a ∈ R n a \in\mathbb{R}^n a ∈ R n e ∈ S n − 1 e \in S^{n-1} e ∈ S n − 1 δ \delta δ
K δ f ( e ) = sup a ∈ R n 1 ∣ T δ ( a , e ) ∣ ∫ T δ ( a , e ) ∣ f ( x ) ∣ d x . K_\delta f(e) = \sup_{a \in\mathbb{R}^n} \frac{1}{|T_\delta(a,e)|} \int_{T_\delta(a,e)} |f(x)| \, dx. K δ f ( e ) = a ∈ R n sup ∣ T δ ( a , e ) ∣ 1 ∫ T δ ( a , e ) ∣ f ( x ) ∣ d x . The stronger Kakeya maximal conjecture asks that, for every ε > 0 \varepsilon> 0 ε > 0 C ε , n C_{\varepsilon,n} C ε , n
∥ K δ f ∥ L n ( S n − 1 ) ≤ C ε , n δ − ε ∥ f ∥ L n ( R n ) \|K_\delta f\|_{L^n(S^{n-1})} \le C_{\varepsilon,n}\delta^{-\varepsilon}\|f\|_{L^n(\mathbb{R}^n)} ∥ K δ f ∥ L n ( S n − 1 ) ≤ C ε , n δ − ε ∥ f ∥ L n ( R n ) for all f ∈ L n ( R n ) f \in L^n(\mathbb{R}^n) f ∈ L n ( R n ) 0 < δ < 1 0 < \delta< 1 0 < δ < 1 [17 ] . For partial maximal estimates, the dimension parameter records the Hausdorff lower bound supplied by that estimate. Theorem 1.1 proves the four-dimensional Hausdorff assertion.
Wolff’s bound dim H K ≥ ( n + 2 ) / 2 \dim_H K \ge(n+2)/2 dim H K ≥ ( n + 2 ) /2 R 4 \mathbb{R}^4 R 4 [14 , 33 ] . Bourgain connected line incidences in separated slices to sumset and difference-set estimates, using Gowers’s quantitative form of the Balog–Szemerédi theorem [5 ] . Katz and Tao developed sums–differences iterations for higher-dimensional bounds [17 ] . In three dimensions, Katz, Łaba, and Tao combined multiscale clustering and local planar structure with the arithmetic method to improve the upper Minkowski lower bound beyond 5 / 2 5/2 5/2 [19 ] . Łaba and Tao then used multiscale geometric structure to obtain upper Minkowski dimension strictly greater than ( n + 2 ) / 2 (n+2)/2 ( n + 2 ) /2 n ≥ 4 n \ge4 n ≥ 4 [21 ] , Theorem 1.3.
The later four-dimensional Hausdorff improvements addressed concentration near algebraic sets. For compact Kakeya sets, the polynomial Wolff estimates of Guth and Zahl, together with the algebraic direction bounds of Zahl and Katz and Rogers, give the lower bound 3 + 1 / 40 3+1/40 3 + 1/40 [14 , 16 , 34 ] . The polynomial Wolff axioms restrict concentration near low-degree algebraic sets; the accompanying multilinear estimates exploit quantitative linear independence of tube directions [14 ] . Katz and Zahl’s planebrush organizes tubes around a common plane and combines this geometry with Guth and Zahl’s trilinear estimate. It gives Hausdorff dimension at least 3.059, while their distinct maximal-function estimate has dimension parameter at least 3.049 [18 ] . Borges, Chan, Chen, Liu, Xi, and Zhan subsequently improved the maximal parameter to ( 159 + 145 ) / 56 ≈ 3.0543 (159+\sqrt{145})/56 \approx3.0543 ( 159 + 145 ) /56 ≈ 3.0543 [4 ] , Theorem 1.2. This improves the maximal estimate while leaving the larger Hausdorff bound 3.059 as a separate conclusion.
Rai Choudhuri proved the bound 13 / 4 13/4 13/4 [26 ] . Wang and Zakharov have since obtained a full-dimensional union estimate for almost AD-regular sticky tube families satisfying convex Wolff axioms and a dense-shading condition [32 ] ; those hypotheses do not cover arbitrary Kakeya sets. In three dimensions, Wang and Zahl proved the Kakeya set conjecture [30 ] , and Guth, Wang, and Zahl subsequently streamlined the proof [15 ] .
The three-dimensional estimate used in our reduction is the weighted full-time plank bound of OpenAI [24 ] , Lemma 2.3, recorded as Lemma 2.15. It is proved there by weighted convex clustering from the union estimate of Guth et al. [15 ] , Theorem 1.1. It supplies spatial boxes with enough incidence mass to support the quadratic local fits described below. The further analytic inputs have distinct roles. Determinant multilinear Kakeya [7 ] controls configurations of independent velocities. The restricted-triple dot-product theorem of Wang and Zahl [31 ] and the multiplicative-convolution estimate of Orponen et al. [25 ] provide planar rigidity. The latter is applied in Proposition 7.6 to construct scalar coefficients with small interval probabilities. That step also uses a graph Balog–Szemerédi–Gowers argument and fixed-iterate sumset calculus [28 , 29 ] . The graph argument retains a substantial set of original incidence edges, which are needed to fit the resulting planar structure back to the trajectories.
Mechanisms of the proof
The proof uses quadratic local fits to capture line concentration that persists across scales. Its basic object is a chart : a collection of line pieces in a moving spatial box and a time interval, on which one quadratic polynomial test stays small. Each chart also carries a mass requirement, measured using selected time-line pairs. This prevents a fit to an exceptional collection of negligible weight from counting as progress. Normalizing inside a chart produces another problem of the same kind, with its time interval rescaled to unit length. Charts can therefore be nested.
The two quantities to balance are the thickness of the quadratic relation and the length of the line pieces on which it holds. Longer intervals at a prescribed thickness give better charts. The reduction in Section 2 turns a hypothetical Hausdorff deficit into a weighted line problem with two competing properties: refining to terminal depth loses less than one power of density per unit depth, yet every substantial terminal chart system must use intervals of a definite shrinking length. The proof rules out this combination.
More precisely, write the lines as ( t , y 0 + t V , w 0 + t W ) (t, y_0 + tV, w_0 + tW) ( t , y 0 + t V , w 0 + t W ) y ∈ R 2 y \in\mathbb{R}^2 y ∈ R 2 ( t , y ) (t, y) ( t , y ) w w w N 0 → ∞ N_0 \to\infty N 0 → ∞ n ( r ) = N 0 − f ( r ) n(r) = N_0^{-f(r)} n ( r ) = N 0 − f ( r ) d t d y \mathrm{d}t\,\mathrm{d}y d t d y
f ( L ) − f ( r ) ≤ d ( L − r ) ( 0 ≤ r ≤ L ) , d < 1 , f(L) - f(r) \le d(L-r) \qquad(0 \le r \le L), \qquad d < 1, f ( L ) − f ( r ) ≤ d ( L − r ) ( 0 ≤ r ≤ L ) , d < 1 , where L L L true when its incidence mass, divided by its interval length, is at least this density times its spatial volume, up to subpower factors. A chart system’s coverage is the selected incidence mass, and mass N 0 − o ( 1 ) N_0^{-o(1)} N 0 − o ( 1 ) N 0 − h + o ( 1 ) N_0^{-h+o(1)} N 0 − h + o ( 1 ) time cost is h h h horizon H ( E ) H(E) H ( E ) E E E H ( E ) > 0 H(E) > 0 H ( E ) > 0
The exponent below one makes a comparison principle available. Local graph representations of the polynomial tests give scalar sheets, which are analytic branches of polynomial equations. The density bound and the lower mass bounds of retained entries supply lists of size at most δ − d + o ( 1 ) \delta^{-d+o(1)} δ − d + o ( 1 ) δ \delta δ d < 1 d < 1 d < 1
To locate a forbidden improvement, we follow nested charts in a nearly extremal weighted problem. Their relative density and time profiles become linear, with time rate k > 0 k > 0 k > 0 narrowness ℓ \ell ℓ ℓ \ell ℓ
There are three geometric regimes. When ℓ = 0 \ell= 0 ℓ = 0
When 0 < ℓ < ∞ 0 < \ell< \infty 0 < ℓ < ∞ Y ′ = − Z + t v Y^\prime= -Z + tv Y ′ = − Z + t v H H H H 2 H^2 H 2 2 k = 1 2k = 1 2 k = 1 2 k < 1 2k < 1 2 k < 1
When ℓ = ∞ \ell= \infty ℓ = ∞
Each regime initially produces local candidates on substantial residual portions of the weighted family. To obtain a contradiction, these fits must become chart assignments with enough total coverage and with finite lists of labels recoverable from the original observations. In the finite-narrowness cases, greedy coverage and the small-list comparison give this passage through Proposition 4.11. The unbounded-narrowness construction recovers coverage and finite label lists after its full-time normalization. In each case, the improved chart system is lifted to the original weighted problem, where it contradicts the extremal choice. The comparison, the finite-narrowness conversion, and the conditional higher-jet argument (Proposition 5.6) are stated separately because their hypotheses do not require a low-entropy original line family.
Organization and scale conventions
Section 2 gives the weighted models, chart normalization, and reduction from an arbitrary Kakeya set. Section 3 proves the scalar-sheet comparison and terminal extension. Section 4 constructs the critical paths and proves the criterion that turns local candidates into forbidden improvements.
Section 5 develops the conditional scalar estimates, and Section 6 treats the isotropic case and its auxiliary scalar model. Section 7 proves the planar rigidity tools and fits their conclusions to the original incidence edges. Sections 8 and 9 use these results to treat finite positive narrowness and its critical time regimes. Section 10 treats unbounded narrowness and completes the proof of Theorem 1.1. Section 11 develops the dimensional and geometric consequences.
Two scale conventions are essential when these steps are combined. Section 4 distinguishes the original resolution N 0 N_0 N 0 N N N N 0 N_0 N 0 N N N
Figure 1. The contradiction from a hypothetical Kakeya counterexample. A critical weighted path has one of three narrowness regimes. Each produces improvements on substantial residuals, which are assembled into a chart system with recoverable labels in the original problem. In the middle branch, 2 k = 1 2k = 1 2 k = 1 2 k ≠ 1 2k \ne1 2 k = 1 2 k < 1 2k < 1 2 k < 1
Weighted models and the reduction from arbitrary Kakeya sets
We encode a line family by a probability law on trajectories and a selection of trajectory–time incidences. A local fit confines the base spatial projection of each whole trajectory piece to a moving box and keeps a quadratic polynomial small along that piece. Its mass is measured on the selected incidences. Theorem 2.17 turns a hypothetical Kakeya counterexample into a weighted problem in which fitting at the final thickness requires a positive power cost in time localization. The infimum of these costs is the horizon.
We first develop the observations and density estimates used to measure these fits. We then define the charts, show how their priors and densities transform under normalization, and prove the reduction from an arbitrary Kakeya set. All subpower factors in this section refer to one exact sequence with resolution tending to infinity; later sequences of exact problems will be introduced separately.
Exact problems and observations
The model has two versions. Version 2 retains the lines in R 4 \mathbb{R}^{4} R 4
Definition 2.1 (Exact problems). An exact problem is a sequence indexed by a real parameter N 0 → ∞ N_0 \to\infty N 0 → ∞ 0 < L < ∞ 0 < L < \infty 0 < L < ∞ 0 ≤ r ≤ L 0 \le r \le L 0 ≤ r ≤ L a r = N 0 − r a_r = N_0^{-r} a r = N 0 − r K 0 ≥ 1 K_0 \ge1 K 0 ≥ 1 log K 0 / log N 0 → 0 \log K_0 / \log N_0 \to0 log K 0 / log N 0 → 0 A ∼ log B A \sim_{\log} B A ∼ l o g B K 0 − 1 B ≤ A ≤ K 0 B K_0^{-1}B \le A \le K_0B K 0 − 1 B ≤ A ≤ K 0 B
There are finitely many auxiliary worlds γ \gamma γ ω γ ≥ 0 \omega_\gamma\ge0 ω γ ≥ 0 ∑ γ ω γ = 1 \sum_\gamma\omega_\gamma= 1 ∑ γ ω γ = 1 γ \gamma γ ν γ \nu_\gamma ν γ t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ]
y ( t ) = y 0 + t V ∈ R j , ∣ y ( t ) ∣ + ∣ V ∣ ≤ K 0 . y(t) = y_0 + tV \in\mathbb{R}^j, \qquad|y(t)| + |V| \le K_0. y ( t ) = y 0 + t V ∈ R j , ∣ y ( t ) ∣ + ∣ V ∣ ≤ K 0 . We use the following two versions.
In version 1, j = 1 j = 1 j = 1 w ( t ) = w 0 + w 1 t + w 2 t 2 w(t) = w_0 + w_1t + w_2t^2 w ( t ) = w 0 + w 1 t + w 2 t 2 sup [ 0 , 1 ] ∣ w ∣ ≤ K 0 \sup_{[0,1]} |w| \le K_0 sup [ 0 , 1 ] ∣ w ∣ ≤ K 0 P ( t , y , w ) = w − F ( t , y ) P(t,y,w) = w - F(t,y) P ( t , y , w ) = w − F ( t , y ) F F F B = 1 B = 1 B = 1 X = w X = w X = w
In version 2, j = 2 j = 2 j = 2 w ( t ) = w 0 + t W w(t) = w_0 + tW w ( t ) = w 0 + t W w w w y y y B > 0 B > 0 B > 0 P ∗ ( t , y , w ) P_*(t,y,w) P ∗ ( t , y , w ) X ( t ) = P ∗ ( t , y ( t ) , w ( t ) ) X(t) = P_*(t,y(t),w(t)) X ( t ) = P ∗ ( t , y ( t ) , w ( t ))
sup [ 0 , 1 ] ∣ X ∣ ≤ K 0 , sup [ 0 , 1 ] ∣ ∂ w P ∗ ∣ ≤ K 0 B , ∣ ∂ w P ∗ ∣ ≥ B / K 0 at selected incidences , ∣ ∂ w w P ∗ ∣ ≤ K 0 B 2 . (2.1) \begin{aligned}
\sup_{[0,1]} |X| &\le K_0, & \sup_{[0,1]} |\partial_w P_*| &\le K_0B,\\
|\partial_w P_*| &\ge B/K_0 \quad\text{at selected incidences}, & |\partial_{ww}P_*| &\le K_0B^2.
\end{aligned}\tag{2.1} [ 0 , 1 ] sup ∣ X ∣ ∣ ∂ w P ∗ ∣ ≤ K 0 , ≥ B / K 0 at selected incidences , [ 0 , 1 ] sup ∣ ∂ w P ∗ ∣ ∣ ∂ w w P ∗ ∣ ≤ K 0 B , ≤ K 0 B 2 . ( 2.1 ) Tests in this version are arbitrary polynomials P ( t , y , w ) P(t,y,w) P ( t , y , w )
In both versions X X X [ 0 , 1 ] [0,1] [ 0 , 1 ]
An incidence selection is an event E γ E_\gamma E γ
d μ γ ( l , t ) = 1 E γ ( l , t ) d ν γ ( l ) d t , μ = ∑ γ ω γ μ γ , μ ( a l l ) ≥ K 0 − 1 . (2.2) d\mu_\gamma(l,t) = 1_{E_\gamma}(l,t)\,d\nu_\gamma(l)\,dt, \qquad\mu= \sum_\gamma\omega_\gamma\mu_\gamma, \qquad\mu(\mathrm{all}) \ge K_0^{-1}.\tag{2.2} d μ γ ( l , t ) = 1 E γ ( l , t ) d ν γ ( l ) d t , μ = γ ∑ ω γ μ γ , μ ( all ) ≥ K 0 − 1 . ( 2.2 ) Thus μ γ \mu_\gamma μ γ
Here and below polynomial bounds are N 0 M N_0^M N 0 M M M M
Definition 2.2 (Tempered data). The numbers of worlds, specified parameter ranges and coefficients, and B B B B − 1 B^{-1} B − 1
Exact-level selections, labels, cells, and frames obey such uniform bounds. Chart widths and time lengths have polynomially bounded reciprocals. Changes of trajectory coordinates are invertible and have polynomially controlled constant Jacobian. We permit polynomially many copies of coefficient space, with an internal discrete trajectory index. This index is not a world or an observed point feature; all assignments concern indexed trajectories.
Measurable arguments may be used to locate a finite family of boxes or polynomial tests. The output is encoded by those constants and geometric tests, not by a description of the intermediate measurable search.
Worlds with extremely small success fractions can be removed. After forming chart worlds, charts with extremely small conditional prior mass can also be removed: there are only polynomially many charts, and the selected measure is bounded by prior times time. The threshold can be a sufficiently small reciprocal polynomial. These observations preserve the convention in Definition 2.2.
Use nested half-open dyadic grids, rounding requested widths upwards within a factor of two. The observation at depth r r r
p r = ( γ , t , y ( t ) , [ w ( t ) ] a r / B ) . (2.3) p_r=(\gamma,t,y(t),[w(t)]_{a_r/B}).\tag{2.3} p r = ( γ , t , y ( t ) , [ w ( t ) ] a r / B ) . ( 2.3 ) Within a world its density is with respect to d t d y dt\,dy d t d y t t t y 0 y_0 y 0 y ( t ) = y 0 + t V y(t)=y_0+tV y ( t ) = y 0 + t V
A density is typically n = N 0 − f n=N_0^{-f} n = N 0 − f τ > 0 \tau>0 τ > 0 [ N 0 − τ n , N 0 τ n ] [N_0^{-\tau}n,N_0^\tau n] [ N 0 − τ n , N 0 τ n ] N 0 − c ( τ ) N_0^{-c(\tau)} N 0 − c ( τ ) c ( τ ) > 0 c(\tau)>0 c ( τ ) > 0 p r p_r p r 0 ≤ r ≤ L 0\le r\le L 0 ≤ r ≤ L n ( r ) = N 0 − f ( r ) n(r)=N_0^{-f(r)} n ( r ) = N 0 − f ( r )
Density selection and homogeneous profiles
The selected observation densities need not initially have uniform sizes. We will restrict to incidences on which their logarithmic sizes stabilize. Restrictions and finite refinements obey a simple mass estimate; a finite-table approximation will make the subsequent density selections compatible with the tempered data.
Lemma 2.3 (Restriction, refinement, and caps). Let g g g unnormalized density of an observation and g ′ g' g ′ 0 < ϵ < 1 0<\epsilon<1 0 < ϵ < 1
∫ { g ′ < ϵ g } g ′ ≤ ϵ ∫ g . (2.4) \int_{\{g'<\epsilon g\}}g'\le\epsilon\int g.\tag{2.4} ∫ { g ′ < ϵ g } g ′ ≤ ϵ ∫ g . ( 2.4 ) If an observation is refined by a discrete label with at most D D D ϵ g \epsilon g ϵ g ϵ D ∫ g \epsilon D\int g ϵD ∫ g
Consequently further inverse-subpower restrictions, and refinements by subpower lists, preserve typical logarithmic densities. Upper density bounds may be integrated over arbitrary base regions and summed over discrete entries after the exceptional observations have been removed.
Proof. Restriction gives 0 ≤ g ′ ≤ g 0\le g'\le g 0 ≤ g ′ ≤ g { g ′ < ϵ g } \{g'<\epsilon g\} { g ′ < ϵ g } (2.4) . For a refinement with densities g c ′ g'_c g c ′ g c ′ < ϵ g g'_c<\epsilon g g c ′ < ϵ g D D D ϵ = N 0 − τ \epsilon=N_0^{-\tau} ϵ = N 0 − τ D D D
For clarity about exceptional sets, write the pushed-forward measure as a good part plus an omitted part, with densities g g o o d , g b a d g_{\mathrm{good}},g_{\mathrm{bad}} g good , g bad g b a d > g g o o d g_{\mathrm{bad}}>g_{\mathrm{good}} g bad > g good 2 ∫ g b a d 2\int g_{\mathrm{bad}} 2 ∫ g bad
A datum is known by p s p_s p s K 0 K_0 K 0 p s p_s p s
Lemma 2.4 (Joint base flattening). Fix tempered discrete labels and predicates of polynomial total count. Given a fixed T > 0 T > 0 T > 0 D D D ( t , y ) (t,y) ( t , y ) ρ ∼ N 0 − D \rho\sim N_0^{-D} ρ ∼ N 0 − D O ( N 0 − T ) O(N_0^{-T}) O ( N 0 − T )
Proof. Use coordinates ( t , y , V ) (t,y,V) ( t , y , V ) ρ \rho ρ N 0 C ρ N_0^C\rho N 0 C ρ C C C
Within-cell averaging has the same bound, up to a dimensional constant: write its L 1 L^1 L 1 ρ \rho ρ D > C + T + 1 D > C+T+1 D > C + T + 1
A measurable selector has the same success up to total variation error on the flattened law. On that law the posterior of the discrete entries is constant inside each cell with its conditioning bins fixed; a constant maximizing list can be chosen there. There are polynomially many such cells and entries. The resulting lookup table is tempered, and its success transfers back by the same total variation estimate.
When two grids overlap with at most K 0 K_0 K 0
Lemma 2.5 (Homogenization). After a tempered further selection of inverse-subpower mass and a subsequence, an exact problem has a homogeneous profile f ( r ) f(r) f ( r ) 1 1 1 ( p r , [ V ] a u ) (p_r,[V]_{a_u}) ( p r , [ V ] a u ) ( r , u ) (r,u) ( r , u ) f ( r , u ) f(r,u) f ( r , u ) 1 1 1 r r r j j j u u u f ( r , 0 ) = f ( r ) f(r,0)=f(r) f ( r , 0 ) = f ( r )
Proof. Choose a countable dense depth grid containing all required endpoints. At stage N 0 N_0 N 0
Polynomial bounds give a polynomial ceiling for each flattened density. Densities below N 0 − C N_0^{-C} N 0 − C C C C ϵ N 0 → 0 \epsilon_{N_0}\to0 ϵ N 0 → 0 m N 0 m_{N_0} m N 0
( O ( ϵ N 0 − 1 ) ) m N 0 = N 0 o ( 1 ) . \left(O(\epsilon_{N_0}^{-1})\right)^{m_{N_0}}=N_0^{o(1)}. ( O ( ϵ N 0 − 1 ) ) m N 0 = N 0 o ( 1 ) . One list carries inverse-subpower mass. Restrict to it. This restriction is a cell table testing the joint discrete observations and is tempered. Its upper bounds persist, and Lemma 2.3 shows that its densities lose a fixed power relative to the preselection values only on power-small mass.
The total variation estimate transfers these conclusions to the exact base densities. Indeed, where two densities differ by a large multiplicative factor, their L 1 L^1 L 1
Refinement from depth r r r r ′ ≥ r r' \ge r r ′ ≥ r K 0 N 0 r ′ − r K_0N_0^{r'-r} K 0 N 0 r ′ − r u u u u ′ ≥ u u' \ge u u ′ ≥ u K 0 N 0 j ( u ′ − u ) K_0N_0^{j(u'-u)} K 0 N 0 j ( u ′ − u )
0 ≤ f ( r ′ , u ) − f ( r , u ) ≤ r ′ − r , 0 \le f(r',u)-f(r,u) \le r'-r, 0 ≤ f ( r ′ , u ) − f ( r , u ) ≤ r ′ − r , 0 ≤ f ( r , u ′ ) − f ( r , u ) ≤ j ( u ′ − u ) . 0 \le f(r,u')-f(r,u) \le j(u'-u). 0 ≤ f ( r , u ′ ) − f ( r , u ) ≤ j ( u ′ − u ) . These inequalities extend the profiles uniquely to all intermediate depths; nesting squeezes their typical densities between nearby tested ones. Since ∣ V ∣ ≤ K 0 |V| \le K_0 ∣ V ∣ ≤ K 0 K 0 K_0 K 0 f ( r , 0 ) = f ( r ) f(r,0)=f(r) f ( r , 0 ) = f ( r )
Other fixed logarithmic quantities with polynomial upper and reciprocal bounds may be homogenized by the same argument. In subsequent cap estimates, fixed tolerances on fixed finite grids are sent to zero sufficiently slowly, and the grids increase sufficiently slowly, that the exceptional mass is negligible compared with the particular inverse-subpower selection in use. This convention concerns the present exact N 0 N_0 N 0
The class used in the contradiction imposes two further bounds. The first limits how rapidly the typical density can decrease as the hidden observation is refined toward the terminal depth. The second limits velocity concentration within a terminal observation.
Definition 2.6 (The cap class). Fix S , η > 0 S,\eta>0 S , η > 0 C ( d ) C(d) C ( d ) d < 1 d<1 d < 1
f ( L ) − f ( r ) ≤ d ( L − r ) ( 0 ≤ r ≤ L ) , (2.5) f(L)-f(r)\le d(L-r) \qquad(0\le r\le L),\tag{2.5} f ( L ) − f ( r ) ≤ d ( L − r ) ( 0 ≤ r ≤ L ) , ( 2.5 ) and, for 0 < u ≤ η L 0<u\le\eta L 0 < u ≤ η L ( p L , [ V ] a u ) (p_L,[V]_{a_u}) ( p L , [ V ] a u )
n ( L ) N 0 − S u + o ( 1 ) . (2.6) n(L)N_0^{-Su+o(1)}.\tag{2.6} n ( L ) N 0 − S u + o ( 1 ) . ( 2.6 ) The o ( 1 ) o(1) o ( 1 ) S , η S,\eta S , η
Lemma 2.7 (Angular caps under coarsening). Suppose (2.6) holds. For 0 ≤ b < L 0\le b<L 0 ≤ b < L and 0 < u ≤ η L 0<u\le\eta L 0 < u ≤ η L the conditional probability
P { [ V ] a u = v ∣ p b } , v = [ V ] a u , \mathbb{P}\{[V]_{a_u}=v\mid p_b\},\qquad v=[V]_{a_u}, P {[ V ] a u = v ∣ p b } , v = [ V ] a u , under the selected incidence law has typical upper bound N 0 − S u + o ( 1 ) N_0^{-Su+o(1)} N 0 − S u + o ( 1 ) evaluated at the sampled angular entry. Normalizing the total incidence mass does not change this conditional distribution. Good restricted measures with subpower slack can be prepared simultaneously on slowly increasing finite depth grids, losing negligibly relative to a specified inverse-subpower selection. Proof. Fix the exponents and a tolerance. Remove the fine entries where the joint angular upper bound or the pure p L p_L p L N 0 − S u + O ( τ ) N_0^{-Su+O(\tau)} N 0 − S u + O ( τ ) (2.4) handles a later inverse-subpower restriction of the denominator.
For finitely many tests their power-small exceptions can be summed. Choose the tolerance and number of tests slowly to obtain the final assertion. These exclusions may be purely analytical; if an output selection needs them, Lemma 2.4 implements the corresponding cell tests. In particular this argument does not assert a bound on the maximum angular probability over all bins of an unpruned law.
Time charts and prior-mass budgets
We now describe the local fits whose time cost defines the horizon. A chart controls each assigned trajectory throughout its time interval. Its incidence selection specifies where the lower derivative bound is required and supplies the mass entering the true floor. This selection need not occupy the whole interval.
Definition 2.8 (Charts, true packets, and known atlases). At depth r ∈ ( 0 , L ] r \in(0,L] r ∈ ( 0 , L ] I I I q = N 0 − h + o ( 1 ) q=N_0^{-h+o(1)} q = N 0 − h + o ( 1 ) h ≥ 0 h \ge0 h ≥ 0 A c A_c A c c c c K 0 − 1 K_0^{-1} K 0 − 1
A chart has an affine spatial center y c ( t ) y_c(t) y c ( t ) D c D_c D c
∥ D c ∥ ≤ q K 0 , J c = ∣ det D c ∣ , sup t ∈ I ∣ D c − 1 ( y ( t ) − y c ( t ) ) ∣ ≤ K 0 ( l ∈ A c ) . (2.7) \lVert D_c\rVert\le qK_0,\qquad J_c=\lvert\det D_c\rvert,\qquad\sup_{t\in I}\left\lvert D_c^{-1}\bigl(y(t)-y_c(t)\bigr)\right\rvert\le K_0\quad(l\in A_c).\tag{2.7} ∥ D c ∥ ≤ q K 0 , J c = ∣ det D c ∣ , t ∈ I sup D c − 1 ( y ( t ) − y c ( t ) ) ≤ K 0 ( l ∈ A c ) . ( 2.7 ) It has a test of the appropriate model class satisfying
sup t ∈ I ∣ P ( t , y ( t ) , w ( t ) ) ∣ ≤ K 0 a r , sup t ∈ I ∣ ∂ w P ( t , y ( t ) , w ( t ) ) ∣ ≤ K 0 B , ∣ ∂ w P ∣ ≥ B / K 0 on its selected incidences , ∣ ∂ w w P ∣ ≤ K 0 B 2 / a r . (2.8) \begin{aligned}
\sup_{t\in I}\lvert P(t,y(t),w(t))\rvert&\le K_0a_r, &
\sup_{t\in I}\lvert\partial_wP(t,y(t),w(t))\rvert&\le K_0B,\\
\lvert\partial_wP\rvert&\ge B/K_0\quad\text{on its selected incidences}, &
\lvert\partial_{ww}P\rvert&\le K_0B^2/a_r.
\end{aligned}\tag{2.8} t ∈ I sup ∣ P ( t , y ( t ) , w ( t ))∣ ∣ ∂ w P ∣ ≤ K 0 a r , ≥ B / K 0 on its selected incidences , t ∈ I sup ∣ ∂ w P ( t , y ( t ) , w ( t ))∣ ∣ ∂ w w P ∣ ≤ K 0 B , ≤ K 0 B 2 / a r . ( 2.8 ) All chart data, assignments, selections, inverse coordinate changes, and reciprocal widths are tempered.
Write
ν c = ν γ ( c ) ( A c ) , m c = μ γ ( c ) ( selected incidences assigned to c ) . \nu_c=\nu_{\gamma(c)}(A_c),\qquad m_c=\mu_{\gamma(c)}(\text{selected incidences assigned to }c). ν c = ν γ ( c ) ( A c ) , m c = μ γ ( c ) ( selected incidences assigned to c ) . A packet is an individual chart together with its label, time interval, geometric and test data, full trajectory assignment A c A_c A c ν γ ( c ) ∣ A c \nu_{\gamma(c)}|_{A_c} ν γ ( c ) ∣ A c ν c \nu_c ν c m c m_c m c true when
m c q ≥ K 0 − 1 n ( r ) J c . (2.9) \frac{m_c}{q}\ge K_0^{-1}n(r)J_c.\tag{2.9} q m c ≥ K 0 − 1 n ( r ) J c . ( 2.9 ) A system is true if every used packet is true. A known atlas at depth r r r p L p_L p L (2.9) . We also allow known atlases at r = 0 r=0 r = 0 a 0 = 1 a_0=1 a 0 = 1
Since y − y c y-y_c y − y c I I I (2.7) also bounds q D c − 1 ( V − y c ′ ) qD_c^{-1}(V-y'_c) q D c − 1 ( V − y c ′ ) K 0 K_0 K 0
m c ≤ q ν c , ∑ c ω γ ( c ) q ν c ≤ 1. (2.10) m_c \leq q\nu_c,\qquad\sum_c \omega_{\gamma(c)}q\nu_c \leq1.\tag{2.10} m c ≤ q ν c , c ∑ ω γ ( c ) q ν c ≤ 1. ( 2.10 ) The sum includes all worlds, intervals, and labels of one system.
The following elementary estimate converts the small-value and lower-derivative tests into a bound on the hidden-coordinate bins.
Lemma 2.9 (Quadratic sublevel sets). Let Q Q Q ε , b , δ > 0 \varepsilon,b,\delta>0 ε , b , δ > 0
{ w ∈ R : ∣ Q ( w ) ∣ ≤ ε , ∣ Q ′ ( w ) ∣ ≥ b } \{w\in\mathbb{R}: |Q(w)|\leq\varepsilon,\ |Q'(w)|\geq b\} { w ∈ R : ∣ Q ( w ) ∣ ≤ ε , ∣ Q ′ ( w ) ∣ ≥ b } has total length at most 4 ε / b 4\varepsilon/b 4 ε / b C ( 1 + ε / ( b δ ) ) C(1+\varepsilon/(b\delta)) C ( 1 + ε / ( b δ )) δ \delta δ C C C
Proof. The condition ∣ Q ′ ∣ ≥ b |Q'|\geq b ∣ Q ′ ∣ ≥ b Q Q Q 2 ε / b 2\varepsilon/b 2 ε / b δ \delta δ □ \square □
The defining true floor compares incidence mass with observation density and spatial volume. The next lemma combines that floor with the assignment budget to obtain comparability with the prior mass ν c \nu_c ν c
Lemma 2.10 (Saturation and label knowledge). A true system may be restricted, retaining inverse-subpower total mass and enlarging subpower slacks, so that every used chart satisfies
m c q ∼ log ν c ∼ log n ( r ) J c , (2.11) \frac{m_c}{q}\sim_{\log}\nu_c\sim_{\log}n(r)J_c,\tag{2.11} q m c ∼ l o g ν c ∼ l o g n ( r ) J c , ( 2.11 ) and its label is known by p r p_r p r
Proof. For a large subpower H H H q ν c > H m c q\nu_c>Hm_c q ν c > H m c
∑ q ν c > H m c ω γ ( c ) m c ≤ H − 1 ∑ c ω γ ( c ) q ν c ≤ H − 1 . \sum_{q\nu_c>Hm_c}\omega_{\gamma(c)}m_c
\leq H^{-1}\sum_c\omega_{\gamma(c)}q\nu_c
\leq H^{-1}. q ν c > H m c ∑ ω γ ( c ) m c ≤ H − 1 c ∑ ω γ ( c ) q ν c ≤ H − 1 . Choose H − 1 H^{-1} H − 1
Analytically remove the exceptional p r p_r p r K 0 n ( r ) K_0n(r) K 0 n ( r ) c c c ∣ P ∣ ≤ K 0 a r |P|\leq K_0a_r ∣ P ∣ ≤ K 0 a r ∣ P w ∣ ≥ B / K 0 |P_w|\geq B/K_0 ∣ P w ∣ ≥ B / K 0 K 0 K_0 K 0 a r / B a_r/B a r / B ε = K 0 a r \varepsilon=K_0a_r ε = K 0 a r b = B / K 0 b=B/K_0 b = B / K 0 δ = a r / B \delta=a_r/B δ = a r / B K 0 q J c K_0qJ_c K 0 q J c m c m_c m c K 0 n ( r ) q J c K_0n(r)qJ_c K 0 n ( r ) q J c (2.11) before a final enlargement of the slack.
To obtain lists, let a chart cross an observation if its time, spatial, value, and event-derivative tests are compatible with that observation. If T ( p r ) T(p_r) T ( p r )
∫ T ( p r ) d μ c a p ≤ K 0 ∑ c ω γ ( c ) n ( r ) q J c ≤ K 0 ∑ c ω γ ( c ) m c . \int T(p_r)\,d\mu_{\mathrm{cap}}
\leq K_0\sum_c\omega_{\gamma(c)}n(r)qJ_c
\leq K_0\sum_c\omega_{\gamma(c)}m_c. ∫ T ( p r ) d μ cap ≤ K 0 c ∑ ω γ ( c ) n ( r ) q J c ≤ K 0 c ∑ ω γ ( c ) m c . Markov’s inequality, with a sufficiently large subpower threshold, leaves subpower lists and loses negligible system mass. The joint flattening lemma implements these lists as tempered tables if necessary. Finally discard charts that lost more than half their selected mass in these operations. Their total loss is at most twice the removed incidence mass, and (2.9) survives with enlarged slack.
Later restrictions can reduce the selected mass of an individual chart. The next lemma gives two ways to continue a count: restore a floor for the current entries by pruning, or charge earlier witnesses using a cap for their own frozen measure.
Lemma 2.11 (Restoring current floors and charging witnesses). Fix a chart layer with the assignment budget (2.10) . Suppose a later selected measure has inverse-subpower mass in the same normalization. For each chart let at most D ≤ K 0 D \le K_0 D ≤ K 0 m ~ c , e \widetilde m_{c,e} m c , e 0 < ϵ < 1 0 < \epsilon< 1 0 < ϵ < 1
m ~ c , e < ϵ q ν c D (2.12) \widetilde m_{c,e} < \frac{\epsilon q\nu_c}{D}\tag{2.12} m c , e < D ϵ q ν c ( 2.12 ) loses at most ϵ \epsilon ϵ ϵ \epsilon ϵ ϵ i \epsilon_i ϵ i ∑ i ϵ i ≤ ϵ \sum_i \epsilon_i \le\epsilon ∑ i ϵ i ≤ ϵ ϵ \epsilon ϵ
If time is further partitioned, the corresponding threshold uses ∣ I ′ ∣ ν c |I'|\nu_c ∣ I ′ ∣ ν c I ′ ⊂ I I' \subset I I ′ ⊂ I q ν c q\nu_c q ν c
For a collection of disjoint witness submeasures of a fixed measure λ \lambda λ U U U λ ( U ) \lambda(U) λ ( U ) D D D D λ ( U ) D\lambda(U) D λ ( U ) λ \lambda λ
Proof. Sum (2.12) over at most D D D (2.10) . For finitely many layers, choose positive tolerances ϵ i \epsilon_i ϵ i ∑ i ϵ i ≤ ϵ \sum_i \epsilon_i \le\epsilon ∑ i ϵ i ≤ ϵ ϵ i \epsilon_i ϵ i i i i i i i ϵ i \epsilon_i ϵ i ϵ \epsilon ϵ ∑ I ′ ⊂ I ∣ I ′ ∣ ν c = q ν c \sum_{I'\subset I}|I'|\nu_c=q\nu_c ∑ I ′ ⊂ I ∣ I ′ ∣ ν c = q ν c
For the last assertion, if the witnesses of the counted labels are disjoint submeasures of a frozen measure λ \lambda λ U U U λ ( U ) \lambda(U) λ ( U ) D D D D λ ( U ) D\lambda(U) D λ ( U ) λ \lambda λ
Lemma 2.12 (Fullness of the largest spatial scale). In a cap-class problem, a chart system known by p L p_L p L whether true or not, has
∥ D c ∥ ∼ log q \lVert D_c\rVert\sim_{\log} q ∥ D c ∥ ∼ l o g q on an inverse-subpower selection, after homogenizing the ratio if necessary.
Proof. The upper bound is part of (2.7) . If the ratio were at most N 0 − α N_0^{-\alpha} N 0 − α α > 0 \alpha>0 α > 0 (2.7) would confine V V V K 0 N 0 − α K_0N_0^{-\alpha} K 0 N 0 − α y c ′ y'_c y c ′ u > 0 u>0 u > 0 u < α / 2 u<\alpha/2 u < α /2 u ≤ η L u\leq\eta L u ≤ η L K 0 K_0 K 0 a u a_u a u p L p_L p L K 0 K_0 K 0 N 0 − S u + o ( 1 ) N_0^{-Su+o(1)} N 0 − S u + o ( 1 ) (2.6) . Summing these bounds and then integrating shows that the alleged mass is power-small. The exceptional entries were removed before this sum. This contradiction excludes every fixed-power deficiency. □
Normalization, composition, and horizon
To iterate the construction, we regard each chart as a new world with its conditional trajectory prior and its own unit time interval. The density transformation must be tracked within that world before child charts can be lifted back to the original problem. The following lemma records both operations and the label-knowledge condition needed for intermediate true floors.
Lemma 2.13 (Chart normalization and lifting). Let a homogeneous exact problem of length L L L 0 ≤ r < L 0\leq r<L 0 ≤ r < L L ′ = L − r L'=L-r L ′ = L − r
n ′ ( L ′ ) ∼ log n ( L ) J c ν c , (2.13) n'(L')\sim_{\log}n(L)\frac{J_c}{\nu_c},\tag{2.13} n ′ ( L ′ ) ∼ l o g n ( L ) ν c J c , ( 2.13 ) n ′ ( s ) ≤ N 0 o ( 1 ) n ( r + s ) J c ν c ( 0 ≤ s < L ′ ) . (2.14) n'(s)\leq N_0^{o(1)}n(r+s)\frac{J_c}{\nu_c}\qquad(0\leq s<L').\tag{2.14} n ′ ( s ) ≤ N 0 o ( 1 ) n ( r + s ) ν c J c ( 0 ≤ s < L ′ ) . ( 2.14 ) If the parent label is already known by p r p_r p r (2.14) . The class C ( d ) C(d) C ( d ) S , η S,\eta S , η
Proof. Put
Z = ∑ c ω γ ( c ) q ν c , ω c ′ = ω γ ( c ) q ν c Z . Z=\sum_c \omega_{\gamma(c)}q\nu_c,\qquad
\omega'_c=\frac{\omega_{\gamma(c)}q\nu_c}{Z}. Z = c ∑ ω γ ( c ) q ν c , ω c ′ = Z ω γ ( c ) q ν c . If the atlas success is M ≥ K 0 − 1 M\geq K_0^{-1} M ≥ K 0 − 1 M ≤ Z ≤ 1 M\leq Z\leq1 M ≤ Z ≤ 1 c = ( γ , I , A c ) c=(\gamma,I,A_c) c = ( γ , I , A c ) ν γ ( ⋅ ∣ A c ) \nu_\gamma(\mathord{\cdot}\mid A_c) ν γ ( ⋅ ∣ A c )
t = t c + q τ , y = y c ( t ) + D c z , 0 ≤ τ ≤ 1. t=t_c+q\tau,\qquad y=y_c(t)+D_c z,\qquad0\leq\tau\leq1. t = t c + q τ , y = y c ( t ) + D c z , 0 ≤ τ ≤ 1. The new selected measure is the pushforward of the old assigned selected measure divided by q ν c q\nu_c q ν c Z Z Z
In version 1, if the parent test is w − F ( t , y ) w-F(t,y) w − F ( t , y )
w ′ = w − F ( t , y ) a r . w'=\frac{w-F(t,y)}{a_r}. w ′ = a r w − F ( t , y ) . This is quadratic along trajectories and bounded by K 0 K_0 K 0 w w w P ∗ ′ = P / a r P'_* = P/a_r P ∗ ′ = P / a r B ′ = B / a r B' = B/a_r B ′ = B / a r
sup ∣ P ∗ ′ ∣ ≤ K 0 , sup ∣ ∂ w P ∗ ′ ∣ ≤ K 0 B ′ , ∣ ∂ w P ∗ ′ ∣ ≥ B ′ / K 0 , ∣ ∂ w w P ∗ ′ ∣ ≤ K 0 ( B ′ ) 2 \sup|P'_*| \le K_0,\quad\sup|\partial_w P'_*| \le K_0B',\quad|\partial_w P'_*| \ge B'/K_0,\quad|\partial_{ww}P'_*| \le K_0(B')^2 sup ∣ P ∗ ′ ∣ ≤ K 0 , sup ∣ ∂ w P ∗ ′ ∣ ≤ K 0 B ′ , ∣ ∂ w P ∗ ′ ∣ ≥ B ′ / K 0 , ∣ ∂ w w P ∗ ′ ∣ ≤ K 0 ( B ′ ) 2 in the required places. All base and velocity bounds follow from (2.7) . At r = 0 r=0 r = 0
These coordinate changes preserve the permitted models. Their maps on coefficient space have constant Jacobian: the base changes are affine, and in version 1 the subtraction of the quadratic F ( t , y ) F(t,y) F ( t , y ) w w w
At fixed old base and chart label, new hidden-bin width at depth s s s a r + s / B a_{r+s}/B a r + s / B
ρ c ′ = q J c q ν c ρ c = J c ν c ρ c . (2.15) \rho'_c = \frac{qJ_c}{q\nu_c}\rho_c = \frac{J_c}{\nu_c}\rho_c.\tag{2.15} ρ c ′ = q ν c q J c ρ c = ν c J c ρ c . ( 2.15 ) The numerator q J c qJ_c q J c q ν c q\nu_c q ν c Z Z Z
Refining p L p_L p L (2.15) , the bounded bin overlap, and Lemma 2.3 to obtain (2.13) . Global exceptional masses are multiplied by Z − 1 ≤ K 0 Z^{-1} \le K_0 Z − 1 ≤ K 0 (2.14) . Knowledge by p r p_r p r
n ′ ( s ) ∼ log n ( r + s ) n ( r ) , n ′ ( 0 ) ∼ log 1. n'(s) \sim_{\log} \frac{n(r+s)}{n(r)},\qquad n'(0) \sim_{\log} 1. n ′ ( s ) ∼ l o g n ( r ) n ( r + s ) , n ′ ( 0 ) ∼ l o g 1. For the angular bound, a new velocity bin of width N 0 − u N_0^{-u} N 0 − u V = y c ′ + D c V ′ / q V = y'_c + D_cV'/q V = y c ′ + D c V ′ / q K 0 N 0 − u K_0N_0^{-u} K 0 N 0 − u K 0 K_0 K 0
ρ j o i n t ′ ≤ N 0 o ( 1 ) n ( L ) J c ν c N 0 − s u ∼ log n ′ ( L ′ ) N 0 − s u \rho'_{\mathrm{joint}} \le N_0^{o(1)}n(L)\frac{J_c}{\nu_c}N_0^{-s u} \sim_{\log} n'(L')N_0^{-s u} ρ joint ′ ≤ N 0 o ( 1 ) n ( L ) ν c J c N 0 − s u ∼ l o g n ′ ( L ′ ) N 0 − s u for u ≤ η L ′ u \le\eta L' u ≤ η L ′
log N 0 n ′ ( s ) n ′ ( L ′ ) ≤ f ( L ) − f ( r + s ) + o ( 1 ) ≤ d ( L ′ − s ) + o ( 1 ) , \log_{N_0}\frac{n'(s)}{n'(L')} \le f(L)-f(r+s)+o(1) \le d(L'-s)+o(1), log N 0 n ′ ( L ′ ) n ′ ( s ) ≤ f ( L ) − f ( r + s ) + o ( 1 ) ≤ d ( L ′ − s ) + o ( 1 ) , which proves the endpoint cap.
For lifting, let a child have relative time length q d q_d q d J d J_d J d m d ′ m'_d m d ′
q c d = q q d , J c d = J c J d , m c d = q ν c m d ′ , m c d q c d = ν c m d ′ q d . (2.16) q_{cd}=q q_d,\qquad J_{cd}=J_cJ_d,\qquad m_{cd}=q\nu_c m'_d,\qquad\frac{m_{cd}}{q_{cd}}=\nu_c\frac{m'_d}{q_d}.\tag{2.16} q c d = q q d , J c d = J c J d , m c d = q ν c m d ′ , q c d m c d = ν c q d m d ′ . ( 2.16 ) At the terminal depth, a child true floor and (2.13) give the old floor K 0 − 1 n ( L ) J c J d K_0^{-1}n(L)J_cJ_d K 0 − 1 n ( L ) J c J d
The tests lift by multiplication by a r a_r a r w ′ − G ( τ , z ) w'-G(\tau,z) w ′ − G ( τ , z )
w − F ( t , y ) − a r G ( τ , z ) , w-F(t,y)-a_rG(\tau,z), w − F ( t , y ) − a r G ( τ , z ) , which is still of the special quadratic form. In version 2, a r Q ( τ , z , w ) a_rQ(\tau,z,w) a r Q ( τ , z , w ) B B B
∣ ∂ w w ( a r Q ) ∣ ≤ K 0 a r ( B / a r ) 2 a s = K 0 B 2 a r + s . \lvert\partial_{ww}(a_rQ)\rvert\le K_0a_r\frac{(B/a_r)^2}{a_s}=K_0\frac{B^2}{a_{r+s}}. ∣ ∂ w w ( a r Q )∣ ≤ K 0 a r a s ( B / a r ) 2 = K 0 a r + s B 2 . The small-value and first-derivative bounds transform in the same way. Spatial matrices and times compose, whole-block fitting is preserved, and assignments may record the entire path with nested time partitions. All operations retain tempered bounds.
In version 2, largest-axis fullness gives a geometric interpretation to
log N 0 q 2 J c , (2.17) \log_{N_0}\frac{q^2}{J_c},\tag{2.17} log N 0 J c q 2 , ( 2.17 ) which we call the narrowness in the specified depth units: it is the additional thinness of the smaller spatial axis, up to subpower factors. Under composition,
q c d 2 J c d = q 2 J c q d 2 J d . \frac{q_{cd}^2}{J_{cd}}=\frac{q^2}{J_c}\frac{q_d^2}{J_d}. J c d q c d 2 = J c q 2 J d q d 2 . Thus narrowness exponents add exactly, without any assumption that the spatial axes are parallel. Statements about a common value of this exponent use a homogeneous subfamily.
Definition 2.14 (Minimum horizon). For a homogeneous exact problem E E E L L L H ( E ) H(E) H ( E ) h ≥ 0 h \ge0 h ≥ 0 L L L E E E
The rest of the section proves two complementary bounds. Every homogeneous exact problem satisfies H ( E ) ≤ L H(E) \le L H ( E ) ≤ L a L a_L a L H ( E ) > 0 H(E) > 0 H ( E ) > 0
We first prove that true terminal systems always exist with horizon at most L L L a L a_L a L
Lemma 2.15 (Weighted full-time plank estimate). Fix a bounded affine-line chart in R 2 \mathbb{R}^2 R 2 ε > 0 \varepsilon> 0 ε > 0 η ε > 0 \eta_\varepsilon> 0 η ε > 0 C ε < ∞ C_\varepsilon< \infty C ε < ∞
M i ( t ) = b i + t u i , 0 ≤ t ≤ 1 , M_i(t)=b_i+tu_i,\qquad0\le t\le1, M i ( t ) = b i + t u i , 0 ≤ t ≤ 1 , be finitely many indexed traces, with b i , u i b_i,u_i b i , u i ω i \omega_i ω i D ≥ 1 D \ge1 D ≥ 1 ρ = D − 1 \rho= D^{-1} ρ = D − 1
P = { ( t , x ) : 0 ≤ t ≤ 1 , ∣ ⟨ x − c − t v , e 1 ⟩ ∣ ≤ a ρ , ∣ ⟨ x − c − t v , e 2 ⟩ ∣ ≤ b ρ } , 1 ≤ a ≤ b ≤ D , \mathcal{P}=\left\{(t,x):\begin{array}{c}0\le t\le1,\\ \lvert\langle x-c-tv,e_1\rangle\rvert\le a\rho,\\ \lvert\langle x-c-tv,e_2\rangle\rvert\le b\rho\end{array}\right\},\qquad1\le a\le b\le D, P = ⎩ ⎨ ⎧ ( t , x ) : 0 ≤ t ≤ 1 , ∣⟨ x − c − t v , e 1 ⟩∣ ≤ a ρ , ∣⟨ x − c − t v , e 2 ⟩∣ ≤ b ρ ⎭ ⎬ ⎫ , 1 ≤ a ≤ b ≤ D , where ( e 1 , e 2 ) (e_1,e_2) ( e 1 , e 2 ) c , v ∈ R 2 c,v\in\mathbb{R}^2 c , v ∈ R 2
∑ i : M i ⊂ P ω i ≤ A a b . (2.18) \sum_{i:M_i\subset\mathcal{P}}\omega_i\le Aab.\tag{2.18} i : M i ⊂ P ∑ ω i ≤ A ab . ( 2.18 ) Fixed dilations of these plank tests are permitted. Mark time bins J i J_i J i ρ \rho ρ # J i ≥ D 1 − η ε \#J_i\ge D^{1-\eta_\varepsilon} # J i ≥ D 1 − η ε E v i s E_{\mathrm{vis}} E vis
∑ i ω i # J i ≤ C ε A D ε # E v i s . (2.19) \sum_i\omega_i\#J_i\le C_\varepsilon A D^\varepsilon\#E_{\mathrm{vis}}.\tag{2.19} i ∑ ω i # J i ≤ C ε A D ε # E vis . ( 2.19 ) Proof. This is the indexed weighted formulation of OpenAI [24 ] , Lemma 2.3. The plank condition concerns containment of the entire trace, not merely its marked portions. The cited statement permits arbitrary positive weights and repeated indices, and has precisely the long-mark hypothesis above.
We record explicitly two extensions of use below. A continuous weighted law of bounded affine traces can be rounded in coefficient space at a mesh much smaller than ρ \rho ρ
Also suppose the chart parameters are bounded by a subpower K 0 K_0 K 0 N 0 − 1 N_0^{-1} N 0 − 1 K 0 − 1 K_0^{-1} K 0 − 1 ε > 0 \varepsilon>0 ε > 0 N 0 N_0 N 0 ε ↓ 0 \varepsilon\downarrow0 ε ↓ 0 ε \varepsilon ε C ε D ε C_\varepsilon D^\varepsilon C ε D ε u , v u,v u , v ρ 2 ∑ M i ⊂ P ω i / ( u v ) \rho^2\sum_{M_i\subset \mathcal{P}}\omega_i/(uv) ρ 2 ∑ M i ⊂ P ω i / ( uv )
Proposition 2.16 (Initial horizon bound). Every homogeneous exact problem satisfies H ( E ) ≤ L H(E)\le L H ( E ) ≤ L
Proof. Choose a dyadic q ∼ a L q\sim a_L q ∼ a L q q q I I I y y y I I I q q q X X X a L a_L a L G G G y y y X X X K 0 q K_0q K 0 q X X X X X X K 0 a L K_0a_L K 0 a L w w w O ( K 0 a L 2 ) O(K_0a_L^2) O ( K 0 a L 2 ) G G G p L p_L p L
Let b G b_G b G X X X w − b G w-b_G w − b G P ∗ − b G P_*-b_G P ∗ − b G R q Rq R q R R R [ 0 , 1 ] [0,1] [ 0 , 1 ] J 0 = ( R q ) j J_0=(Rq)^j J 0 = ( R q ) j G G G n ( L ) J 0 n(L)J_0 n ( L ) J 0 G G G
Choose a common mesh ρ = D m − 1 ∼ N 0 − D \rho=D_m^{-1}\sim N_0^{-D} ρ = D m − 1 ∼ N 0 − D D m D_m D m D D D
Fix 0 < ξ < 1 0<\xi<1 0 < ξ < 1 ρ ≤ r i ≲ 1 \rho\le r_i\lesssim1 ρ ≤ r i ≲ 1 1 ≤ i ≤ j 1\le i\le j 1 ≤ i ≤ j
N 0 − ξ n ( L ) J 0 ∏ i = 1 j r i . (2.20) N_0^{-\xi}n(L)J_0\prod_{i=1}^{j}r_i.\tag{2.20} N 0 − ξ n ( L ) J 0 i = 1 ∏ j r i . ( 2.20 ) Assign all such available trajectories to that chart and remove them from the group. Fixed constants in the width cutoffs are chosen to allow subsequent enlargements.
This procedure has polynomially many steps, uniformly for 0 < ξ < 1 0<\xi<1 0 < ξ < 1 J 0 ∏ i r i J_0\prod_i r_i J 0 ∏ i r i G G G
Suppose the algorithm covers less than half the initial selected mass along a subsequence. In the residual remove cells that fail the original group floor with tolerance ξ / 4 \xi/4 ξ /4 K 0 − 1 K_0^{-1} K 0 − 1
Write m G m_G m G m G / K 0 > 0 m_G/K_0>0 m G / K 0 > 0
N 0 − ξ / 4 n ( L ) J 0 ρ j + 1 . N_0^{-\xi/4}n(L)J_0\rho^{j+1}. N 0 − ξ /4 n ( L ) J 0 ρ j + 1 . The number of such cells is consequently at most
m G N 0 − ξ / 4 n ( L ) J 0 ρ j + 1 . (2.21) \frac{m_G}{N_0^{-\xi/4}n(L)J_0\rho^{j+1}}.\tag{2.21} N 0 − ξ /4 n ( L ) J 0 ρ j + 1 m G . ( 2.21 ) The mass used in this count is the original whole-cell witness. A residual cell need not retain that floor.
Mark all time bins with positive surviving labeled duration on each used trace. There are at least D m / K 0 D_m/K_0 D m / K 0 ϵ > 0 \epsilon>0 ϵ > 0
For j = 1 j=1 j = 1 [ 0 , 1 ] [0,1] [ 0 , 1 ] O ( ρ − 1 ) O(\rho^{-1}) O ( ρ − 1 ) (2.21) ,
∑ i ω i # J i # E v i s ≳ K 0 − 1 N 0 − ξ / 4 n ( L ) J 0 ρ 2 , (2.22) \frac{\sum_i \omega_i \#J_i}{\#E_{\mathrm{vis}}} \gtrsim K_0^{-1}N_0^{-\xi/4}n(L)J_0\rho^2,\tag{2.22} # E vis ∑ i ω i # J i ≳ K 0 − 1 N 0 − ξ /4 n ( L ) J 0 ρ 2 , ( 2.22 ) since the numerator is at least m G D m / K 0 m_GD_m/K_0 m G D m / K 0
Taking the supremum plank constant in Equation (2.18) , the weighted estimate therefore supplies a full-time normalized plank of physical spatial area u v uv uv
used trajectory weight in the plank u v ≳ ε K 0 − 1 N 0 − ξ / 4 D m − ε n ( L ) J 0 . (2.23) \frac{\text{used trajectory weight in the plank}}{uv} \gtrsim_\varepsilon K_0^{-1}N_0^{-\xi/4}D_m^{-\varepsilon}n(L)J_0.\tag{2.23} uv used trajectory weight in the plank ≳ ε K 0 − 1 N 0 − ξ /4 D m − ε n ( L ) J 0 . ( 2.23 ) The harmless fixed enlargement also contains the unrounded traces.
In the padded case, project the rectangle onto the true spatial coordinate, and denote its projected width by W W W u v uv uv C u v / W Cuv/W C uv / W u u u v v v θ \theta θ u ∣ cos θ ∣ + v ∣ sin θ ∣ u|\cos\theta|+v|\sin\theta| u ∣ cos θ ∣ + v ∣ sin θ ∣
2 min ( u ∣ sin θ ∣ , v ∣ cos θ ∣ ) , 2\min\left(\frac{u}{|\sin\theta|},\frac{v}{|\cos\theta|}\right), 2 min ( ∣ sin θ ∣ u , ∣ cos θ ∣ v ) , with the evident interpretation at zero denominators. Their product is at most a fixed multiple of u v uv uv (2.23) into the same lower bound for original trace weight divided by W W W
Every counted original residual trace has labeled duration at least K 0 − 1 K_0^{-1} K 0 − 1 N 0 N_0 N 0
D ε < ξ / 4. D\varepsilon<\xi/4. D ε < ξ /4. The two fixed losses then total less than ξ / 2 \xi/2 ξ /2
Consequently at least half the selected mass is covered. Let ξ ↓ 0 \xi\downarrow0 ξ ↓ 0 ξ \xi ξ ε \varepsilon ε ξ < 1 \xi<1 ξ < 1 (2.20) is now the true floor with a subpower loss, and q ∼ a L q\sim a_L q ∼ a L h = L h=L h = L
From an arbitrary Kakeya counterexample to positive horizon
Theorem 2.17 (Reduction to a bad weighted problem). If a set K ⊂ R 4 K\subset\mathbb{R}^4 K ⊂ R 4 dim H K < 4 \dim_{\mathrm H}K<4 dim H K < 4 0 < d < 1 0<d<1 0 < d < 1 S , η > 0 S,\eta>0 S , η > 0 E E E C ( d ) C(d) C ( d ) H ( E ) > 0 H(E)>0 H ( E ) > 0
Proof. We give the selection from K K K
A finite selection from outer measure. Fix a bounded open graph-slope chart of directions, with slopes s ∈ R 3 s\in\mathbb{R}^3 s ∈ R 3
x ( t ) = b + t s , x = ( y , w ) . x(t)=b+ts,\qquad x=(y,w). x ( t ) = b + t s , x = ( y , w ) . A unit segment in one of these directions has a nondegenerate time projection. Every such segment contains the graph over some interval with rational endpoints, and its intercept belongs to a bounded box with integer endpoints. These are countably many choices. Countable outer subadditivity therefore supplies one interval and one intercept box for which the set S 0 S_0 S 0 a > 0 a>0 a > 0 [ 0 , 1 ] [0,1] [ 0 , 1 ]
Choose 0 < σ < 1 0<\sigma<1 0 < σ < 1 dim H K < 4 − σ \dim_{\mathrm H} K<4-\sigma dim H K < 4 − σ
∑ i r i 4 − σ ≤ 1. \sum_i r_i^{4-\sigma}\le1. i ∑ r i 4 − σ ≤ 1. Only balls meeting the fixed bounded region containing the eligible graph segments are needed for the following argument; their centers are bounded as well. For k ≥ 1 k\ge1 k ≥ 1 U k U_k U k 2 − k − 1 < r i ≤ 2 − k 2^{-k-1}<r_i\le2^{-k} 2 − k − 1 < r i ≤ 2 − k C 2 k ( 4 − σ ) C2^{k(4-\sigma)} C 2 k ( 4 − σ )
Choose a fixed c > 0 c>0 c > 0 ∑ k ≥ 1 c k − 2 < 1 \sum_{k\ge1}ck^{-2}<1 ∑ k ≥ 1 c k − 2 < 1 k k k
∣ { t ∈ [ 0 , 1 ] : b + t s ∈ U k } ∣ ≥ c k − 2 , \lvert\{t\in[0,1]:b+ts\in U_k\}\rvert\ge ck^{-2}, ∣{ t ∈ [ 0 , 1 ] : b + t s ∈ U k }∣ ≥ c k − 2 , because the sum of these time measures is at least one. Define S k ⊂ S 0 S_k\subset S_0 S k ⊂ S 0 S k S_k S k S 0 ⊂ ⋃ k S k S_0\subset\bigcup_k S_k S 0 ⊂ ⋃ k S k c ′ > 0 c'>0 c ′ > 0 c ′ ∑ k ≥ 1 k − 2 < 1 c'\sum_{k\ge1}k^{-2}<1 c ′ ∑ k ≥ 1 k − 2 < 1 k k k
∣ S k ∣ ∗ ≥ c ′ a k − 2 . (2.24) \lvert S_k\rvert^*\ge c' ak^{-2}.\tag{2.24} ∣ S k ∣ ∗ ≥ c ′ a k − 2 . ( 2.24 ) Indeed the reverse strict inequality for every k k k ∣ S 0 ∣ ∗ < a \lvert S_0\rvert^*<a ∣ S 0 ∣ ∗ < a S k S_k S k (2.24) . As the maximum cover radius tends to zero, every possible such k k k
Put δ = 2 − k \delta=2^{-k} δ = 2 − k δ \delta δ S k S_k S k S k S_k S k δ \delta δ M M M
c k − 2 δ − 3 ≤ M ≤ C δ − 3 . (2.25) ck^{-2}\delta^{-3}\le M\le C\delta^{-3}.\tag{2.25} c k − 2 δ − 3 ≤ M ≤ C δ − 3 . ( 2.25 ) Choose one successful witness for each of these finitely many slopes. This is the only witness choice needed, and uses no measurable selection.
Smoothing and three density bounds. Let N 0 = δ − 1 N_0=\delta^{-1} N 0 = δ − 1 L = 1 L=1 L = 1 P ∗ = w P_*=w P ∗ = w B = 1 B=1 B = 1 δ \delta δ U k U_k U k c k − 2 ck^{-2} c k − 2
This is a tempered version-2 problem. The mixture may be represented by the internally indexed copies of Definition 2.2; its count and density heights are polynomial, its supports are boxes, and the finite union of enlarged balls is a bounded-degree polynomial predicate of polynomial size. The separated slope centers and (2.25) give the full three-slope density ceiling
d ν s d s ≤ C k 2 ≤ K 0 . (2.26) \frac{d\nu_s}{ds}\le Ck^2\le K_0.\tag{2.26} d s d ν s ≤ C k 2 ≤ K 0 . ( 2.26 ) Conditional on the full slope, the y y y C δ − 2 C\delta^{-2} C δ − 2
Homogenize jointly for 0 ≤ r ≤ 1 0 \le r \le1 0 ≤ r ≤ 1 0 ≤ u ≤ 4 0 \le u \le4 0 ≤ u ≤ 4 u u u V V V
Write
n ( r , u ) = N 0 − f ( r , u ) , m ( r , u ) = N 0 r n ( r , u ) , g ( r , u ) = r − f ( r , u ) . n(r,u) = N_0^{-f(r,u)}, \qquad m(r,u) = N_0^r n(r,u), \qquad g(r,u) = r - f(r,u). n ( r , u ) = N 0 − f ( r , u ) , m ( r , u ) = N 0 r n ( r , u ) , g ( r , u ) = r − f ( r , u ) . The resulting limiting exponents obey
g ( 1 , 0 ) ≥ σ , g ( 0 , 0 ) ≤ 0 , g ( r , u ) ≤ r + 2 − 2 u . (2.27) g(1,0) \ge\sigma, \qquad g(0,0) \le0, \qquad g(r,u) \le r + 2 - 2u.\tag{2.27} g ( 1 , 0 ) ≥ σ , g ( 0 , 0 ) ≤ 0 , g ( r , u ) ≤ r + 2 − 2 u . ( 2.27 ) For the first inequality, at width δ \delta δ O ( 1 ) O(1) O ( 1 ) w w w ( t , y ) (t,y) ( t , y ) O ( δ 3 ) O(\delta^3) O ( δ 3 ) C δ σ − 1 C\delta^{\sigma-1} C δ σ − 1 n ( 1 , 0 ) n(1,0) n ( 1 , 0 ) n ( 1 , 0 ) ≥ K 0 − 1 δ 1 − σ n(1,0) \ge K_0^{-1}\delta^{1-\sigma} n ( 1 , 0 ) ≥ K 0 − 1 δ 1 − σ g ( 1 , 0 ) ≥ σ g(1,0) \ge\sigma g ( 1 , 0 ) ≥ σ
For the third inequality, the V V V K 0 K_0 K 0 (2.26) over the bounded remaining slope. A V V V K 0 N 0 − 2 u K_0N_0^{-2u} K 0 N 0 − 2 u t t t y ( t ) y(t) y ( t ) C δ − 2 C\delta^{-2} C δ − 2 w w w K 0 N 0 2 − 2 u K_0N_0^{2-2u} K 0 N 0 2 − 2 u g ( r , u ) g(r,u) g ( r , u )
For the second inequality, the projected y y y ρ = D m − 1 \rho= D_m^{-1} ρ = D m − 1
ν { l : y l ( [ 0 , 1 ] ) ⊂ P } ≤ K 0 a b ρ 2 (2.28) \nu\{l : y_l([0,1]) \subset \mathcal{P}\} \le K_0ab\rho^2\tag{2.28} ν { l : y l ([ 0 , 1 ]) ⊂ P } ≤ K 0 ab ρ 2 ( 2.28 ) for spatial radii a ρ a\rho a ρ b ρ b\rho b ρ t = 0 , 1 t = 0,1 t = 0 , 1 2 a ρ 2a\rho 2 a ρ 2 b ρ 2b\rho 2 b ρ K 0 K_0 K 0
Suppose instead that n ( 0 , 0 ) n(0,0) n ( 0 , 0 ) α > 0 \alpha> 0 α > 0 ρ ∼ N 0 − D \rho\sim N_0^{-D} ρ ∼ N 0 − D N 0 α ρ 3 N_0^{\alpha}\rho^3 N 0 α ρ 3 α \alpha α w w w K 0 N 0 − α ρ − 3 K_0N_0^{-\alpha}\rho^{-3} K 0 N 0 − α ρ − 3 K 0 − 1 K_0^{-1} K 0 − 1 m r e t ≥ K 0 − 1 m_{\mathrm{ret}} \ge K_0^{-1} m ret ≥ K 0 − 1 ρ \rho ρ
∑ i ω i # J i ≥ m r e t ρ . \sum_i \omega_i\#J_i \ge\frac{m_{\mathrm{ret}}}{\rho}. i ∑ ω i # J i ≥ ρ m ret . Dividing by the visited-cell bound, and enlarging the subpower factor, yields the raw weighted average multiplicity
∑ i ω i # J i # E v i s ≥ m r e t / ρ K 0 N 0 − α ρ − 3 ≥ K 0 − 1 N 0 α ρ 2 . \frac{\sum_i \omega_i\#J_i}{\#E_{\mathrm{vis}}}
\ge\frac{m_{\mathrm{ret}}/\rho}{K_0N_0^{-\alpha}\rho^{-3}}
\ge K_0^{-1}N_0^\alpha\rho^2. # E vis ∑ i ω i # J i ≥ K 0 N 0 − α ρ − 3 m ret / ρ ≥ K 0 − 1 N 0 α ρ 2 . On the other hand, Lemma 2.15 and (2.28) bound it by C ε K 0 D m ε ρ 2 C_\varepsilon K_0D_m^\varepsilon\rho^2 C ε K 0 D m ε ρ 2 D ε < α / 2 D\varepsilon<\alpha/2 D ε < α /2 N 0 N_0 N 0 g ( 0 , 0 ) ≤ 0 g(0,0) \le0 g ( 0 , 0 ) ≤ 0
Selecting an endpoint and an angular scale. The bounds on g g g R , S > 0 R,S > 0 R , S > 0 R + 4 S < σ / 2 R + 4S < \sigma/2 R + 4 S < σ /2 R , S < 1 / 10 R,S < 1/10 R , S < 1/10
g ( r , u ) − R r + S u on [ 0 , 1 ] × [ 0 , 4 ] . (2.29) g(r,u) - Rr + Su \quad\text{on } [0,1] \times[0,4].\tag{2.29} g ( r , u ) − R r + S u on [ 0 , 1 ] × [ 0 , 4 ] . ( 2.29 ) Continuity follows from homogenization. At a maximizer ( r 0 , u 0 ) (r_0,u_0) ( r 0 , u 0 )
r 0 > 0 , u 0 < 4 , g ( r 0 , u 0 ) ≥ σ / 2. (2.30) r_0 > 0, \qquad u_0 < 4, \qquad g(r_0,u_0) \ge\sigma/2.\tag{2.30} r 0 > 0 , u 0 < 4 , g ( r 0 , u 0 ) ≥ σ /2. ( 2.30 ) Indeed the value at ( 1 , 0 ) (1,0) ( 1 , 0 ) σ − R \sigma-R σ − R r = 0 r=0 r = 0 u u u g ( 0 , 0 ) ≤ 0 g(0,0)\le0 g ( 0 , 0 ) ≤ 0 4 S 4S 4 S u = 4 u=4 u = 4 (2.27) bounds it by r − 6 − R r + 4 S < 0 r-6-Rr+4S<0 r − 6 − R r + 4 S < 0 g ( r 0 , u 0 ) ≥ σ − R − 4 S g(r_0,u_0)\ge\sigma-R-4S g ( r 0 , u 0 ) ≥ σ − R − 4 S
Partition the y y y V V V B \mathcal{B} B a u 0 a_{u_0} a u 0 ν ( B ) \nu(\mathcal{B}) ν ( B ) y y y y B ( t ) y_{\mathcal{B}}(t) y B ( t ) y y y a u 0 a_{u_0} a u 0 w w w ( t , y ) (t,y) ( t , y ) V V V y 0 = y − t V y_0=y-tV y 0 = y − t V B \mathcal{B} B ( p s , [ V ] a u 0 ) (p_s,[V]_{a_{u_0}}) ( p s , [ V ] a u 0 ) s s s
For clarity, let ρ B ( s ) \rho_{\mathcal{B}}(s) ρ B ( s ) y = y B ( t ) + a u 0 y n e w y=y_{\mathcal{B}}(t)+a_{u_0}y_{\mathrm{new}} y = y B ( t ) + a u 0 y new a u 0 2 a_{u_0}^2 a u 0 2 ν ( B ) \nu(\mathcal{B}) ν ( B )
ρ n e w , B ( s ) = a u 0 2 ν ( B ) ρ B ( s ) . \rho_{\mathrm{new},\mathcal{B}}(s)=\frac{a_{u_0}^2}{\nu(\mathcal{B})}\rho_{\mathcal{B}}(s). ρ new , B ( s ) = ν ( B ) a u 0 2 ρ B ( s ) . This is the specialization of (2.15) to a full-time crop. The actual rounded grids have bounded overlap. Combining that overlap and the bounded crop-label ambiguity with Lemma 2.3 gives the typical equivalence, for 0 ≤ s ≤ r 0 0\le s\le r_0 0 ≤ s ≤ r 0
n n e w ( s ) ∼ log n ( s , u 0 ) a u 0 2 ν ( B ) . (2.31) n_{\mathrm{new}}(s)\sim_{\log} n(s,u_0)\frac{a_{u_0}^2}{\nu(\mathcal{B})}.\tag{2.31} n new ( s ) ∼ l o g n ( s , u 0 ) ν ( B ) a u 0 2 . ( 2.31 ) A new angular bin of depth u u u u 0 + u u_0+u u 0 + u
Maximality in (2.29) gives
f ( r 0 , u 0 ) − f ( s , u 0 ) ≤ ( 1 − R ) ( r 0 − s ) , 0 ≤ s ≤ r 0 , f ( r 0 , u 0 + u ) − f ( r 0 , u 0 ) ≥ S u , 0 ≤ u ≤ 4 − u 0 . (2.32) \begin{aligned}
f(r_0,u_0)-f(s,u_0)&\le(1-R)(r_0-s), &&0\le s\le r_0,\\
f(r_0,u_0+u)-f(r_0,u_0)&\ge Su, &&0\le u\le4-u_0.
\end{aligned}\tag{2.32} f ( r 0 , u 0 ) − f ( s , u 0 ) f ( r 0 , u 0 + u ) − f ( r 0 , u 0 ) ≤ ( 1 − R ) ( r 0 − s ) , ≥ S u , 0 ≤ s ≤ r 0 , 0 ≤ u ≤ 4 − u 0 . ( 2.32 ) Consequently the new exact problem, with length L = r 0 L=r_0 L = r 0 C ( 1 − R ) C(1-R) C ( 1 − R ) S S S η = ( 4 − u 0 ) / ( 2 r 0 ) > 0 \eta=(4-u_0)/(2r_0)>0 η = ( 4 − u 0 ) / ( 2 r 0 ) > 0
It remains to prove positive horizon. We return a true terminal chart’s mass floor to the original prior, then bound the prior mass of all trajectories that can satisfy its geometric tests. Comparing these bounds will force its time interval to be short.
The original-prior floor of a terminal chart. Consider any true terminal chart in this new problem, including one obtained after a further permitted selection or on a subsequence. Let its time length be q q q J n e w J_{\mathrm{new}} J new
J p h y s = a u 0 2 J n e w . J_{\mathrm{phys}}=a_{u_0}^2J_{\mathrm{new}}. J phys = a u 0 2 J new . Multiplying its true floor by ν ( B ) \nu(\mathcal{B}) ν ( B ) (2.31) , its selected mass in the original, unnormalized prior satisfies
m o l d , c q ≥ K 0 − 1 n ( r 0 , u 0 ) J p h y s = K 0 − 1 m ( r 0 , u 0 ) a r 0 J p h y s . (2.33) \frac{m_{\mathrm{old},c}}{q}\ge K_0^{-1}n(r_0,u_0)J_{\mathrm{phys}}=K_0^{-1}m(r_0,u_0)a_{r_0}J_{\mathrm{phys}}.\tag{2.33} q m old , c ≥ K 0 − 1 n ( r 0 , u 0 ) J phys = K 0 − 1 m ( r 0 , u 0 ) a r 0 J phys . ( 2.33 ) This calculation cancels ν ( B ) \nu(\mathcal{B}) ν ( B )
Every trajectory contributing positive selected time to this floor satisfies the whole-block spatial, small-value, and upper-derivative tests in the original coordinates, and satisfies ∣ P w ∣ ≥ K 0 − 1 |P_w|\ge K_0^{-1} ∣ P w ∣ ≥ K 0 − 1 m o l d , c / q m_{\mathrm{old},c}/q m old , c / q
The slope-volume bound. Fix this chart. Let Σ ⊂ R 3 \Sigma\subset\mathbb{R}^{3} Σ ⊂ R 3 s s s b b b
sup t ∈ I ∣ D p h y s − 1 ( y b , s ( t ) − y c ( t ) ) ∣ ≤ K 0 , sup t ∈ I ∣ P ( t , b + t s ) ∣ ≤ K 0 a r 0 , \sup_{t\in I}\left|D_{\mathrm{phys}}^{-1}\left(y_{b,s}(t)-y_{c}(t)\right)\right|\le K_{0},\qquad\sup_{t\in I}\left|P(t,b+ts)\right|\le K_{0}a_{r_{0}}, t ∈ I sup D phys − 1 ( y b , s ( t ) − y c ( t ) ) ≤ K 0 , t ∈ I sup ∣ P ( t , b + t s ) ∣ ≤ K 0 a r 0 , and
sup t ∈ I ∣ P w ( t , b + t s ) ∣ ≤ K 0 , sup t ∈ I ∣ P w ( t , b + t s ) ∣ ≥ K 0 − 1 . \sup_{t\in I}\left|P_{w}(t,b+ts)\right|\le K_{0},\qquad\sup_{t\in I}\left|P_{w}(t,b+ts)\right|\ge K_{0}^{-1}. t ∈ I sup ∣ P w ( t , b + t s ) ∣ ≤ K 0 , t ∈ I sup ∣ P w ( t , b + t s ) ∣ ≥ K 0 − 1 . Here P ( t , b + t s ) P(t,b+ts) P ( t , b + t s ) ( y , w ) (y,w) ( y , w ) D p h y s D_{\mathrm{phys}} D phys y y y
The set Σ \Sigma Σ Σ \Sigma Σ (2.33) .
We claim that
∣ Σ ∣ ≤ K 0 a r 0 J p h y s q 3 . (2.34) |\Sigma|\le K_{0}\frac{a_{r_{0}}J_{\mathrm{phys}}}{q^{3}}.\tag{2.34} ∣Σ∣ ≤ K 0 q 3 a r 0 J phys . ( 2.34 ) The original full-slope density ceiling will then convert this geometric bound into an upper bound for the prior mass of an eligible chart. To prove the claim, we select one eligible intercept for each slope and compare the volume swept out by these lines with the volume allowed by the chart tests.
The definition of Σ \Sigma Σ b b b t t t C 1 C^{1} C 1 b = b ( s ) b=b(s) b = b ( s ) C 1 C^{1} C 1 Σ \Sigma Σ [1 ] . Their bounds depend on the formula format, not on the numerical magnitudes of the chart coefficients.
For each t t t F t ( s ) = b ( s ) + t s F_{t}(s)=b(s)+ts F t ( s ) = b ( s ) + t s C 1 C^{1} C 1
det D F t ( s ) = det ( D b ( s ) + t Id ) , (2.35) \det DF_{t}(s)=\det(Db(s)+t\operatorname{Id}),\tag{2.35} det D F t ( s ) = det ( D b ( s ) + t Id ) , ( 2.35 ) a monic cubic in t t t F t F_{t} F t t t t C 1 C^{1} C 1 D b Db D b N 0 N_{0} N 0
Times with a large Jacobian and derivative. For a fixed slope s s s I I I c q cq c q (2.35) . The total removed length is at most 6 c q 6cq 6 c q c q cq c q
∣ det D F t ( s ) ∣ ≥ c 3 q 3 . \left|\det DF_{t}(s)\right|\ge c^{3}q^{3}. ∣ det D F t ( s ) ∣ ≥ c 3 q 3 . Also, g s ( t ) = P w ( t , b ( s ) + t s ) g_{s}(t)=P_{w}(t,b(s)+ts) g s ( t ) = P w ( t , b ( s ) + t s ) sup I ∣ g s ∣ ≥ K 0 − 1 \sup_{I}|g_{s}|\ge K_{0}^{-1} sup I ∣ g s ∣ ≥ K 0 − 1 ϵ > 0 \epsilon>0 ϵ > 0
∣ { t ∈ I : ∣ g s ( t ) ∣ < ϵ / K 0 } ∣ ≤ 4 ϵ q . (2.36) \left|\left\{t\in I:|g_{s}(t)|<\epsilon/K_{0}\right\}\right|\le4\epsilon q.\tag{2.36} ∣ { t ∈ I : ∣ g s ( t ) ∣ < ϵ / K 0 } ∣ ≤ 4 ϵ q . ( 2.36 ) To verify this, let M = sup I ∣ g s ∣ M=\sup_{I}|g_{s}| M = sup I ∣ g s ∣ g s g_{s} g s I I I M / 2 M/2 M /2 ∣ g s ∣ ≥ M / 2 |g_{s}|\ge M/2 ∣ g s ∣ ≥ M /2 M / ( 2 q ) M/(2q) M / ( 2 q ) ( − ϵ / K 0 , ϵ / K 0 ) (-\epsilon/K_{0},\epsilon/K_{0}) ( − ϵ / K 0 , ϵ / K 0 ) 4 ϵ q 4\epsilon q 4 ϵ q
Choose c , ϵ c,\epsilon c , ϵ q / 4 q/4 q /4 Σ t \Sigma_t Σ t C 1 C^1 C 1
∫ I ∣ Σ t ∣ d t ≥ q 2 ∣ Σ ∣ . (2.37) \int_I |\Sigma_t|\,dt \ge\frac{q}{2}|\Sigma|.\tag{2.37} ∫ I ∣ Σ t ∣ d t ≥ 2 q ∣Σ∣. ( 2.37 ) The two good-time conditions therefore overlap uniformly for each slope; no independence of them is asserted.
The area formula and the uniform regular-fiber bound give
q 4 ∣ Σ ∣ ≲ q 3 ∫ I ∣ Σ t ∣ d t ≲ ∫ I ∣ F t ( Σ t ) ∣ d t (2.38) q^4|\Sigma| \lesssim q^3\int_I|\Sigma_t|\,dt \lesssim\int_I|F_t(\Sigma_t)|\,dt\tag{2.38} q 4 ∣Σ∣ ≲ q 3 ∫ I ∣ Σ t ∣ d t ≲ ∫ I ∣ F t ( Σ t ) ∣ d t ( 2.38 ) ≤ K 0 q J p h y s a r 0 . (2.39) \le K_0qJ_{\mathrm{phys}}a_{r_0}.\tag{2.39} ≤ K 0 q J phys a r 0 . ( 2.39 ) For the last inequality, the allowed y y y K 0 J p h y s K_0J_{\mathrm{phys}} K 0 J phys t , y t,y t , y
∣ P ( t , y , w ) ∣ ≤ K 0 a r 0 , ∣ P w ( t , y , w ) ∣ ≥ ϵ / K 0 |P(t,y,w)|\le K_0a_{r_0},\qquad|P_w(t,y,w)|\ge\epsilon/K_0 ∣ P ( t , y , w ) ∣ ≤ K 0 a r 0 , ∣ P w ( t , y , w ) ∣ ≥ ϵ / K 0 confine w w w K 0 a r 0 K_0a_{r_0} K 0 a r 0 ( t , y ) (t,y) ( t , y ) I I I (2.39) by q 4 q^4 q 4 (2.34) .
Comparison with the mass floor. Use the original full-slope density ceiling Equation (2.26). The original prior mass of all geometrically eligible trajectories is at most K 0 ∣ Σ ∣ K_0|\Sigma| K 0 ∣Σ∣ a r 0 J p h y s > 0 a_{r_0}J_{\mathrm{phys}}>0 a r 0 J phys > 0
m ( r 0 , u 0 ) ≤ K 0 q − 3 . m(r_0,u_0)\le K_0q^{-3}. m ( r 0 , u 0 ) ≤ K 0 q − 3 . By (2.30) , m ( r 0 , u 0 ) ≥ N 0 σ / 2 m(r_0,u_0)\ge N_0^{\sigma/2} m ( r 0 , u 0 ) ≥ N 0 σ /2 q = N 0 − h + o ( 1 ) q=N_0^{-h+o(1)} q = N 0 − h + o ( 1 ) h ≥ σ / 6 h\ge\sigma/6 h ≥ σ /6 H ( E ) ≥ σ / 6 > 0 H(E)\ge\sigma/6>0 H ( E ) ≥ σ /6 > 0
We call a cap-class problem with H ( E ) > 0 H(E)>0 H ( E ) > 0 S , η > 0 S,\eta>0 S , η > 0
Comparison of algebraic sheets and extension of terminal time
We prove a comparison principle for two small lists of scalar functions and apply it to extend the terminal charts of an exact problem in time. The comparison forces sheets that agree on many sampled incidences to agree on a whole neighborhood. Algebraic continuation then carries these local comparisons to a common complex point, where finite jets provide labels for representative equations. This is how a fit on a longer time block can remain known from the old terminal observation.
The comparison uses fresh conditional samplings. Successful paths are kept as unnormalized submeasures; the sampling kernels are never conditioned on a later step succeeding.
We use real quantifier elimination and semialgebraic cell decomposition with parameters in fixed dimension and degree: formulas of fixed format define semialgebraic sets, and the numbers of connected components and isolated points of their fibers have bounds uniform in their real coefficients. A one-variable semialgebraic function which is finite and bounded on compact subintervals of [ 2 , ∞ ) [2,\infty) [ 2 , ∞ ) [1 ] , Chapters 3, 5, 7, and 14.
A holomorphic function will be called an algebraic sheet of degree at most D D D
P ( z , f ( z ) ) = 0 P(z,f(z))=0 P ( z , f ( z )) = 0 for a nonzero polynomial P P P D D D P P P
The next lemma belongs to the Bernstein–Remez theory of algebraic functions; compare Roytwarf and Yomdin [27 ] , Sections 3.3–3.5. We include a proof of the formulation used here.
Lemma 3.1 (Algebraic analytic norm comparison). Let Ω ⊂ C n \Omega\subset\mathbb{C}^n Ω ⊂ C n ball or polydisc , and let B ⊂ Ω ∩ R n B \subset\Omega\cap\mathbb{R}^n B ⊂ Ω ∩ R n K ⋐ Ω K \Subset\Omega K ⋐ Ω D D D ∥ f ∥ B = 1 \lVert f\rVert_B=1 ∥ f ∥ B = 1
for every fixed integer q q q max ∣ α ∣ ≤ q ∥ ∂ α f ∥ K ≤ C q \max_{|\alpha|\leq q}\lVert\partial^\alpha f\rVert_K\leq C_q max ∣ α ∣ ≤ q ∥ ∂ α f ∥ K ≤ C q
at every z ∈ K z\in K z ∈ K
max ∣ α ∣ ≤ D ∣ ∂ α f ( z ) ∣ ≥ c K > 0. \max_{|\alpha|\leq D}|\partial^\alpha f(z)|\geq c_K>0. ∣ α ∣ ≤ D max ∣ ∂ α f ( z ) ∣ ≥ c K > 0. Consequently, let B 1 ⊂ Ω ∩ R n B_1\subset\Omega\cap\mathbb{R}^n B 1 ⊂ Ω ∩ R n E ⊂ B E\subset B E ⊂ B θ = ∣ E ∣ / ∣ B ∣ > 0 \theta=|E|/|B|>0 θ = ∣ E ∣/∣ B ∣ > 0
∥ f ∥ B 1 + max ∣ α ∣ ≤ q ∥ ∂ α f ∥ B 1 ≤ C q θ − C ∥ f ∥ E (3.1) \lVert f\rVert_{B_1}+\max_{|\alpha|\leq q}\lVert\partial^\alpha f\rVert_{B_1}\leq C_q\theta^{-C}\lVert f\rVert_E\tag{3.1} ∥ f ∥ B 1 + ∣ α ∣ ≤ q max ∥ ∂ α f ∥ B 1 ≤ C q θ − C ∥ f ∥ E ( 3.1 ) The statements are homogeneous without normalization. They are unchanged, with the corresponding scaled derivatives, by affine translation and rescaling of the base, including translation to a parallel real slice through a complex point. A sufficiently long finite jet at the center of an interior ball controls the norm on that ball.
Proof. We first prove local boundedness of the normalized family, allowing the degree in the function variable to drop in a limit. Suppose f ν f_\nu f ν P ν ( z , W ) P_\nu(z,W) P ν ( z , W ) P P P W W W P = P ( z ) P=P(z) P = P ( z ) x ∈ B x\in B x ∈ B f ν ( x ) f_\nu(x) f ν ( x )
P ( x ) = lim ν → ∞ P ν ( x , f ν ( x ) ) = 0. P(x)=\lim_{\nu\to\infty}P_\nu(x,f_\nu(x))=0. P ( x ) = ν → ∞ lim P ν ( x , f ν ( x )) = 0. A polynomial vanishing on the real interior of B B B
Write a ( z ) a(z) a ( z ) P P P W W W Z = { a = 0 } Z=\{a=0\} Z = { a = 0 } Ω ∖ Z \Omega\setminus Z Ω ∖ Z P ( z , ⋅ ) P(z,\cdot) P ( z , ⋅ ) W W W P ν P_\nu P ν ν \nu ν P ν P_\nu P ν
Choose a point in the interior of B ∖ Z B\setminus Z B ∖ Z f ν f_\nu f ν Ω ∖ Z \Omega\setminus Z Ω ∖ Z f ν f_\nu f ν Ω ∖ Z \Omega\setminus Z Ω ∖ Z
It remains to cross Z Z Z Z Z Z a a a Z Z Z Z Z Z Ω ∖ Z \Omega\setminus Z Ω ∖ Z f ν f_\nu f ν Ω \Omega Ω
Cauchy’s estimates and compactness yield a subsequence converging holomorphically on compacta, together with all fixed derivatives, to a function f f f B B B ∥ f ∥ B = 1 \lVert f\rVert_B = 1 ∥ f ∥ B = 1 f f f P ( z , f ( z ) ) = 0 P(z,f(z)) = 0 P ( z , f ( z )) = 0 P P P W W W a 0 ( z ) a_0(z) a 0 ( z )
a 0 ( z ) = − ∑ k ≥ 1 a k ( z ) f ( z ) k . a_0(z)=-\sum_{k\geq1}a_k(z)f(z)^k. a 0 ( z ) = − k ≥ 1 ∑ a k ( z ) f ( z ) k . If f f f r r r a 0 a_0 a 0 r r r a 0 a_0 a 0 D D D r ≤ D r\leq D r ≤ D K K K
For completeness, these jet bounds imply a quantitative sublevel estimate. At each point of a fixed real interior ball, some directional derivative of order at most D D D ℜ f \Re f ℜ f ℑ f \Im f ℑ f p ≥ 1 p\geq1 p ≥ 1 m m m p + 1 p+1 p + 1 m m m
∣ { s : ∣ f ( s ) ∣ ≤ u } ∣ ≤ C u 1 / p . \bigl|\{s:|f(s)|\leq u\}\bigr|\leq Cu^{1/p}. { s : ∣ f ( s ) ∣ ≤ u } ≤ C u 1/ p . Integrating over transverse slices and using a finite cover gives, for some fixed c > 0 c>0 c > 0
∣ { x ∈ B 1 : ∣ f ( x ) ∣ ≤ u ∥ f ∥ B 1 } ∣ ≤ C u c ∣ B 1 ∣ , 0 < u < 1. (3.2) \bigl|\{x\in B_1:|f(x)|\leq u\lVert f\rVert_{B_1}\}\bigr|\leq Cu^c|B_1|,\qquad0<u<1.\tag{3.2} { x ∈ B 1 : ∣ f ( x ) ∣ ≤ u ∥ f ∥ B 1 } ≤ C u c ∣ B 1 ∣ , 0 < u < 1. ( 3.2 ) The same compactness argument compares the norms on the fixed interior balls involved here. Apply (3.2) with B B B B 1 B_1 B 1 E ⊂ B E\subset B E ⊂ B ∥ f ∥ B \lVert f\rVert_B ∥ f ∥ B C θ − C ∥ f ∥ E C\theta^{-C}\lVert f\rVert_E C θ − C ∥ f ∥ E (3.1) on B 1 B_1 B 1
We record closure properties needed when using this lemma. Sums, differences, and fixed-degree polynomial evaluations of a fixed number of algebraic sheets have polynomial relations of bounded degree. Over the rational function field in z z z
Local comparisons will eventually be recorded at a common complex point. The next lemma controls the cost of moving a sheet along a path that avoids its branch points, uniformly even when the equation coefficients degenerate.
Lemma 3.2 (Polynomial cost of continuation). Fix the base dimension, polynomial degrees, number of sheets, and a bound on the number of pieces of a broken straight path. Work in a fixed bounded complex base region. The sheets may be affine functions, roots of linear equations, or simple roots of quadratic equations in the function variable whose quadratic coefficient is constant. Differences and fixed-degree polynomial evaluations of these sheets are allowed. Suppose the linear coefficients needed for linear roots and the discriminants needed for quadratic roots are nonzero throughout an A − 1 A^{-1} A − 1 A ≥ 2 A \ge2 A ≥ 2 c / A c/A c / A A C A^C A C C C C
Proof. There are two separate uniformity arguments. For a fixed A A A O ( A ) O(A) O ( A ) A − 1 A^{-1} A − 1 C O ( A ) C^{O(A)} C O ( A ) A A A [ 2 , ∞ ) [2,\infty) [ 2 , ∞ )
We next show that the best such bound has a fixed semialgebraic description. Split all complex parameters into real and imaginary parts. The equation coefficients, the boundedly many path vertices, and the endpoint branch choices form a fixed-dimensional parameter space. Nonvanishing on the tube is expressed by real quantification over a segment parameter and a displacement of size less than A − 1 A^{-1} A − 1
For a square root of a nonvanishing discriminant, continuation of the endpoint signs can be encoded with finitely many crossings. Choose a generic ray from zero, avoiding the discriminant values at the path vertices. Along each straight piece the discriminant is a polynomial in a real segment parameter. Except when its argument is constant, there are only boundedly many critical directions: differentiate the argument, whose numerator is a real polynomial of bounded degree. Choose the ray away from those directions; a constant-argument piece can also be avoided. Crossings of the ray are then transversal and their number is bounded by the degree. Relative to a square root of the ray direction, label an endpoint square root by the sign of its imaginary part. Each crossing changes this sign, and the parity of the bounded list of crossings specifies exactly the continued endpoint root. Crossing parameters, their order, transversality, and the parity condition can all be described by polynomial conditions and real quantifiers of fixed complexity.
To evaluate a norm on an endpoint ball, append one straight segment from its center to a variable point of that ball and use the same description of continuation. Quantification over that point and its selected roots describes the norm inequalities. It follows from quantifier elimination that the supremum of the endpoint ratio over all admissible data is a one-variable semialgebraic function of A A A A C A^C A C O ( A ) O(A) O ( A )
Reversing the path proves the reverse comparison. Applying Lemma 3.1 on the endpoint balls, and accounting for their radius in derivatives, proves the jet assertions. □
In particular, a tiny constant quadratic coefficient causes no loss outside the stated bound. For example, the roots of ε W 2 + z W + 1 \varepsilon W^2+zW+1 ε W 2 + z W + 1 A A A
Two-color matching
The next lemma compares close values with separated derivative data. At value accuracy N − 1 N^{-1} N − 1 N d + o ( 1 ) N^{d+o(1)} N d + o ( 1 ) d < 1 d<1 d < 1
Lemma 3.3 (Two-color matching). Fix D ≥ 1 D\ge1 D ≥ 1 J > D J>D J > D d 0 < 1 d_0<1 d 0 < 1 N → ∞ N\to\infty N → ∞ C J + 1 C^{J+1} C J + 1 N d + o ( 1 ) N^{d+o(1)} N d + o ( 1 ) d ≤ d 0 d\le d_0 d ≤ d 0 J + 1 J+1 J + 1 K = N o ( 1 ) K=N^{o(1)} K = N o ( 1 ) ( i 1 , i 2 , t ) (i_1,i_2,t) ( i 1 , i 2 , t ) κ = 1 , 2 \kappa=1,2 κ = 1 , 2 ( i κ , t ) (i_\kappa,t) ( i κ , t )
K π κ , i κ d t , ∑ i π κ , i = 1 , (3.3) K\pi_{\kappa,i_\kappa}\,dt,\qquad\sum_i\pi_{\kappa,i}=1,\tag{3.3} K π κ , i κ d t , i ∑ π κ , i = 1 , ( 3.3 ) and every retained event satisfies
∣ f 1 , i 1 ( t ) − f 2 , i 2 ( t ) ∣ ≤ K N − 1 , (3.4) \lvert f_{1,i_1}(t)-f_{2,i_2}(t)\rvert\le KN^{-1},\tag{3.4} ∣ f 1 , i 1 ( t ) − f 2 , i 2 ( t )∣ ≤ K N − 1 , ( 3.4 ) max 1 ≤ j ≤ D ∣ f 1 , i 1 ( j ) ( t ) − f 2 , i 2 ( j ) ( t ) ∣ ≥ K − 1 . (3.5) \max_{1\le j\le D}\lvert f_{1,i_1}^{(j)}(t)-f_{2,i_2}^{(j)}(t)\rvert\ge K^{-1}.\tag{3.5} 1 ≤ j ≤ D max ∣ f 1 , i 1 ( j ) ( t ) − f 2 , i 2 ( j ) ( t )∣ ≥ K − 1 . ( 3.5 ) The impossibility is uniform with sufficiently small fixed power losses in place of all displayed subpower losses, for these fixed orders and d ≤ d 0 d\le d_0 d ≤ d 0
Proof. Suppose such a sequence exists. We record the functions’ jets at successive time and value scales. The list size bounds the number of finest records. On most nearby scales, an alternating path along functions from the two lists has a polynomial endpoint map with a nonzero Jacobian. Its image forces more records than this bound allows. The resampling argument below controls the law of these paths while retaining their full dependence on the preceding choices. This use of alternating paths and an endpoint Jacobian is analogous to the method of refinements for incidence operators; compare Christ [8 ] , Sections 3–4.
All probability restrictions in the proof have subpower mass, until the bounded-length switching experiment is introduced. Normalizing such a restriction changes (3.3) only by a subpower factor. We enlarge K K K
Jet scales and nested states. Logarithmically pigeonhole the derivative gaps through order J J J β j \beta_j β j j j j N − β j + o ( 1 ) N^{-\beta_j+o(1)} N − β j + o ( 1 )
β j ≥ 0 , β 0 ≥ 1 , β j = 0 for some 1 ≤ j ≤ D . \beta_j\ge0,\qquad\beta_0\ge1,\qquad\beta_j=0\text{ for some }1\le j\le D. β j ≥ 0 , β 0 ≥ 1 , β j = 0 for some 1 ≤ j ≤ D . At value scale N − a N^{-a} N − a N − r ( a ) N^{-r(a)} N − r ( a ) N − a N^{-a} N − a j j j r ( a ) ≥ ( a − β j ) / j r(a)\ge(a-\beta_j)/j r ( a ) ≥ ( a − β j ) / j j j j N − ( a − j r ( a ) ) N^{-(a-jr(a))} N − ( a − j r ( a )) a a a 0 ≤ a ≤ 1 0 \leq a \leq1 0 ≤ a ≤ 1
r ( a ) = max 1 ≤ j ≤ J a − β j j , l j ( a ) = max 0 ≤ x ≤ a { x − j r ( x ) } . (3.6) r(a)=\max_{1\leq j\leq J}\frac{a-\beta_j}{j},\qquad l_j(a)=\max_{0\leq x\leq a}\{x-jr(x)\}.\tag{3.6} r ( a ) = 1 ≤ j ≤ J max j a − β j , l j ( a ) = 0 ≤ x ≤ a max { x − j r ( x )} . ( 3.6 ) Then r ( 0 ) = 0 r(0)=0 r ( 0 ) = 0 r r r
l 0 ( a ) = a , 0 ≤ l j ( a ) ≤ β j , l j ( a ) − l j + i ( a ) ≤ i r ( a ) . (3.7) l_0(a)=a,\qquad0\leq l_j(a)\leq\beta_j,\qquad l_j(a)-l_{j+i}(a)\leq ir(a).\tag{3.7} l 0 ( a ) = a , 0 ≤ l j ( a ) ≤ β j , l j ( a ) − l j + i ( a ) ≤ i r ( a ) . ( 3.7 ) The last inequality follows by evaluating l j + i l_{j+i} l j + i l j l_j l j r ( x ) ≥ x / D r(x)\geq x/D r ( x ) ≥ x / D a > 0 a>0 a > 0
( J + 1 − j ) r ( a ) > l j ( a ) . (3.8) (J+1-j)r(a)>l_j(a).\tag{3.8} ( J + 1 − j ) r ( a ) > l j ( a ) . ( 3.8 ) Indeed, if a maximizing point x x x l j ( a ) l_j(a) l j ( a ) ( J + 1 ) r ( x ) > x (J+1)r(x)>x ( J + 1 ) r ( x ) > x r r r x = 0 x=0 x = 0 l j ( a ) = 0 l_j(a)=0 l j ( a ) = 0 r ( a ) > 0 r(a)>0 r ( a ) > 0
On an open piece where the unique active term in r r r m m m
r ( a ) = ( a − β m ) / m , (3.9) r(a)=(a-\beta_m)/m,\tag{3.9} r ( a ) = ( a − β m ) / m , ( 3.9 ) l j ( a ) = a − j r ( a ) ( 0 ≤ j ≤ m ) , (3.10) l_j(a)=a-jr(a)\qquad(0\leq j\leq m),\tag{3.10} l j ( a ) = a − j r ( a ) ( 0 ≤ j ≤ m ) , ( 3.10 ) l j ( a ) > a − j r ( a ) ( j > m ) . (3.11) l_j(a)>a-jr(a)\qquad(j>m).\tag{3.11} l j ( a ) > a − j r ( a ) ( j > m ) . ( 3.11 ) For j ≤ m j\leq m j ≤ m x − j r ( x ) x-jr(x) x − j r ( x ) j > m j>m j > m β m \beta_m β m r r r [ 0 , 1 ] [0,1] [ 0 , 1 ]
We choose the finite grid before constructing its states or selecting their atom masses. Fix an integer C > 5 J C>5J C > 5 J
l m + i ( a ) + i r ( a ) − β m > 0 ( 1 ≤ i ≤ J − m ) l_{m+i}(a)+ir(a)-\beta_m>0\qquad(1\leq i\leq J-m) l m + i ( a ) + i r ( a ) − β m > 0 ( 1 ≤ i ≤ J − m ) and all needed remainder gaps are uniformly positive. Choose a common small step b b b A 0 A_0 A 0 [ a , a + b ] [a,a+b] [ a , a + b ] a + C b a+Cb a + C b 1 1 1 a + b a+b a + b a + C b a+Cb a + C b N N N b b b
For a level a a a F a F_a F a I a I_a I a N − r ( a ) N^{-r(a)} N − r ( a ) 0 , … , J 0,\ldots,J 0 , … , J I a I_a I a N − l j ( a ) N^{-l_j(a)} N − l j ( a ) I a I_a I a j j j
K ∑ i = 0 J − j N − l j + i ( a ) − i r ( a ) + K N − ( J + 1 − j ) r ( a ) ≤ K N − l j ( a ) . (3.12) K\sum_{i=0}^{J-j}N^{-l_{j+i}(a)-ir(a)}+KN^{-(J+1-j)r(a)}\leq KN^{-l_j(a)}.\tag{3.12} K i = 0 ∑ J − j N − l j + i ( a ) − i r ( a ) + K N − ( J + 1 − j ) r ( a ) ≤ K N − l j ( a ) . ( 3.12 ) Here Equations (3.7) and (3.8) handle, respectively, the binned terms and the remainder. Applying the same calculation to the two arms, starting from the event gaps and l j ≤ β j l_j\leq\beta_j l j ≤ β j K K K
An index from one color and I 1 I_1 I 1
∣ supp F 1 ∣ ≤ N d + r ( 1 ) + o ( 1 ) . (3.13) |\operatorname{supp} F_1| \le N^{d+r(1)+o(1)}.\tag{3.13} ∣ supp F 1 ∣ ≤ N d + r ( 1 ) + o ( 1 ) . ( 3.13 ) For this fixed grid, make a further subpower selection on which the masses of occurring atoms, measured before this selection , are N − u a + o ( 1 ) N^{-u_a+o(1)} N − u a + o ( 1 ) (3.13) . The exponents are nonnegative and nondecreasing, and
u 1 ≤ d + r ( 1 ) . (3.14) u_1 \le d+r(1).\tag{3.14} u 1 ≤ d + r ( 1 ) . ( 3.14 ) If a retained coarse atom at a a a c > a c>a c > a
N u c − u a + o ( 1 ) . (3.15) N^{u_c-u_a+o(1)}.\tag{3.15} N u c − u a + o ( 1 ) . ( 3.15 ) To see this without any assertion about later conditional probabilities, denote the pre-selection law by p p r e p_{\mathrm{pre}} p pre p p r e p_{\mathrm{pre}} p pre N − u c − o ( 1 ) N^{-u_c-o(1)} N − u c − o ( 1 ) N − u a + o ( 1 ) N^{-u_a+o(1)} N − u a + o ( 1 ) (3.15) . Write P \mathbb{P} P
The upper bound (3.14) will contradict the growth forced by matching. More precisely, on almost all small steps in an active m m m
u a + b − u a ≥ b ( 1 + 1 m − o b ( 1 ) ) . u_{a+b}-u_a \ge b\left(1+\frac{1}{m}-o_b(1)\right). u a + b − u a ≥ b ( 1 + m 1 − o b ( 1 ) ) . Summing these increments, then decreasing b b b d + r ( 1 ) d+r(1) d + r ( 1 ) 1 + r ( 1 ) 1+r(1) 1 + r ( 1 )
A palette with little entropy growth. We use the entropy chain rule and conditional mutual information in their standard forms; see Cover and Thomas [9 ] . Fix D 1 > 5 D_1>5 D 1 > 5 Z Z Z m m m I A 0 I_{A_0} I A 0 N − β m − D 1 b N^{-\beta_m-D_1b} N − β m − D 1 b l m ( A 0 ) = β m l_m(A_0)=\beta_m l m ( A 0 ) = β m
H ( Z ∣ F A 0 ) ≤ 2 D 1 b log N + O ( 1 ) . H(Z\mid F_{A_0})\le2D_1b\log N+O(1). H ( Z ∣ F A 0 ) ≤ 2 D 1 b log N + O ( 1 ) . At a step retained in the grid construction, let
L a = log P ( Z ∣ F a + C b ) P ( Z ∣ F a ) L_a=\log\frac{\mathbb{P}(Z\mid F_{a+Cb})}{\mathbb{P}(Z\mid F_a)} L a = log P ( Z ∣ F a ) P ( Z ∣ F a + C b ) at the sampled palette value. Conditional mutual information and the entropy chain rule give
∑ a E L a ≤ C H ( Z ∣ F A 0 ) . \sum_a \mathbb{E}L_a\le C H(Z\mid F_{A_0}). a ∑ E L a ≤ C H ( Z ∣ F A 0 ) . The negative part has bounded expectation. Indeed, for probability vectors p , q p,q p , q
∑ p ( z ) < q ( z ) p ( z ) log q ( z ) p ( z ) ≤ 1 e ∑ z q ( z ) ≤ 1 e . \sum_{p(z)<q(z)}p(z)\log\frac{q(z)}{p(z)}\le\frac{1}{e}\sum_zq(z)\le\frac{1}{e}. p ( z ) < q ( z ) ∑ p ( z ) log p ( z ) q ( z ) ≤ e 1 z ∑ q ( z ) ≤ e 1 . Apply this conditional on the fine state, with p p p q q q b − 1 b^{-1} b − 1 E ( L a ) + \mathbb{E}(L_a)_+ E ( L a ) + O ( b 2 log N + 1 ) O(b^2\log N+1) O ( b 2 log N + 1 ) O ( b 3 / 4 ) + o N ( 1 ) O(b^{3/4})+o_N(1) O ( b 3/4 ) + o N ( 1 ) L a > b 5 / 4 log N L_a>b^{5/4}\log N L a > b 5/4 log N O ( b 1 / 2 ) + o N ( 1 ) O(b^{1/2})+o_N(1) O ( b 1/2 ) + o N ( 1 ) b 1 / 4 b^{1/4} b 1/4
P ( Z = z ∣ F a + C b ) ≤ N b 5 / 4 P ( Z = z ∣ F a ) (3.16) \mathbb{P}(Z=z\mid F_{a+Cb})\le N^{b^{5/4}}\mathbb{P}(Z=z\mid F_a)\tag{3.16} P ( Z = z ∣ F a + C b ) ≤ N b 5/4 P ( Z = z ∣ F a ) ( 3.16 ) at the sampled value, apart from probability tending to zero with b b b
The switching experiment. Fix a good step. We will make m + 1 m+1 m + 1 m m m (3.15) ; comparing those volumes will force the required increment. Start with E 0 ∼ P E^0 \sim\mathbb{P} E 0 ∼ P I a I_a I a p + 1 p+1 p + 1 E ~ p \widetilde{E}^p E p P \mathbb{P} P
F a + C b ( E ~ p ) = F a + C b ( E p ) . F_{a+Cb}(\widetilde{E}^p)=F_{a+Cb}(E^p). F a + C b ( E p ) = F a + C b ( E p ) . Write t p = t ( E p ) t_p=t(E^p) t p = t ( E p ) t ~ p = t ( E ~ p ) \widetilde{t}_p=t(\widetilde{E}^p) t p = t ( E p ) I a + C b I_{a+Cb} I a + C b t p t_p t p κ p \kappa_p κ p
φ p = f κ p , i κ p ( E ~ p ) . \varphi_p=f_{\kappa_p,i_{\kappa_p}}(\widetilde{E}^p). φ p = f κ p , i κ p ( E p ) . Draw E p + 1 E^{p+1} E p + 1 t p + 1 t_{p+1} t p + 1 P \mathbb{P} P I a I_a I a φ p \varphi_p φ p t p t_p t p t p + 1 t_{p+1} t p + 1
Before any failures are removed, E p E^p E p E ~ p \widetilde{E}^p E p 2 p P 2^p\mathbb{P} 2 p P P \mathbb{P} P
The possible coarse states stay in a subpower neighbor set of the start. A refresh preserves F a F_a F a a a a I a I_a I a K K K m + 1 m+1 m + 1 K K K K K K
Let t a t_a t a I a I_a I a J J J T T T p + 1 p+1 p + 1 φ = φ p \varphi=\varphi_p φ = φ p
s = ( t − t a ) N r ( a ) , Y j = N l j ( a ) ∂ t j ( φ − T ) ( t ) , 0 ≤ j < m . s=(t-t_a)N^{r(a)}, \qquad Y_j=N^{l_j(a)}\partial_t^j(\varphi-T)(t), \qquad0\leq j<m. s = ( t − t a ) N r ( a ) , Y j = N l j ( a ) ∂ t j ( φ − T ) ( t ) , 0 ≤ j < m . Along a traversal,
d Y j d s = Y j + 1 ( j < m − 1 ) , d Y m − 1 d s = U + O ( N − 2 b ) , (3.17) \frac{dY_j}{ds}=Y_{j+1}\quad(j<m-1), \qquad\frac{dY_{m-1}}{ds}=U+O(N^{-2b}),\tag{3.17} d s d Y j = Y j + 1 ( j < m − 1 ) , d s d Y m − 1 = U + O ( N − 2 b ) , ( 3.17 ) where U U U K K K N β m N^{\beta_m} N β m m m m t a t_a t a
K ∑ i = 1 J − m N β m − l m + i ( a ) − i r ( a ) + K N β m − ( J + 1 − m ) r ( a ) . K\sum_{i=1}^{J-m}N^{\beta_m-l_{m+i}(a)-ir(a)}+KN^{\beta_m-(J+1-m)r(a)}. K i = 1 ∑ J − m N β m − l m + i ( a ) − i r ( a ) + K N β m − ( J + 1 − m ) r ( a ) . The strict trimmed gaps make this smaller than N − 2 b N^{-2b} N − 2 b b b b
Choose U U U A 0 A_0 A 0 T T T I A 0 I_{A_0} I A 0 t a t_a t a O ( N − D 1 b ) O(N^{-D_1b}) O ( N − D 1 b ) m m m
l m + i ( A 0 ) + i r ( A 0 ) − β m > 0. l_{m+i}(A_0)+ir(A_0)-\beta_m>0. l m + i ( A 0 ) + i r ( A 0 ) − β m > 0. The remainder has the same strict gap as before. This discretizes U U U (3.17) using only the start, the coarse state, the palette, and the color. The two controls available in a fresh pair differ by at least K − 1 K^{-1} K − 1 m m m β m \beta_m β m 1 / ( 2 K ) 1/(2K) 1/ ( 2 K ) K K K
At a switch, the old and new arms have K K K a + C b a+Cb a + C b t p t_p t p t ~ p \widetilde{t}_p t p t p t_p t p (3.12) there gives a jump in Y j Y_j Y j
K N l j ( a ) − l j ( a + C b ) = K N − ( 1 − j / m ) C b ≤ N − 2 b , j < m . K N^{l_j(a)-l_j(a+Cb)} = K N^{-(1-j/m)Cb} \leq N^{-2b}, \qquad j < m. K N l j ( a ) − l j ( a + C b ) = K N − ( 1 − j / m ) C b ≤ N − 2 b , j < m . Here C > 5 J C > 5J C > 5 J
Kill a step if Equation (3.16) fails at its refresh. Also kill it if its chosen color-index/interval pair has P \mathbb{P} P π κ , i ∣ I a ∣ / log N \pi_{\kappa,i}|I_a|/\log N π κ , i ∣ I a ∣/ log N (3.3) . For each color the sum of these latter thresholds over all indices and intervals is 1 / log N 1/\log N 1/ log N
K π κ , i π κ , i ∣ I a ∣ / log N ≤ K ∣ I a ∣ . \frac{K\pi_{\kappa,i}}{\pi_{\kappa,i}|I_a|/\log N} \leq\frac{K}{|I_a|}. π κ , i ∣ I a ∣/ log N K π κ , i ≤ ∣ I a ∣ K . Consequently the m m m ∣ I a ∣ / K |I_a|/K ∣ I a ∣/ K b b b N N N
Domination conditional on the entire past. Fix the exact start E 0 E^0 E 0 T T T Λ \Lambda Λ Λ ( R ) ≤ K \Lambda(\mathbb{R}) \leq K Λ ( R ) ≤ K
Let H p \mathcal{H}_p H p H p \mathcal{H}_p H p (3.16) dominates that law by the corresponding coarse palette law. The chosen arm can depend on the full history and the full refreshed event, but its point mass is bounded by the sum of the two candidate-control point masses. Finally, conditional on that complete refreshed event and its actual chosen index, the retained new-time density is at most K d s K\,ds K d s unnormalized successful step kernel satisfies
Q H p ( d U d s ) ≤ K N b 5 / 4 Λ ( d U ) d s for every admissible H p (3.18) Q_{\mathcal{H}_p}(dU\,ds) \leq K N^{b^{5/4}}\Lambda(dU)\,ds\quad\text{for every admissible }\mathcal{H}_p\tag{3.18} Q H p ( d U d s ) ≤ K N b 5/4 Λ ( d U ) d s for every admissible H p ( 3.18 ) The same Λ \Lambda Λ
For a nonnegative necessary test depending only on the start and the successive controls and times, Equation (3.18) may be iterated backwards. Integrate the final step with its full previous history fixed. The upper integral then depends only on the previous displayed controls and times, so repeat with the previous step. No successful kernel is renormalized and no kernel is conditioned on future survival. This is why possible hidden dependence of future events on the refreshed line identity does not change the product upper bound.
The endpoint map and its target. Write U 1 , … , U m + 1 U_1,\ldots,U_{m+1} U 1 , … , U m + 1 s 1 , … , s m + 1 s_1,\ldots,s_{m+1} s 1 , … , s m + 1 s = s m + 1 s=s_{m+1} s = s m + 1 s 0 s_0 s 0 (3.17) with exact constant controls and no switching errors. Its endpoint differs from the actual low jets by O ( N − 2 b ) O(N^{-2b}) O ( N − 2 b ) s s s j j j
1 ( m − j ) ! [ U 1 ( s − s 0 ) m − j + ∑ p = 1 m ( U p + 1 − U p ) ( s − s p ) m − j ] . (3.19) \frac{1}{(m-j)!}\left[U_1(s-s_0)^{m-j}+\sum_{p=1}^{m}(U_{p+1}-U_p)(s-s_p)^{m-j}\right].\tag{3.19} ( m − j )! 1 [ U 1 ( s − s 0 ) m − j + p = 1 ∑ m ( U p + 1 − U p ) ( s − s p ) m − j ] . ( 3.19 ) The omitted part depends only on the initial jets and s s s
For fixed controls the map from ( s 1 , … , s m , s ) (s_1,\ldots,s_m,s) ( s 1 , … , s m , s ) (3.19) with respect to s p s_p s p
− U p + 1 − U p ( m − j − 1 ) ! ( s − s p ) m − j − 1 . -\frac{U_{p+1}-U_p}{(m-j-1)!}(s-s_p)^{m-j-1}. − ( m − j − 1 )! U p + 1 − U p ( s − s p ) m − j − 1 . Thus the absolute Jacobian determinant is exactly
∏ p = 1 m ∣ U p + 1 − U p ∣ ∏ 1 ≤ p < q ≤ m ∣ s p − s q ∣ ∏ k = 0 m − 1 k ! . (3.20) \frac{\prod_{p=1}^{m}|U_{p+1}-U_p|\prod_{1\leq p<q\leq m}|s_p-s_q|}{\prod_{k=0}^{m-1}k!}.\tag{3.20} ∏ k = 0 m − 1 k ! ∏ p = 1 m ∣ U p + 1 − U p ∣ ∏ 1 ≤ p < q ≤ m ∣ s p − s q ∣ . ( 3.20 ) It is at least K − 1 K^{-1} K − 1
Set c = a + b c=a+b c = a + b F c F_c F c I c I_c I c I c I_c I c T T T j < m j<m j < m
K N l j ( a ) ( ∑ k = j J N − l k ( c ) − ( k − j ) r ( c ) + N − ( J + 1 − j ) r ( c ) ) ≤ K N l j ( a ) − l j ( c ) = K N − ( 1 − j / m ) b . (3.21) KN^{l_j(a)}\left(\sum_{k=j}^{J}N^{-l_k(c)-(k-j)r(c)}+N^{-(J+1-j)r(c)}\right)
\leq KN^{l_j(a)-l_j(c)}=KN^{-(1-j/m)b}.\tag{3.21} K N l j ( a ) k = j ∑ J N − l k ( c ) − ( k − j ) r ( c ) + N − ( J + 1 − j ) r ( c ) ≤ K N l j ( a ) − l j ( c ) = K N − ( 1 − j / m ) b . ( 3.21 ) This uses the child’s interval, not the whole coarse interval. It includes every higher-jet uncertainty and the Taylor remainder. The simulation error N − 2 b N^{-2b} N − 2 b O ( N − b / m ) O(N^{-b/m}) O ( N − b / m )
K N − b ( 1 m + ∑ j = 0 m − 1 ( 1 − j / m ) ) = K N − b ( 1 m + m + 1 2 ) . (3.22) KN^{-b\left(\frac{1}{m}+\sum_{j=0}^{m-1}(1-j/m)\right)}
=KN^{-b\left(\frac{1}{m}+\frac{m+1}{2}\right)}.\tag{3.22} K N − b ( m 1 + ∑ j = 0 m − 1 ( 1 − j / m ) ) = K N − b ( m 1 + 2 m + 1 ) . ( 3.22 ) Its center may move rapidly with the final time; Fubini’s theorem uses only the widths of the fibers in (3.21) .
Take the union of these targets over both colors and all retained children of every possible coarse neighbor of the start. By (3.15) , it has at most N u a + b − u a + o ( 1 ) N^{u_{a+b}-u_a+o(1)} N u a + b − u a + o ( 1 )
For each fixed separated control tuple, the change-of-variables formula, (3.20) , and the bounded regular-fiber multiplicity bound the time integral by K K K Λ m + 1 \Lambda^{m+1} Λ m + 1 Λ \Lambda Λ (3.18) . The conditional successful probability is at most
N u a + b − u a − b ( 1 m + m + 1 2 ) + ( m + 1 ) b 5 / 4 + o N ( 1 ) . N^{u_{a+b}-u_a-b\left(\frac{1}{m}+\frac{m+1}{2}\right)+(m+1)b^{5/4}+o_N(1)}. N u a + b − u a − b ( m 1 + 2 m + 1 ) + ( m + 1 ) b 5/4 + o N ( 1 ) . The averaged lower bound for success gives, on every good step,
u a + b − u a ≥ b ( 1 + 1 m − o b ( 1 ) ) . (3.23) u_{a+b}-u_a \ge b\left(1+\frac{1}{m}-o_b(1)\right).\tag{3.23} u a + b − u a ≥ b ( 1 + m 1 − o b ( 1 ) ) . ( 3.23 ) since ( m + 1 ) / 2 ≥ 1 (m+1)/2 \ge1 ( m + 1 ) /2 ≥ 1
For this fixed trimmed grid, first let N → ∞ N \to\infty N → ∞ (3.23) ; all omitted increments are nonnegative. Only after this fixed-grid limit do we let b → 0 b \to0 b → 0 b / m b/m b / m ∫ 0 1 r ′ ( a ) d a = r ( 1 ) \int_0^1 r'(a)\,da=r(1) ∫ 0 1 r ′ ( a ) d a = r ( 1 ) b b b (3.14) implies d ≥ 1 d \ge1 d ≥ 1
Finally, if no uniform small fixed-power tolerance existed, choose counterexamples with loss exponents tending to zero and N N N d 0 d_0 d 0
Corollary 3.4 (Local matching of scalar graphs). Fix the base dimension, algebraic degrees, real comparison balls with fixed holomorphic margins, and d 0 < 1 d_0<1 d 0 < 1 ε ∗ > 0 \varepsilon_* > 0 ε ∗ > 0 M , Δ > 0 M,\Delta>0 M , Δ > 0 R = M / Δ R=M/\Delta R = M /Δ R R R 0 < ε ≤ ε ∗ 0<\varepsilon\le\varepsilon_* 0 < ε ≤ ε ∗ d ≤ d 0 d\le d_0 d ≤ d 0
each list has at most R d + ε R^{d+\varepsilon} R d + ε M R ε MR^\varepsilon M R ε
a subprobability law on pairs of indices and a common real base point in the comparison domain has total mass at least R − ε R^{-\varepsilon} R − ε R ε π κ , i d z R^\varepsilon\pi_{\kappa,i}\,dz R ε π κ , i d z π κ , i \pi_{\kappa,i} π κ , i
on its events, the two values differ by at most Δ R ε \Delta R^\varepsilon Δ R ε M R − ε MR^{-\varepsilon} M R − ε
Fixed multiplicative constants may be absorbed by decreasing ε ∗ \varepsilon_* ε ∗ R R R
Proof. The positive pair mass makes both lists nonempty. Choose a member f ∗ f_* f ∗ ∥ f i − f ∗ ∥ ≤ 2 M R ε \lVert f_i-f_*\rVert\le2MR^\varepsilon ∥ f i − f ∗ ∥ ≤ 2 M R ε f ∗ f_* f ∗ M M M R C ε R^{C\varepsilon} R C ε R − C ε R^{-C\varepsilon} R − C ε R R R ε \varepsilon ε R − 1 + ε R^{-1+\varepsilon} R − 1 + ε
Slice parallel to that direction. Fubini’s theorem supplies a fiber interval with sliced pair mass at least R − C ε R^{-C\varepsilon} R − C ε R C ε R^{C\varepsilon} R C ε R C ε R^{C\varepsilon} R C ε N = R N = R N = R ε \varepsilon ε ε ∗ \varepsilon_* ε ∗
Trading terminal thickness for time
We now apply scalar-sheet matching to a true terminal chart system. The aim is to enlarge its time intervals by a factor N 0 e N_0^e N 0 e C e Ce C e
Lemma 3.5 (Terminal time extension). Fix d 0 < 1 d_0 < 1 d 0 < 1 S , η > 0 S, \eta> 0 S , η > 0 C < ∞ C < \infty C < ∞ c > 0 c > 0 c > 0 C ( d ) C(d) C ( d ) d ≤ d 0 d \le d_0 d ≤ d 0 L L L h > 0 h > 0 h > 0
0 < e < min ( h , c L ) 0 < e < \min(h,cL) 0 < e < min ( h , c L ) there is, on a retained subsequence, a known atlas at horizon exponent h − e h-e h − e p L p_L p L a L ′ a_{L'} a L ′
0 < L ′ < L , L − L ′ ≤ C e . 0 < L' < L,\qquad L-L' \le Ce. 0 < L ′ < L , L − L ′ ≤ C e . The resulting charts, trajectory assignments, tests, and selected incidences are tempered in the original exact sequence. The constant C C C
We recall the quadratic identities used to replace implicit tests by graphs. Write z z z P ( z , w ) P(z,w) P ( z , w ) c = P w w c=P_{ww} c = P w w
D = P w 2 − 2 c P \mathcal{D}=P_w^2-2cP D = P w 2 − 2 c P is independent of w w w P P P P w P_w P w c c c P w ≠ 0 P_w\ne0 P w = 0 2 ∣ c P ∣ ≤ 1 2 ∣ P w ∣ 2 2|cP|\le\frac{1}{2}|P_w|^2 2∣ c P ∣ ≤ 2 1 ∣ P w ∣ 2 D > 0 \mathcal{D}>0 D > 0 f f f P w ( z , f ) P_w(z,f) P w ( z , f ) P w ( z , w ) P_w(z,w) P w ( z , w )
∣ w − f ∣ = 2 ∣ P ( w ) ∣ ∣ P w ( w ) ∣ + D ≤ 2 ∣ P ( w ) ∣ ∣ P w ( w ) ∣ . (3.24) |w-f|=\frac{2|P(w)|}{|P_w(w)|+\sqrt{\mathcal{D}}}\le\frac{2|P(w)|}{|P_w(w)|}.\tag{3.24} ∣ w − f ∣ = ∣ P w ( w ) ∣ + D 2∣ P ( w ) ∣ ≤ ∣ P w ( w ) ∣ 2∣ P ( w ) ∣ . ( 3.24 ) For c = 0 c=0 c = 0 c ≠ 0 c\ne0 c = 0 F c t r F_{\mathrm{ctr}} F ctr
w − F c t r ( z ) = P w ( z , w ) P w w . (3.25) w-F_{\mathrm{ctr}}(z)=\frac{P_w(z,w)}{P_{ww}}.\tag{3.25} w − F ctr ( z ) = P w w P w ( z , w ) . ( 3.25 ) These formulas follow by completing the square. A nonvanishing discriminant on a simply connected complex ball supplies a simple holomorphic branch there; a linear equation with nonzero coefficient supplies its branch by division.
Proof. Throughout this proof N = N 0 N=N_0 N = N 0 δ = N − L \delta=N^{-L} δ = N − L w w w B w Bw B w δ \delta δ N O ( e ) N^{O(e)} N O ( e ) L L L h h h e e e e e e e / L e/L e / L
The proof has three parts. We first compare the old equations at two incidences in a longer time block. We then record their agreement by finitely many derivatives at a common complex point and choose one real representative equation for each derivative bin. Finally, we fix one source time per block, use the source incidence to assign trajectories to these bins, and normalize the representative equations to obtain the new charts.
Comparing the old terminal equations.
Prepare the true terminal system by Lemma 2.10, so its labels are known by p L p_L p L q 0 = N − h + o ( 1 ) q_0 = N^{-h+o(1)} q 0 = N − h + o ( 1 ) q 0 q_0 q 0 q 0 g q_0g q 0 g g = N − β g = N^{-\beta} g = N − β β ≥ 0 \beta\ge0 β ≥ 0 g = 1 g = 1 g = 1 q 0 q_0 q 0
n Q = n ( L ) q 0 j g . (3.26) n_Q = n(L)q_0^j g.\tag{3.26} n Q = n ( L ) q 0 j g . ( 3.26 ) Each used old label has selected incidence mass at least n Q / K 0 n_Q/K_0 n Q / K 0 q 0 q_0 q 0
Choose a coarser dyadic partition into blocks I I I q 1 q_1 q 1 q 1 / q 0 q_1/q_0 q 1 / q 0 N e N^e N e e < h e < h e < h γ \gamma γ I I I ν γ \nu_\gamma ν γ t 0 t_0 t 0 t t t I I I ω γ q 1 \omega_\gamma q_1 ω γ q 1
Let s I ( ℓ ) s_I(\ell) s I ( ℓ ) ℓ \ell ℓ I I I
∑ γ , I ω γ q 1 ∫ s I ( ℓ ) 2 d ν γ ( ℓ ) . \sum_{\gamma,I}\omega_\gamma q_1 \int s_I(\ell)^2\,d\nu_\gamma(\ell). γ , I ∑ ω γ q 1 ∫ s I ( ℓ ) 2 d ν γ ( ℓ ) . By Jensen’s inequality it is at least the square of the old selected mass, and hence is bounded below by an inverse-subpower factor. Since 0 ≤ s I ≤ 1 0 \le s_I \le1 0 ≤ s I ≤ 1
Write A A A A ′ A' A ′ t 0 t_0 t 0 t t t q 0 q_0 q 0 P A P_A P A P A ′ P_{A'} P A ′ A A A
∣ P A , w ( t 0 , y ( t 0 ) , w ( t 0 ) ) ∣ ≥ K 0 − 1 . \lvert P_{A,w}(t_0,y(t_0),w(t_0))\rvert\ge K_0^{-1}. ∣ P A , w ( t 0 , y ( t 0 ) , w ( t 0 ))∣ ≥ K 0 − 1 . The label A ′ A' A ′ t t t t t t
Here is the form of the labels we will construct. From A A A D D D D D D α D \alpha_D α D α D \alpha_D α D b b b c 0 = ( D , b ) c_0=(D,b) c 0 = ( D , b ) A ′ A' A ′ c 0 c_0 c 0
At this stage t 0 t_0 t 0 c 0 c_0 c 0
In one world and new block the old label count is at most
K 0 N e n Q . (3.27) \frac{K_0N^e}{n_Q}.\tag{3.27} n Q K 0 N e . ( 3.27 ) Indeed the new block contains O ( N e ) O(N^{e}) O ( N e ) K 0 / n Q K_0/n_Q K 0 / n Q ν γ \nu_\gamma ν γ
Polynomial extrapolation from an old subblock to the new block gives, for either old test on its assigned trajectory,
sup I ∣ P ∣ ≤ δ N C 0 e , sup I ∣ P w ∣ ≤ N C 0 e , (3.28) \sup_I |P| \le\delta N^{C_0 e}, \qquad\sup_I |P_w| \le N^{C_0 e},\tag{3.28} I sup ∣ P ∣ ≤ δ N C 0 e , I sup ∣ P w ∣ ≤ N C 0 e , ( 3.28 ) after increasing a fixed C 0 C_0 C 0 K 0 O ( N e ) K_0O(N^e) K 0 O ( N e )
For comparison at the source, we also need to evaluate the witness test away from its own incidence. The derivative of each test is at least N − e N^{-e} N − e K 0 − 1 K_0^{-1} K 0 − 1 I I I M M M ε M \varepsilon M εM O ( ε ) O(\varepsilon) O ( ε ) ε = K 0 N − e \varepsilon=K_0N^{-e} ε = K 0 N − e
Cap the one-time densities at both vertices on a fixed thickness-depth grid of spacing at most e e e 0 , L 0,L 0 , L
Also cap the joint terminal/angular density at angular depth u = R e u=Re u = R e
n ( L ) N e − S R e , n(L)N^{e-SRe}, n ( L ) N e − S R e , where R R R R e ≤ η L Re\le\eta L R e ≤ η L
A common spatial frame. In version two, consider a pair whose old narrow normals differ in sine angle by more than g N R e gN^{Re} g N R e g N R e < 1 gN^{Re}<1 g N R e < 1 K 0 g K_0g K 0 g K 0 N − R e K_0N^{-Re} K 0 N − R e
We bound the exceptional pair mass using the angular cap before any conditioning on that exception. Fix A A A
q 0 2 g N C 0 e . q_0^2gN^{C_0e}. q 0 2 g N C 0 e . At each exact base, apply Lemma 2.9 with the value width from (3.28) , the opposite-time derivative floor N − e N^{-e} N − e δ \delta δ N O ( e ) N^{O(e)} N O ( e ) C 0 C_0 C 0 R R R N C 0 e N^{C_0e} N C 0 e A ′ A' A ′ K 0 K_0 K 0 K 0 K_0 K 0 R e Re R e A A A
K 0 N e n Q ( q 0 2 g N C 0 e ) ( N C 0 e ) ( n ( L ) N ( 1 − S R ) e ) . \frac{K_0N^e}{n_Q}\left(q_0^2gN^{C_0e}\right)\left(N^{C_0e}\right)\left(n(L)N^{(1-SR)e}\right). n Q K 0 N e ( q 0 2 g N C 0 e ) ( N C 0 e ) ( n ( L ) N ( 1 − S R ) e ) . The four factors are, respectively, the old-label count, spatial area, hidden-bin count, and joint density cap; the partner-label and angular-bin multiplicities are absorbed in K 0 K_0 K 0 ∫ I d t / q 1 = 1 \int_I dt/q_1 = 1 ∫ I d t / q 1 = 1 n Q = n ( L ) q 0 2 g n_Q = n(L)q_0^2g n Q = n ( L ) q 0 2 g
K 0 N ( 2 + 2 C 0 − S R ) e . (3.29) K_0N^{(2+2C_0-SR)e}.\tag{3.29} K 0 N ( 2 + 2 C 0 − S R ) e . ( 3.29 ) Choose R R R
Put g # = min ( 1 , g N R e ) g^\# = \min(1,gN^{Re}) g # = min ( 1 , g N R e ) g # = 1 g^\# = 1 g # = 1 D D D A A A q 1 , q 1 g # q_1,q_1g^\# q 1 , q 1 g # q 1 q_1 q 1 g # < 1 g^\# < 1 g # < 1 g # g^\# g # g # g^\# g # q 1 , q 1 g # q_1,q_1g^\# q 1 , q 1 g # g # = 1 g^\# = 1 g # = 1 q 1 q_1 q 1 I I I
Moreover, A ′ A' A ′ K 0 K_0 K 0 D D D A A A A ′ A' A ′ A ′ A' A ′ K 0 q 1 K_0q_1 K 0 q 1 K 0 q 1 g K_0q_1g K 0 q 1 g K 0 K_0 K 0 K 0 g K_0g K 0 g g # g^\# g # K 0 q 1 g # K_0q_1g^\# K 0 q 1 g # K 0 g # K_0g^\# K 0 g #
Use coordinates z = ( τ , ξ ) z=(\tau,\xi) z = ( τ , ξ ) D D D I I I
N C 1 e q 0 j g . (3.30) N^{C_1e}q_0^jg.\tag{3.30} N C 1 e q 0 j g . ( 3.30 ) Here C 1 C_1 C 1 R R R z 0 z_0 z 0 t 0 t_0 t 0
Let J D J_D J D ( A , D ) (A,D) ( A , D ) ( A ′ , D ) (A',D) ( A ′ , D ) z 0 = ( τ 0 , ξ 0 ) z_0=(\tau_0,\xi_0) z 0 = ( τ 0 , ξ 0 ) d τ 0 d ξ 0 d\tau_0\,d\xi_0 d τ 0 d ξ 0
N C 1 e n Q . (3.31) N^{C_1e}n_Q.\tag{3.31} N C 1 e n Q . ( 3.31 ) Indeed, the source's own test or the witness's extrapolated test at t 0 t_0 t 0 N C 0 e N^{C_0e} N C 0 e N e n ( L ) N^en(L) N e n ( L )
d t 0 d y q 1 = J D d τ 0 d ξ 0 . \frac{dt_0\,dy}{q_1}=J_D\,d\tau_0\,d\xi_0. q 1 d t 0 d y = J D d τ 0 d ξ 0 . Consequently, the density is at most J D N e n ( L ) N C 0 e ≤ N C 1 e n Q J_DN^en(L)N^{C_0e}\leq N^{C_1e}n_Q J D N e n ( L ) N C 0 e ≤ N C 1 e n Q C 1 C_1 C 1 (3.30) . No conditioning on the label and no new lower mass bound for that entry is used.
We will repeatedly use the following counting consequence. Suppose a group construction gives at most N C 1 e N^{C_1e} N C 1 e
n Q N − C ′ e n_QN^{-C'e} n Q N − C ′ e costs a power-small total amount if C ′ C' C ′ (3.27) , in each world/block, and then summing with the original weights. Apply this first to the entries ( A , D ) (A,D) ( A , D ) ( A ′ , D ) (A',D) ( A ′ , D )
Stable simple graphs and comparison mass. We now construct the graphs whose agreement will later be recorded at the anchor. Write P A , D P_{A,D} P A , D D D D w ↦ B w w \mapsto Bw w ↦ B w P A ′ , D P_{A',D} P A ′ , D w w w P P P C A C_A C A (3.28) . If
∣ P w w ∣ ≥ δ − 1 N − C A e , \lvert P_{ww}\rvert\ge\delta^{-1}N^{-C_Ae}, ∣ P w w ∣ ≥ δ − 1 N − C A e , replace this test for comparison by P ~ = w − F c t r ( z ) \widetilde{P}=w-F_{\mathrm{ctr}}(z) P = w − F ctr ( z ) (3.25) and the derivative upper bound give
∣ w − F c t r ( z ) ∣ ≤ δ N O ( e ) \lvert w-F_{\mathrm{ctr}}(z)\rvert\le\delta N^{O(e)} ∣ w − F ctr ( z )∣ ≤ δ N O ( e ) throughout the new block. Otherwise let P ~ = P \widetilde{P}=P P = P P ~ A , D \widetilde{P}_{A,D} P A , D P ~ A ′ , D \widetilde{P}_{A',D} P A ′ , D D D D C A C_A C A ∣ P w w P ∣ \lvert P_{ww}P\rvert ∣ P w w P ∣ ∣ P w ∣ 2 \lvert P_w\rvert^2 ∣ P w ∣ 2 z 0 , w ( t 0 ) z_0,w(t_0) z 0 , w ( t 0 ) ∣ P w ∣ ≥ N − e \lvert P_w\rvert\ge N^{-e} ∣ P w ∣ ≥ N − e
N − O ( e ) ≤ D ( z 0 ) : = P ~ w 2 − 2 P ~ w w P ~ ≤ N O ( e ) . (3.32) N^{-O(e)}\le\mathcal{D}(z_0):=\widetilde{P}_w^2-2\widetilde{P}_{ww}\widetilde{P}\le N^{O(e)}.\tag{3.32} N − O ( e ) ≤ D ( z 0 ) := P w 2 − 2 P w w P ≤ N O ( e ) . ( 3.32 ) For replaced tests the discriminant is one.
For each converted equation, the coefficients of its discriminant polynomial in z z z N O ( e ) N^{O(e)} N O ( e ) n Q N − C ′ e n_QN^{-C'e} n Q N − C ′ e n Q N C 1 e n_QN^{C_1e} n Q N C 1 e N − ( C ′ + C 1 ) e N^{-(C'+C_1)e} N − ( C ′ + C 1 ) e (3.32) holds. Polynomial Remez therefore bounds the coefficients and hence the discriminant and its derivatives on a fixed complex enlargement. This argument uses the original heavy entry, before later joint deletions.
Fix a net of real balls U U U N − C U e N^{-C_Ue} N − C U e z 0 z_0 z 0 C U C_U C U (3.32) . For each pair, a net center within half a radius of z 0 z_0 z 0 8 U 8U 8 U 8 U 8U 8 U w ( t 0 ) w(t_0) w ( t 0 ) (3.24) applies; for a replaced test use its affine graph. In either case,
∣ w ( t 0 ) − f ( z 0 ) ∣ ≤ δ N O ( e ) . (3.33) \lvert w(t_0)-f(z_0)\rvert\le\delta N^{O(e)}.\tag{3.33} ∣ w ( t 0 ) − f ( z 0 )∣ ≤ δ N O ( e ) . ( 3.33 ) We claim that, apart from power-small pair mass, the two chosen sheets differ in norm on 2 U 2U 2 U
δ N C ∗ e (3.34) \delta N^{C_*e}\tag{3.34} δ N C ∗ e ( 3.34 ) for a fixed sufficiently large C ∗ C_* C ∗ M M M 2 M 2M 2 M M ≥ δ N C ∗ e M\ge\delta N^{C_*e} M ≥ δ N C ∗ e M , D , U M,D,U M , D , U M M M U U U O ( M ) O(M) O ( M )
Each old label has at most N O ( e ) N^{O(e)} N O ( e ) C ∗ C_* C ∗ D D D N O ( e ) N^{O(e)} N O ( e ) M M M N N N O ( log N ) O(\log N) O ( log N ) z 0 z_0 z 0 (3.33) . Implicit differentiation, using the derivative lower bound from (3.32) , bounds its fixed jets polynomially. Affine replacements use an inverse curvature only above their cutoff. Finite-jet norm control bounds the whole comparison norm. The polynomial exponent here may depend on the exact problem; its logarithmic contribution is absorbed in N e N^e N e N N N
Let m G m_G m G w ( t 0 ) w(t_0) w ( t 0 ) O ( M ) O(M) O ( M ) (3.33) and choosing C ∗ C_* C ∗ M ≤ 1 M \le1 M ≤ 1 M ≤ a r ≤ M N e M \le a_r \le MN^e M ≤ a r ≤ M N e
n ( r ) ≤ n ( L ) ( a r / δ ) d n(r) \le n(L)(a_r/\delta)^d n ( r ) ≤ n ( L ) ( a r / δ ) d With (3.30) , this proves
m G ≤ N O ( e ) n Q ( M / δ ) d (3.35) m_G \le N^{O(e)}n_Q(M/\delta)^d\tag{3.35} m G ≤ N O ( e ) n Q ( M / δ ) d ( 3.35 ) When M > 1 M > 1 M > 1 w w w d ≥ 0 d \ge0 d ≥ 0 (3.35) .
In each color omit group indices, consisting of a label and branch, whose marginal mass before the joint deletion is less than n Q N − C ′ e n_QN^{-C'e} n Q N − C ′ e C ′ C' C ′ N − 2 e N^{-2e} N − 2 e
≤ N O ( e ) ( M / δ ) d \le N^{O(e)}(M/\delta)^d ≤ N O ( e ) ( M / δ ) d by (3.35) . If the total mass of these large-discrepancy groups were at least N − e N^{-e} N − e m G ′ ≥ m G / 2 m'_G \ge m_G/2 m G ′ ≥ m G /2 n Q / m G ′ n_Q/m'_G n Q / m G ′ 2 N C ′ e 2N^{C'e} 2 N C ′ e (3.31) , after rescaling the ball, then bounds each index-base density by N O ( e ) N^{O(e)} N O ( e )
The lists now agree in value to δ N O ( e ) \delta N^{O(e)} δ N O ( e ) M M M N O ( e ) N^{O(e)} N O ( e ) M / δ M/\delta M / δ C ∗ C_* C ∗ C ∗ C_* C ∗ (3.34) outside power-small mass.
Representative equations at a common anchor
The graphs now agree near their sampled source point, but those comparison balls vary with the point. We next record their agreement at one common complex point for each frame D D D
Common complex anchors. For each D D D α D \alpha_D α D σ D \sigma_D σ D z 0 z_0 z 0
z = z 0 + s ( α D − z 0 ) z = z_0 + s(\alpha_D-z_0) z = z 0 + s ( α D − z 0 ) and the broken path 0 → σ D → 1 0 \to\sigma_D \to1 0 → σ D → 1 s s s
For each retained pair, with probability 1 − N − Ω ( e ) 1 - N^{-\Omega(e)} 1 − N − Ω ( e ) N − O ( e ) N^{-O(e)} N − O ( e ) N − O ( e ) N^{-O(e)} N − O ( e ) z 0 z_0 z 0 α D \alpha_D α D N − B e N^{-Be} N − B e B B B s s s N O ( e ) N^{O(e)} N O ( e ) 0 , 1 0,1 0 , 1 N − O ( e ) N^{-O(e)} N − O ( e )
Remove disks of a much smaller radius N − B e N^{-Be} N − B e B B B σ D \sigma_D σ D σ D \sigma_D σ D N − e N^{-e} N − e N − O ( e ) N^{-O(e)} N − O ( e ) z z z
Choose common endpoint comparison balls of radius N − O ( e ) N^{-O(e)} N − O ( e ) (3.34) to the interior complex source domain, and then use Lemma 3.2. At the anchor their difference, and all required fixed jets in D D D z z z δ N O ( e ) \delta N^{O(e)} δ N O ( e )
For each D D D α D \alpha_D α D
δ N C b e , (3.36) \delta N^{C_b e},\tag{3.36} δ N C b e , ( 3.36 ) where C b C_b C b
Given A ′ A' A ′ D D D D D D P ~ A ′ , D \widetilde{P}_{A',D} P A ′ , D p L p_L p L N O ( e ) N^{O(e)} N O ( e )
A representative test for each bin. The source label A A A A ′ A' A ′ p L p_L p L D D D P ^ \widehat{P} P P ~ A , D \widetilde{P}_{A,D} P A , D N − O ( e ) N^{-O(e)} N − O ( e ) α D \alpha_D α D D D D t 0 t_0 t 0
On a common anchor ball with an inverse-power radius, the representative branch f ^ \widehat{f} f f f f
∥ f − f ^ ∥ ≤ δ N O ( e ) . \lVert f-\widehat{f}\rVert\le\delta N^{O(e)}. ∥ f − f ∥ ≤ δ N O ( e ) . The derivative of P ^ \widehat{P} P f ^ \widehat{f} f N O ( e ) N^{O(e)} N O ( e ) ∣ P ^ w w ∣ ≤ δ − 1 |\widehat{P}_{ww}|\le\delta^{-1} ∣ P w w ∣ ≤ δ − 1 w w w
∣ P ^ ( z , f ( z ) ) ∣ ≤ δ N O ( e ) , ∣ P ^ w ( z , f ( z ) ) ∣ ≤ N O ( e ) . (3.37) |\widehat{P}(z,f(z))|\le\delta N^{O(e)},\qquad|\widehat{P}_{w}(z,f(z))|\le N^{O(e)}.\tag{3.37} ∣ P ( z , f ( z )) ∣ ≤ δ N O ( e ) , ∣ P w ( z , f ( z )) ∣ ≤ N O ( e ) . ( 3.37 ) For example, the quadratic error in the first estimate is bounded by δ − 1 ( δ N O ( e ) ) 2 = δ N O ( e ) \delta^{-1}(\delta N^{O(e)})^2=\delta N^{O(e)} δ − 1 ( δ N O ( e ) ) 2 = δ N O ( e )
Return these evaluations to the source along the source-safe path. Each is a fixed-degree polynomial evaluation of the continued branch of P ~ A , D \widetilde{P}_{A,D} P A , D
On a real segment near t 0 t_0 t 0 N − O ( e ) N^{-O(e)} N − O ( e )
∣ w − f ( z ) ∣ ≤ δ N O ( e ) . |w-f(z)|\le\delta N^{O(e)}. ∣ w − f ( z ) ∣ ≤ δ N O ( e ) . For a replaced source this is its affine-center estimate. For an unreplaced source, P A , D , w P_{A,D,w} P A , D , w (3.28) , and has its own-time lower bound ∣ P A , D , w ( z 0 , w ( t 0 ) ) ∣ ≥ K − 1 |P_{A,D,w}(z_0,w(t_0))|\ge K^{-1} ∣ P A , D , w ( z 0 , w ( t 0 )) ∣ ≥ K − 1 A A A A ′ A' A ′ N − e N^{-e} N − e z z z C A C_A C A ∣ P w w P ∣ ≪ ∣ P w ∣ 2 |P_{ww}P|\ll|P_w|^2 ∣ P w w P ∣ ≪ ∣ P w ∣ 2 (3.24) thus applies continuously to the same source branch chosen at t 0 t_0 t 0 t 0 t_0 t 0
Use ∣ P ^ w w ∣ ≤ δ − 1 |\widehat{P}_{ww}|\le\delta^{-1} ∣ P w w ∣ ≤ δ − 1 f ( z ) f(z) f ( z ) w w w (3.37) . Polynomial extrapolation along the line then gives, on the entire new block,
sup I ∣ P ^ ( z , w ) ∣ ≤ δ N C 2 e , sup I ∣ P ^ w ( z , w ) ∣ ≤ N C 2 e . (3.38) \sup_I|\widehat{P}(z,w)|\le\delta N^{C_2e},\qquad\sup_I|\widehat{P}_w(z,w)|\le N^{C_2e}.\tag{3.38} I sup ∣ P ( z , w ) ∣ ≤ δ N C 2 e , I sup ∣ P w ( z , w ) ∣ ≤ N C 2 e . ( 3.38 ) for a fixed C 2 C_2 C 2 w − F ( t , y ) w-F(t,y) w − F ( t , y )
We also need a lower bound for the representative discriminant at the real source:
∣ D ^ ( z 0 ) ∣ = ∣ P ^ w 2 − 2 P ^ w w P ^ ∣ ≥ N − C 3 e . (3.39) |\widehat{\mathcal{D}}(z_0)|=|\widehat{P}_w^2-2\widehat{P}_{ww}\widehat{P}|\ge N^{-C_3e}.\tag{3.39} ∣ D ( z 0 ) ∣ = ∣ P w 2 − 2 P w w P ∣ ≥ N − C 3 e . ( 3.39 ) This fails only on power-small pair mass, with an exponent C 3 C_3 C 3 z z z B r e a l B_{\mathrm{real}} B real b a n c b_{\mathrm{anc}} b anc
∥ D ^ ∥ B r e a l ≥ N − b a n c e . \lVert\widehat{\mathcal{D}}\rVert_{B_{\mathrm{real}}}\ge N^{-b_{\mathrm{anc}}e}. ∥ D ∥ B real ≥ N − b anc e . Let θ > 0 \theta>0 θ > 0
∣ { z ∈ B r e a l : ∣ D ^ ( z ) ∣ < N − C 3 e } ∣ ≤ C N − θ ( C 3 − b a n c ) e . \bigl|\{z\in B_{\mathrm{real}}:|\widehat{\mathcal{D}}(z)|<N^{-C_3e}\}\bigr|\le C N^{-\theta(C_3-b_{\mathrm{anc}})e}. { z ∈ B real : ∣ D ( z ) ∣ < N − C 3 e } ≤ C N − θ ( C 3 − b anc ) e . For summation, fix a source label A A A D D D P ~ A , D \widetilde{P}_{A,D} P A , D (3.31) for this source, then sum using (3.27) . In each world and block the bad pair mass is at most
K 0 N e n Q ( n Q N C 1 e ) C N − θ ( C 3 − b a n c ) e ≤ K 0 N [ C 1 + 1 − θ ( C 3 − b a n c ) ] e . \begin{aligned}
&\frac{K_0N^e}{n_Q}(n_QN^{C_1e})CN^{-\theta(C_3-b_{\mathrm{anc}})e} \\
&\le K_0N^{[C_1+1-\theta(C_3-b_{\mathrm{anc}})]e}.
\end{aligned} n Q K 0 N e ( n Q N C 1 e ) C N − θ ( C 3 − b anc ) e ≤ K 0 N [ C 1 + 1 − θ ( C 3 − b anc )] e . The bounded representative count is absorbed in K 0 K_0 K 0 ω γ q 1 \omega_\gamma q_1 ω γ q 1
C 3 > b a n c + C 1 + 2 θ . C_3>b_{\mathrm{anc}}+\frac{C_1+2}{\theta}. C 3 > b anc + θ C 1 + 2 . This makes the exception power-small using only constants already fixed. In particular it is independent of the later curvature threshold. The charge is made while t 0 t_0 t 0 t 0 t_0 t 0
Construction of the longer atlas.
The representative equations now satisfy the whole-block bounds (3.38) . Outside power-small pair mass, their discriminants also satisfy (3.39) at the source. We use these facts to construct trajectory assignments on the longer blocks. The source label will determine an assignment, but will not be part of its output label.
Fixing the source and assigning anchor bins. Let a \mathbf{a} a S γ , I ( ℓ , t 0 , t ; a ) \mathcal{S}_{\gamma,I}(\ell,t_0,t;\mathbf{a}) S γ , I ( ℓ , t 0 , t ; a ) (3.39) . For ϑ = ( a , t 0 ) \vartheta=(\mathbf{a},t_0) ϑ = ( a , t 0 )
s γ , I ( ϑ ) = ∫ I ∫ 1 S γ , I ( ℓ , t 0 , t ; a ) d t q 1 d ν γ ( ℓ ) . s_{\gamma,I}(\vartheta)=\int_I\int\mathbf{1}_{\mathcal{S}_{\gamma,I}(\ell,t_0,t;\mathbf{a})}\frac{dt}{q_1}\,d\nu_\gamma(\ell). s γ , I ( ϑ ) = ∫ I ∫ 1 S γ , I ( ℓ , t 0 , t ; a ) q 1 d t d ν γ ( ℓ ) . Sample ϑ \vartheta ϑ t 0 ∈ I t_0\in I t 0 ∈ I ϑ γ , I = ( a γ , I , t γ , I s r c ) \vartheta_{\gamma,I}=(\mathbf{a}_{\gamma,I},t^{\mathrm{src}}_{\gamma,I}) ϑ γ , I = ( a γ , I , t γ , I src )
M f r : = ∑ γ , I ω γ q 1 s γ , I ( ϑ γ , I ) ≥ ∑ γ , I ω γ q 1 E ϑ s γ , I ( ϑ ) = N − o ( 1 ) . M_{\mathrm{fr}}:=\sum_{\gamma,I}\omega_\gamma q_1s_{\gamma,I}(\vartheta_{\gamma,I})\ge\sum_{\gamma,I}\omega_\gamma q_1\mathbb{E}_{\vartheta}s_{\gamma,I}(\vartheta)=N^{-o(1)}. M fr := γ , I ∑ ω γ q 1 s γ , I ( ϑ γ , I ) ≥ γ , I ∑ ω γ q 1 E ϑ s γ , I ( ϑ ) = N − o ( 1 ) . The lower bound concerns this weighted sum; individual blocks may have much smaller mass. In a fixed world and block, write t 0 = t γ , I s r c t_0=t^{\mathrm{src}}_{\gamma,I} t 0 = t γ , I src
d μ γ , I f r ( ℓ , t ) = 1 S γ , I ( ℓ , t 0 , t ; a γ , I ) d ν γ ( ℓ ) d t q 1 . d\mu^{\mathrm{fr}}_{\gamma,I}(\ell,t)=\mathbf{1}_{\mathcal{S}_{\gamma,I}(\ell,t_0,t;\mathbf{a}_{\gamma,I})}\,d\nu_\gamma(\ell)\frac{dt}{q_1}. d μ γ , I fr ( ℓ , t ) = 1 S γ , I ( ℓ , t 0 , t ; a γ , I ) d ν γ ( ℓ ) q 1 d t . It remains an unnormalized restriction of the original product prior. We now define geometric assignments containing its successful trajectories.
At the fixed source time, require old selected source-chart membership. This determines the old label A A A D D D P ~ A , D \widetilde{P}_{A,D} P A , D b b b (3.38) , and (3.39) at t 0 t_0 t 0 P ^ D , b \widehat{P}_{D,b} P D , b
c 0 = ( D , b ) , c_0=(D,b), c 0 = ( D , b ) , merging all source labels that give the same pair. The fixed source time and disjoint old source assignments give at most one label per trajectory in each new block. In particular this assignment is independent of the later time t t t P ^ D , b \hat{P}_{D,b} P ^ D , b
For these assignments retain the original later incidences whose old terminal label A ′ A' A ′ D D D b b b M f r M_{\mathrm{fr}} M fr d ν γ d t / q 1 d\nu_\gamma\,dt/q_1 d ν γ d t / q 1 p L p_L p L A ′ A' A ′ c 0 c_0 c 0 p L p_L p L
It remains to turn each representative equation into an admissible real test. The two curvature cases below preserve these assignments; only the low-curvature case refines their labels.
Normalizing the representative tests. The bounds (3.38) and the choice of C 3 C_3 C 3 C 4 > C 3 / 2 + 2 C_4>C_3/2+2 C 4 > C 3 /2 + 2
C s > C 2 + C 4 + 1 , C_s>C_2+C_4+1, C s > C 2 + C 4 + 1 , and finally choose
T > max ( C 2 + C 3 + 1 , C s + C 4 + 1 ) . (3.40) T>\max(C_2+C_3+1,\ C_s+C_4+1).\tag{3.40} T > max ( C 2 + C 3 + 1 , C s + C 4 + 1 ) . ( 3.40 ) First consider a representative with
∣ P ^ w w ∣ ≤ δ − 1 N − T e . |\hat{P}_{ww}|\leq\delta^{-1}N^{-Te}. ∣ P ^ w w ∣ ≤ δ − 1 N − T e . On every assigned trajectory, the product of curvature and value in (3.38) is at most N ( C 2 − T ) e N^{(C_2-T)e} N ( C 2 − T ) e (3.39) . At the real source,
P ^ w 2 = D ^ + 2 P ^ w w P ^ . \hat{P}_w^2=\widehat{\mathcal{D}}+2\hat{P}_{ww}\hat{P}. P ^ w 2 = D + 2 P ^ w w P ^ . It follows that the discriminant is positive and ∣ P ^ w ( z 0 , w ( t 0 ) ) ∣ ≥ N − C 4 e |\hat{P}_w(z_0,w(t_0))|\geq N^{-C_4e} ∣ P ^ w ( z 0 , w ( t 0 )) ∣ ≥ N − C 4 e
N − C 4 e ≤ M ∂ : = sup I ∣ P ^ w ( z , w ) ∣ ≤ N C 2 e . N^{-C_4e}\leq M_{\partial}:=\sup_I|\hat{P}_w(z,w)|\leq N^{C_2e}. N − C 4 e ≤ M ∂ := I sup ∣ P ^ w ( z , w ) ∣ ≤ N C 2 e . Let v v v M ∂ M_{\partial} M ∂ c = ( D , b , v ) c=(D,b,v) c = ( D , b , v ) P ^ D , b / v \hat{P}_{D,b}/v P ^ D , b / v v v v O ( log N ) O(\log N) O ( log N ) p L p_L p L v = 1 v=1 v = 1
The normalized derivative has block maximum between fixed positive constants. The value bound is at most δ N ( C 2 + C 4 ) e \delta N^{(C_2+C_4)e} δ N ( C 2 + C 4 ) e
δ − 1 N ( − T + C 4 ) e . \delta^{-1}N^{(-T+C_4)e}. δ − 1 N ( − T + C 4 ) e . By (3.40) , these fit the thickness
δ s = δ N C s e . \delta_s=\delta N^{C_se}. δ s = δ N C s e . The derivative upper bound is of order one. For its incidence lower bound, omit later times where its absolute value is below an inverse-subpower number κ \kappa κ O ( κ ) O(\kappa) O ( κ ) O ( κ ) O(\kappa) O ( κ ) κ = N − o ( 1 ) \kappa= N^{-o(1)} κ = N − o ( 1 ) κ = o ( M f r ) \kappa= o(M_{\mathrm{fr}}) κ = o ( M fr )
For the remaining representatives,
∣ P ^ w w ∣ > δ − 1 N − T e |\widehat{P}_{ww}| > \delta^{-1}N^{-Te} ∣ P w w ∣ > δ − 1 N − T e use the real affine critical center and the test w − F c t r ( z ) w-F_{\mathrm{ctr}}(z) w − F ctr ( z ) c = c 0 c = c_0 c = c 0 (3.25) and (3.38) ,
∣ w − F c t r ( z ) ∣ ≤ δ N ( T + C 2 ) e |w-F_{\mathrm{ctr}}(z)| \le\delta N^{(T+C_2)e} ∣ w − F ctr ( z ) ∣ ≤ δ N ( T + C 2 ) e on the entire new block. This fits thickness δ h = δ N ( T + C 2 + 1 ) e \delta_h = \delta N^{(T+C_2+1)e} δ h = δ N ( T + C 2 + 1 ) e
Choose a curvature case carrying at least half the remaining mass, and pass to a subsequence if necessary to obtain a common thickness exponent. All constants were chosen in the order C A C_A C A C 2 C_2 C 2 C 3 , C 4 , C s , T C_3,C_4,C_s,T C 3 , C 4 , C s , T C 3 C_3 C 3 T T T
Let C C C e < c L e < cL e < c L c > 0 c > 0 c > 0 L − C e > 0 L-Ce > 0 L − C e > 0 R e ≤ η L Re \le\eta L R e ≤ η L a L ′ aL' a L ′ 0 < L ′ < L 0 < L' < L 0 < L ′ < L L − L ′ ≤ C e L-L' \le Ce L − L ′ ≤ C e q 1 q_1 q 1 h − e h-e h − e
Finite encoding and return to the original coordinates. The assignments just constructed use polynomially many old labels, coarse boxes, anchor bins, and representative table entries. The chosen source branches and their propagated jets have polynomial ranges, as established above; unused far roots need not be stored. The continuation predicates have bounded semialgebraic complexity by Lemma 3.2, and fixed-order implicit differentiation gives the same property for the endpoint jets. Nearest-root choices, geometric inequalities, and affine block maxima also have fixed-degree descriptions. A whole-block bound quantifies only over one time variable for each individual test. Combining these conditions with the old predicates and finite tables therefore preserves temperedness.
All equation and coordinate coefficients have polynomial bounds. The affine-center constructions invert curvatures only above their specified cutoffs, and the low-curvature normalization divides only by numbers at least a constant times N − C 4 e N^{-C_4e} N − C 4 e
Finally compose the tests with the real affine coordinate map z = z ( t , y ) z=z(t,y) z = z ( t , y ) w ↦ B w w\mapsto Bw w ↦ B w w − F ( t , y ) w-F(t,y) w − F ( t , y ) B B B q 1 q_1 q 1 p L p_L p L
Critical paths and actual improvements
The preceding extension theorem permits a choice of exact problems in which the relative density exponent and the optimal time cost evolve linearly. We construct these paths first. We then pass to short portions of them, keeping an explicit exclusion in the original resolution. The last part of the section shows how local candidates return to that original exclusion with sufficient mass, geometric bounds and terminal label knowledge.
Extremal exact problems
Fix the model version and j j j S S S η \eta η
d ∗ = inf { d < 1 : ∃ E ∈ C ( d ) , H ( E ) > 0 } , k ( d ) = sup E ∈ C ( d ) H ( E ) / L , k = lim d ↓ d ∗ k ( d ) . (4.1) d_* = \inf\{d < 1 : \exists E \in C(d),\ H(E) > 0\}, \qquad k(d) = \sup_{E \in C(d)} H(E)/L, \qquad k = \lim_{d \downarrow d_*} k(d).\tag{4.1} d ∗ = inf { d < 1 : ∃ E ∈ C ( d ) , H ( E ) > 0 } , k ( d ) = E ∈ C ( d ) sup H ( E ) / L , k = d ↓ d ∗ lim k ( d ) . ( 4.1 ) Here 0 ≤ d ∗ < 1 0 \le d_* < 1 0 ≤ d ∗ < 1 k ( d ) k(d) k ( d ) N 0 N_0 N 0 N 0 L N_0^L N 0 L L L L C ( d ) C(d) C ( d ) L L L
Lemma 4.1 (Composition of horizon costs). Let E ∈ C ( d ) E \in C(d) E ∈ C ( d ) L L L r < L r < L r < L h h h F F F
H ( E ) ≤ h + H ( F ) . (4.2) H(E) \le h + H(F).\tag{4.2} H ( E ) ≤ h + H ( F ) . ( 4.2 ) If H o u t H_{\mathrm{out}} H out E E E r > 0 r > 0 r > 0
H ( E ) ≤ H o u t + k ( d ) ( L − r ) . (4.3) H(E) \le H_{\mathrm{out}} + k(d)(L-r).\tag{4.3} H ( E ) ≤ H out + k ( d ) ( L − r ) . ( 4.3 ) These statements allow all the subsequences and incidence restrictions in the definition of the respective infima.
Proof. Choose a true terminal system in F F F H ( F ) H(F) H ( F ) (4.2) .
Coarsening to r r r [ 0 , r ] [0,r] [ 0 , r ] p r p_r p r r r r p r p_r p r E E E p r p_r p r
Take these outer true systems with costs tending to H o u t H_{\mathrm{out}} H out C ( d ) C(d) C ( d ) L − r L-r L − r k ( d ) ( L − r ) k(d)(L-r) k ( d ) ( L − r ) H ( E ) H(E) H ( E ) □ \square □
Lemma 4.2 (A positive critical time rate). The rates in (4.1) satisfy 0 < k ≤ 1 0 < k \le1 0 < k ≤ 1
Proof. The upper bound follows from Proposition 2.16. For a bad problem E E E e > 0 e > 0 e > 0 H ( E ) > 0 H(E) > 0 H ( E ) > 0 h < H ( E ) + e / 4 h < H(E) + e/4 h < H ( E ) + e /4 h − e h-e h − e C e Ce C e C ( d ) C(d) C ( d ) (4.2) , its horizon is at least
H ( E ) − ( h − e ) > 3 e / 4. (4.4) H(E) - (h-e) > 3e/4.\tag{4.4} H ( E ) − ( h − e ) > 3 e /4. ( 4.4 ) Its horizon-to-length ratio is therefore at least 3 / ( 4 C ) 3/(4C) 3/ ( 4 C ) d d d d ∗ d_\ast d ∗ d 0 < 1 d_0 < 1 d 0 < 1 d ≤ d 0 d \le d_0 d ≤ d 0 j j j S S S η \eta η k k k
Scale units to L = 1 L = 1 L = 1 exact problems E i ∈ C ( d i ) E_i \in C(d_i) E i ∈ C ( d i ) d i → d ∗ d_i \to d_\ast d i → d ∗ H ( E i ) → k H(E_i) \to k H ( E i ) → k lim sup i k ( d i ) ≤ k \limsup_i k(d_i) \le k lim sup i k ( d i ) ≤ k d ∗ d_\ast d ∗ i i i N 0 N_0 N 0 i i i ( i , N 0 ) (i,N_0) ( i , N 0 )
In version 2, fix one maximizing sequence. Consider true terminal systems whose horizon exponents tend to k k k E i E_i E i (2.17) , in length-one units. Define ω \omega ω lim sup \limsup lim sup i i i
Put ℓ = inf ω \ell= \inf\omega ℓ = inf ω ℓ = 0 \ell= 0 ℓ = 0
Proposition 4.3 (Critical paths). Fix the model version and its parameters j j j S S S η \eta η d ∗ d_\ast d ∗ k k k ℓ \ell ℓ as above, the following conclusions hold.
Every maximizing sequence of length-one exact problems satisfies
sup 0 ≤ r ≤ 1 ∣ f i ( r ) − f i ( 0 ) − d ∗ r ∣ ⟶ 0. (4.5) \sup_{0\le r\le1} \left|f_i(r)-f_i(0)-d_\ast r\right| \longrightarrow0.\tag{4.5} 0 ≤ r ≤ 1 sup ∣ f i ( r ) − f i ( 0 ) − d ∗ r ∣ ⟶ 0. ( 4.5 ) Only the relative profiles occur; no bound on f i ( 0 ) f_i(0) f i ( 0 ) is required.
For every fixed finite ordered grid of target depths ending at one, there are nested true systems at actual depths tending to those targets. Their cumulative horizons tend to k r kr k r and their segment horizons, in their own length units, tend to k k k Both the coarsened and remaining problems used at each step are maximizing after their individual length normalizations. Full layer assignments, priors and incidence witnesses can be retained in addition to the final path incidences.
In version 2 with ℓ < ∞ \ell< \infty ℓ < ∞ for every ζ > 0 \zeta> 0 ζ > 0 one may choose the base maximizing sequence and these paths so that cumulative narrowness at a target r r r and narrowness increments between targets, equal the corresponding linear values with slope ℓ \ell ℓ up to O ( ζ ) + o i ( 1 ) O(\zeta)+o_i(1) O ( ζ ) + o i ( 1 ) The base sequence has terminal optimal narrowness supremum at most ℓ + ζ \ell+\zeta ℓ + ζ
If ℓ = ∞ \ell= \infty ℓ = ∞ every newly obtained maximizing sequence admits, on subsequences, true terminal systems with relative horizons tending to k k k and relative narrowness tending to infinity. Thus arbitrarily large narrowness increments can be required on each of finitely many successive positive-length segments.
Proof. Relative profiles. We first establish the relative-profile assertion,
f i ( r ) − f i ( 0 ) ⟶ d ∗ r (4.6) f_i(r) - f_i(0) \longrightarrow d_* r\tag{4.6} f i ( r ) − f i ( 0 ) ⟶ d ∗ r ( 4.6 ) uniformly.
Take any uniform subsequential limit of the relative profiles by their increment bounds. If it has a secant on an interior [ a , b ] [a,b] [ a , b ] d ∗ d_* d ∗ 0 < d ′ ′ < d ∗ 0 < d^{\prime\prime} < d_* 0 < d ′′ < d ∗ f i ( x ) − d ′ ′ x f_i(x) - d^{\prime\prime}x f i ( x ) − d ′′ x [ a , b ] [a,b] [ a , b ] r i r_i r i a a a i i i r i r_i r i d ′ ′ d^{\prime\prime} d ′′ a a a x < a x < a x < a
f i ( r i ) − f i ( x ) ≤ r i − x − ( 1 − d ′ ′ ) ( r i − a ) . (4.7) f_i(r_i) - f_i(x) \le r_i - x - (1-d^{\prime\prime})(r_i-a).\tag{4.7} f i ( r i ) − f i ( x ) ≤ r i − x − ( 1 − d ′′ ) ( r i − a ) . ( 4.7 ) Let G i G_i G i H ( G i ) = 0 H(G_i) = 0 H ( G i ) = 0 i i i C i , c i C_i,c_i C i , c i
0 < e < min ( H ( G i ) , c i r i , r i − a C i ) 0 < e < \min\left(H(G_i),c_i r_i,\frac{r_i-a}{C_i}\right) 0 < e < min ( H ( G i ) , c i r i , C i r i − a ) and a true terminal system with horizon h < H ( G i ) + e / 4 h < H(G_i) + e/4 h < H ( G i ) + e /4 h ≥ H ( G i ) > e h \ge H(G_i) > e h ≥ H ( G i ) > e h − e h-e h − e t i t_i t i
0 < r i − t i ≤ C i e < r i − a , 0 < r_i-t_i \le C_i e < r_i-a, 0 < r i − t i ≤ C i e < r i − a , so t i > a t_i > a t i > a
Normalize through this atlas to obtain a homogeneous remaining problem F F F L F = r i − t i L_F = r_i-t_i L F = r i − t i f F f_F f F
f F ( L F ) − f F ( s ) ≤ f i ( r i ) − f i ( t i + s ) ≤ d ′ ′ ( L F − s ) , 0 ≤ s ≤ L F . f_F(L_F) - f_F(s) \le f_i(r_i) - f_i(t_i+s) \le d^{\prime\prime}(L_F-s), \qquad0 \le s \le L_F. f F ( L F ) − f F ( s ) ≤ f i ( r i ) − f i ( t i + s ) ≤ d ′′ ( L F − s ) , 0 ≤ s ≤ L F . The second inequality uses only the slope cap on [ a , r i ] [a,r_i] [ a , r i ] F ∈ C ( d ′ ′ ) F \in C(d^{\prime\prime}) F ∈ C ( d ′′ ) (4.2) ,
H ( F ) ≥ H ( G i ) − ( h − e ) > 3 e / 4 > 0 , H(F) \ge H(G_i) - (h-e) > 3e/4 > 0, H ( F ) ≥ H ( G i ) − ( h − e ) > 3 e /4 > 0 , contradicting the definition of d ∗ d_* d ∗ G i G_i G i e e e i i i
Using these zero outer costs back in the original problems gives H ( E i ) ≤ k ( d i ) ( 1 − r i ) H(E_i) \le k(d_i)(1-r_i) H ( E i ) ≤ k ( d i ) ( 1 − r i ) k > 0 k > 0 k > 0 d ∗ = 0 d_* = 0 d ∗ = 0
Nested true systems. Fix a target 0 < r < 1 0 < r < 1 0 < r < 1 ϵ i > 0 \epsilon_i > 0 ϵ i > 0 f i ( x ) − ( d ∗ + ϵ i ) x f_i(x) - (d_*+\epsilon_i)x f i ( x ) − ( d ∗ + ϵ i ) x [ 0 , r ] [0,r] [ 0 , r ] r i → r r_i \to r r i → r d ∗ + ϵ i d_*+\epsilon_i d ∗ + ϵ i
The coarsened problems at r i r_i r i
H o u t ≤ k ( d ∗ + ϵ i ) r i , H ( E i ) ≤ H o u t + k ( d i ) ( 1 − r i ) , (4.8) H_{\mathrm{out}} \le k(d_*+\epsilon_i)r_i,\qquad H(E_i) \le H_{\mathrm{out}} + k(d_i)(1-r_i),\tag{4.8} H out ≤ k ( d ∗ + ϵ i ) r i , H ( E i ) ≤ H out + k ( d i ) ( 1 − r i ) , ( 4.8 ) so H o u t / r i → k H_{\mathrm{out}}/r_i \to k H out / r i → k
Take true systems on these outer problems with horizon exponents tending to k k k N 0 1 − r i N_0^{1-r_i} N 0 1 − r i H ( E i ) − ( k + o i ( 1 ) ) r i H(E_i)-(k+o_i(1))r_i H ( E i ) − ( k + o i ( 1 )) r i k ( d i ) ( 1 − r i ) k(d_i)(1-r_i) k ( d i ) ( 1 − r i )
We can therefore iterate over any fixed finite ordered grid of target depths ending at one. At each step use the next target fraction of the remaining length, choosing that fixed fraction from the limiting remaining length. The maximizing property just proved permits the next application; at fraction one use the full remaining maximizing sequence. This gives nested true systems whose actual depths tend to the targets and whose segment horizons have the required limits, with the final node at the true endpoint.
Each normalization divides by its individual length, and returning to old units multiplies by that length, not merely its limit. Only finitely many subsequences of i i i
Intermediate true packets lift through true parents by the equality case of Lemma 2.13. At each stage homogenize and prepare them with the bounds following Definition 2.8. Keep their full layer assignments, priors and incidence witnesses, as well as the final path incidences common to all layer selections. The final path alone need not saturate an earlier chart. In a parent, for example, the trajectory prior remains conditioned on its full assignment after further incidence selections. Child trajectory sets use this fixed prior and lift to subsets of the parent assignment.
Labels may include their whole histories; list knowledge composes via the parent’s knowledge and the bounded overlaps of the corresponding hidden grids once chart and exact base are specified. Time lengths and spatial Jacobians multiply. All these paths for a fixed finite grid and i i i
Narrowness optimization.
In version 2 with ℓ < ∞ \ell< \infty ℓ < ∞ ζ > 0 \zeta> 0 ζ > 0 ω ≤ ℓ + ζ \omega\le\ell+\zeta ω ≤ ℓ + ζ k k k lim inf \liminf lim inf ℓ − ζ \ell-\zeta ℓ − ζ ℓ \ell ℓ
Compose these systems through the final depth. True lifting gives a terminal true system with cumulative horizon exponent tending to k k k lim sup ≤ ω \limsup\le\omega lim sup ≤ ω actual segment length in original units. The segment lower bounds therefore give the cumulative lower bound at a node of target r r r ℓ + ζ + o i ( 1 ) \ell+\zeta+o_i(1) ℓ + ζ + o i ( 1 ) ℓ r + O ( ζ ) + o i ( 1 ) \ell_r+O(\zeta)+o_i(1) ℓ r + O ( ζ ) + o i ( 1 ) k r kr k r
If instead ℓ = ∞ \ell=\infty ℓ = ∞ k k k i i i ω \omega ω
□
Corollary 4.4 (Depth-zero normalization and vanishing-depth nodes). A sequence of depth-zero, horizon-zero known normalizations of a maximizing sequence is again maximizing , provided the normalizations have the actual subpower success and terminal knowledge required in Lemma 2.13. This assertion does not require bounded absolute density exponents.
On any maximizing sequence one may choose true nodes at positive depths b i → 0 b_i \to0 b i → 0
Proof. Let F i F_i F i C ( d i ) C(d_i) C ( d i )
H ( E i ) ≤ H ( F i ) ≤ k ( d i ) . (4.9) H(E_i) \le H(F_i) \le k(d_i).\tag{4.9} H ( E i ) ≤ H ( F i ) ≤ k ( d i ) . ( 4.9 ) Both outer bounds tend to k k k F i F_i F i (4.5) applies to f F i ( r ) − f F i ( 0 ) f_{F_i}(r) - f_{F_i}(0) f F i ( r ) − f F i ( 0 )
For every fixed b ∈ ( 0 , 1 ) b \in(0,1) b ∈ ( 0 , 1 ) b i → b b_i \to b b i → b k b + o i ( 1 ) kb + o_i(1) k b + o i ( 1 ) b = 1 / m b = 1/m b = 1/ m m m m m m m 1 / m 1/m 1/ m b b b b i > 0 b_i > 0 b i > 0 b i → 0 b_i \to0 b i → 0 m m m
Lemma 4.5 (Excluded improvements). There are two forbidden outputs of known atlases at an interior path node (including the parent where a short piece starts) as i → ∞ i \to\infty i → ∞ r i , h i r_i, h_i r i , h i p L p_L p L Δ i > 0 \Delta_i > 0 Δ i > 0
A thickness-depth improvement by Δ i > 0 \Delta_i > 0 Δ i > 0 r i r_i r i h i + k Δ i / 2 h_i + k\Delta_i/2 h i + k Δ i /2
In the finite-ℓ \ell ℓ Δ i ≫ ζ > 0 \Delta_i \gg\zeta> 0 Δ i ≫ ζ > 0
Proof. The first on an infinite subsequence would give by normalization and lifting
H ( E i ) ≤ h i + k Δ i / 2 + k ( d i ) ( 1 − r i − Δ i ) ≤ k − k Δ i / 2 + o i ( 1 ) . (4.10) H(E_i) \le h_i + k\Delta_i/2 + k(d_i)(1-r_i-\Delta_i) \le k-k\Delta_i/2+o_i(1).\tag{4.10} H ( E i ) ≤ h i + k Δ i /2 + k ( d i ) ( 1 − r i − Δ i ) ≤ k − k Δ i /2 + o i ( 1 ) . ( 4.10 ) impossible. In the second the remaining problems after the output atlas still maximize, by the same composition reasoning (they have remaining cost at least H ( E i ) − h i H(E_i)-h_i H ( E i ) − h i ≥ ℓ − ζ \ge\ell-\zeta ≥ ℓ − ζ k k k ℓ + ζ \ell+\zeta ℓ + ζ > 4 ζ >4\zeta > 4 ζ
A tangent diagonal with an original-resolution exclusion
We next extract short portions of these paths. The local estimates will use tangent units, while their final outputs must still violate an exclusion in the original exact problem.
Take stages n → ∞ n \to\infty n → ∞ ϵ n → 0 \epsilon_n \to0 ϵ n → 0 n n n D n ϵ n D_n\epsilon_n D n ϵ n ≳ 1 / ( D n ϵ n ) \gtrsim1/(D_n\epsilon_n) ≳ 1/ ( D n ϵ n ) o n ( 1 ) o_n(1) o n ( 1 ) s = ( offset from start ) / ϵ n s = (\text{offset from start})/\epsilon_n s = ( offset from start ) / ϵ n
Use a finest indicated coarse velocity precision of depth ϵ n U n \epsilon_n U_n ϵ n U n original problem coordinates, where U n , D n → ∞ U_n,D_n \to\infty U n , D n → ∞ U n D n ϵ n → 0 U_nD_n\epsilon_n \to0 U n D n ϵ n → 0
Choose a forbidden gain threshold γ n ϵ n \gamma_n\epsilon_n γ n ϵ n γ n → 0 \gamma_n \to0 γ n → 0 ζ ≪ γ n ϵ n \zeta\ll\gamma_n\epsilon_n ζ ≪ γ n ϵ n i i i N 0 N_0 N 0
N = N 0 ϵ n → ∞ , K = N o ( 1 ) (4.11) N = N_0^{\epsilon_n} \to\infty,\qquad K = N^{o(1)}\tag{4.11} N = N 0 ϵ n → ∞ , K = N o ( 1 ) ( 4.11 ) along the stages. All node depths, horizon exponents and profiles needed on the finite grids can use errors o n ( ϵ n ) o_n(\epsilon_n) o n ( ϵ n ) O ( ζ ) O(\zeta) O ( ζ )
Thus later labels at specified s s s K K K on the diagonal , possibly much too large as slacks for an output.
At a fixed stage n n n i i i γ n ϵ n \gamma_n\epsilon_n γ n ϵ n
K p a t h = N 0 o N 0 ∣ i ( 1 ) (4.12) K_{\mathrm{path}} = N_0^{o_{N_0\mid i}(1)}\tag{4.12} K path = N 0 o N 0 ∣ i ( 1 ) ( 4.12 ) include its true-path geometric and list bounds, cumulative products, preparation costs and reciprocal selected path mass. Retain an original tempered final path submeasure of mass p p a t h ≥ K p a t h − 1 \mathfrak{p}_{\mathrm{path}} \ge K_{\mathrm{path}}^{-1} p path ≥ K path − 1
Lemma 4.6 (The finite-sample exclusion). Fix this n n n i i i M # M^\# M # For all sufficiently late N 0 N_0 N 0
One of the improvements in Lemma 4.5, with its buffered gain at least γ n ϵ n \gamma_n\epsilon_n γ n ϵ n
Counts, coefficient magnitudes, reciprocal widths and lengths, piece and condition counts are at most N 0 M # N_0^{M^\#} N 0 M # M # M^\# M #
All geometric slacks and original p L p_L p L ( K p a t h log N 0 ) n (K_{\mathrm{path}}\log N_0)^n ( K path log N 0 ) n p p a t h / log N 0 \mathfrak{p}_{\mathrm{path}}/\log N_0 p path / log N 0
Proof. Otherwise there are infinitely many such outputs along this one exact sequence. The fixed complexity bound makes them uniformly tempered. Their slacks, list sizes and reciprocal masses are subpower in the original resolution N 0 N_0 N 0 n , i n,i n , i N 0 o ( 1 ) N_0^{o(1)} N 0 o ( 1 )
Let M p a t h ≥ 1 M_{\mathrm{path}} \ge1 M path ≥ 1
M # = 2 ( M p a t h + n ) 2 . (4.13) M^{\#}=2^{(M_{\mathrm{path}}+n)^2}.\tag{4.13} M # = 2 ( M path + n ) 2 . ( 4.13 ) The order of the choices is
finite stage and grid ⟶ outer exact problem and its path ⟶ ( M p a t h , M # ) ⟶ N 0 . (4.14) \text{finite stage and grid}\longrightarrow\text{outer exact problem and its path}\longrightarrow(M_{\mathrm{path}},M^{\#})\longrightarrow N_0.\tag{4.14} finite stage and grid ⟶ outer exact problem and its path ⟶ ( M path , M # ) ⟶ N 0 . ( 4.14 ) Choose N 0 N_0 N 0 M # M^{\#} M # G n G_n G n
G n log N 0 ≤ 1 n , N 0 − 1 K p a t h log N 0 ≤ 1 n , (4.15) \frac{G_n}{\log N_0}\le\frac{1}{n},\qquad N_0^{-1}K_{\mathrm{path}}\log N_0\le\frac{1}{n},\tag{4.15} log N 0 G n ≤ n 1 , N 0 − 1 K path log N 0 ≤ n 1 , ( 4.15 ) log ( ( K p a t h log N 0 ) n ) ϵ n log N 0 ≤ 1 n . (4.16) \frac{\log\left((K_{\mathrm{path}}\log N_0)^n\right)}{\epsilon_n\log N_0}\le\frac{1}{n}.\tag{4.16} ϵ n log N 0 log ( ( K path log N 0 ) n ) ≤ n 1 . ( 4.16 ) Each condition is possible in the fixed exact sequence. All other finite-stage approximation requirements are enforced at the same time. In particular the required original profile and horizon errors are o n ( ϵ n ) o_n(\epsilon_n) o n ( ϵ n ) K = N o ( 1 ) K=N^{o(1)} K = N o ( 1 )
There will be a countable menu of fixed candidate scenarios. A scenario will be identified only after the diagonal and a successful subsequence have been chosen. The implementation below proves that, for every fixed scenario S \mathcal{S} S P S ( M p a t h ) P_\mathcal{S}(M_{\mathrm{path}}) P S ( M path ) P S ( M ) ≤ C S ( 1 + M ) d S P_\mathcal{S}(M)\le C_\mathcal{S}(1+M)^{d_\mathcal{S}} P S ( M ) ≤ C S ( 1 + M ) d S
sup M ≥ 1 { d S log 2 ( 1 + M ) − M 2 } < ∞ (4.17) \sup_{M\ge1}\left\{d_\mathcal{S}\log_2(1+M)-M^2\right\}<\infty\tag{4.17} M ≥ 1 sup { d S log 2 ( 1 + M ) − M 2 } < ∞ ( 4.17 ) implies P S ( M ) ≤ 2 ( M + n ) 2 P_\mathcal{S}(M)\le2^{(M+n)^2} P S ( M ) ≤ 2 ( M + n ) 2 M ≥ 1 M\ge1 M ≥ 1 n n n S \mathcal{S} S S \mathcal{S} S n n n H H H
Preparing the tangent laws
The path geometry gives the scales of the local problems. We now prepare the measures on which the local arguments will find improvements. Two operations have different roles. Density and angular exceptions are removed simultaneously at every potential parent before we choose an entropy window. After that choice the parent probabilities and velocity palettes are fixed; graph preparations and later selections restrict those laws without renormalizing them.
Proposition 4.7 (Prepared tangent data). Assume ℓ < ∞ \ell<\infty ℓ < ∞ o ( 1 ) o(1) o ( 1 ) d = d ∗ d=d_* d = d ∗ N = N 0 ϵ n N=N_0^{\epsilon_n} N = N 0 ϵ n K = N o ( 1 ) K=N^{o(1)} K = N o ( 1 ) μ p r e p = ∑ A p A λ A 0 \mu_{\mathrm{prep}}=\sum_A p_A\lambda_A^0 μ prep = ∑ A p A λ A 0 p A = μ p r e p { A } p_A=\mu_{\mathrm{prep}}\{A\} p A = μ prep { A } λ A 0 \lambda_A^0 λ A 0 A A A λ A 0 ≤ K ν A × d t \lambda_A^0\le K \nu_A\times dt λ A 0 ≤ K ν A × d t ν A \nu_A ν A d t dt d t
At every required bounded tangent depth s s s Q s Q_s Q s a s = N − s a_s = N^{-s} a s = N − s q s ∼ log , N N − k s q_s \sim_{\log,N} N^{-ks} q s ∼ l o g , N N − k s J s ∼ log , N q s j N − ℓ s J_s \sim_{\log,N} q_s^jN^{-\ell s} J s ∼ l o g , N q s j N − ℓ s q s q_s q s ν Q ∼ log , N a s d J s \nu_Q \sim_{\log,N} a_s^dJ_s ν Q ∼ l o g , N a s d J s
For the restricted incidence laws, pure hidden and native-signal bins of width p p p K p d Kp^d K p d Q s Q_s Q s K q s ν Q Kq_s\nu_Q K q s ν Q
Let v = V v=V v = V ℓ = 0 \ell=0 ℓ = 0 0 < ℓ < ∞ 0<\ell<\infty 0 < ℓ < ∞ Z , Y Z,Y Z , Y v = Z ′ v=Z' v = Z ′ π A \pi_A π A λ A 0 \lambda_A^0 λ A 0 π A ( B p ) ≤ K p S \pi_A(B_p)\le Kp^S π A ( B p ) ≤ K p S E E E
∑ A p A I λ A 0 ( [ v ] ; E ) = o ( log N ) . \sum_A p_AI_{\lambda_A^0}([v];E)=o(\log N). A ∑ p A I λ A 0 ([ v ] ; E ) = o ( log N ) . Thus the information bound is averaged with the fixed parent weights.
Required finite-mesh point states and path labels have lists at the original p L p_L p L K K K K K K w w w K a s Ka_s K a s K K K
When 0 < ℓ < ∞ 0<\ell<\infty 0 < ℓ < ∞ N 0 N_0 N 0
Proof.
Coordinates and masks at every potential parent. For now retain all potential parent nodes in the finite grid. If A A A b b b q A q_A q A J A J_A J A ν o l d ( A ) \nu_{\mathrm{old}}(A) ν old ( A )
ν o l d ( A ) ∼ log n o l d ( b ) J A . \nu_{\mathrm{old}}(A)\sim_{\log} n_{\mathrm{old}}(b)J_A. ν old ( A ) ∼ l o g n old ( b ) J A . Normalize its base coordinates as in Lemma 2.13. The parent prior ν A \nu_A ν A [ 0 , 1 ] [0,1] [ 0 , 1 ] B = 1 B=1 B = 1 X = P ∗ X=P_* X = P ∗
At width p = N − r p=N^{-r} p = N − r b + ϵ n r b+\epsilon_n r b + ϵ n r J A / ν o l d ( A ) J_A/\nu_{\mathrm{old}}(A) J A / ν old ( A )
K J A ν o l d ( A ) n o l d ( b + ϵ n r ) ≤ K n o l d ( b + ϵ n r ) n o l d ( b ) ≤ K p d . K\frac{J_A}{\nu_{\mathrm{old}}(A)}n_{\mathrm{old}}(b+\epsilon_n r)
\le K\frac{n_{\mathrm{old}}(b+\epsilon_n r)}{n_{\mathrm{old}}(b)}
\le Kp^d. K ν old ( A ) J A n old ( b + ϵ n r ) ≤ K n old ( b ) n old ( b + ϵ n r ) ≤ K p d . The last inequality uses the relative-profile errors o n ( ϵ n ) o_n(\epsilon_n) o n ( ϵ n ) i i i X X X
Prepare angular ceilings on the same finite collection of parents. Refinement of the original terminal observation by a known parent label has typical density at least n ( L ) / K p a t h n(L)/K_{\mathrm{path}} n ( L ) / K path p p p K p Kp K p K p S Kp^{S} K p S
We also prepare the spatial fullness that will be needed in original units. For 0 < ℓ < ∞ 0<\ell<\infty 0 < ℓ < ∞ N 0 N_0 N 0 D s D_s D s q s q_s q s
ρ s = ∥ D s T e Z ∥ ≥ q s / K p a t h . \rho_s=\lVert D_s^{\mathsf T}e_Z\rVert\ge q_s/K_{\mathrm{path}}. ρ s = ∥ D s T e Z ∥ ≥ q s / K path . Indeed ∥ D s ∥ / q s \lVert D_s\rVert/q_s ∥ D s ∥ / q s
All density exceptions have power-small mass at fixed tolerances in each exact problem. The original order n , i , N 0 n,i,N_0 n , i , N 0 o ( 1 ) o(1) o ( 1 )
The fixed probability law and the entropy window. Normalize the surviving path once and call its probability law μ p r e p \mu_{\mathrm{prep}} μ prep A A A s A s_A s A ν A × d t \nu_A\times dt ν A × d t s A > 0 s_A>0 s A > 0
λ A 0 = ( ν A × d t ) ∣ p r e p a r e d p a t h i n A s A , p A = μ p r e p { A } . \lambda_A^0=\frac{(\nu_A\times dt)|_{\mathrm{prepared\ path\ in\ }A}}{s_A},
\qquad
p_A=\mu_{\mathrm{prep}}\{A\}. λ A 0 = s A ( ν A × d t ) ∣ prepared path in A , p A = μ prep { A } . At any one parent node, μ p r e p = ∑ A p A λ A 0 \mu_{\mathrm{prep}}=\sum_A p_A\lambda_A^0 μ prep = ∑ A p A λ A 0 p A p_A p A q A ν o l d ( A ) s A q_A\nu_{\mathrm{old}}(A)s_A q A ν old ( A ) s A
Parents with too small a success fraction, or too small a surviving denominator relative to their pre-mask mass, have negligible total weight: sum the former costs ∑ A ω o l d q A ν o l d ( A ) ≤ 1 \sum_A\omega_{\mathrm{old}}q_A\nu_{\mathrm{old}}(A)\le1 ∑ A ω old q A ν old ( A ) ≤ 1
λ A 0 ≤ K ν A × d t , π A ( B p ) ≤ K p S . (4.18) \lambda_A^0\le K\nu_A\times dt,
\qquad
\pi_A(B_p)\le Kp^S.\tag{4.18} λ A 0 ≤ K ν A × d t , π A ( B p ) ≤ K p S . ( 4.18 ) where π A \pi_A π A λ A 0 \lambda_A^0 λ A 0 s A s_A s A K K K
Choose the window using the single law μ p r e p \mu_{\mathrm{prep}} μ prep V ∗ V_\ast V ∗ ϵ n U n \epsilon_nU_n ϵ n U n
I ( V ∗ ; history through the window end ∣ history through its start ) I(V_\ast;\text{history through the window end}\mid\text{history through its start}) I ( V ∗ ; history through the window end ∣ history through its start ) are nonnegative and sum to at most H ( V ∗ ) H(V_\ast) H ( V ∗ ) H ( V ∗ ) ≤ O ( U n log N ) + o ( log N ) H(V_\ast)\le O(U_n\log N)+o(\log N) H ( V ∗ ) ≤ O ( U n log N ) + o ( log N ) ≳ ( D n ϵ n ) − 1 \gtrsim(D_n\epsilon_n)^{-1} ≳ ( D n ϵ n ) − 1 o ( log N ) o(\log N) o ( log N ) U n D n ϵ n → 0 U_nD_n\epsilon_n\to0 U n D n ϵ n → 0
Use the normalized coordinates of this window’s parent. In version 1 and for ℓ = 0 \ell=0 ℓ = 0 v = V v=V v = V v = Z ′ v=Z' v = Z ′ v v v K K K V ∗ V_* V ∗ log K = o ( log N ) \log K=o(\log N) log K = o ( log N ) ℓ = 0 \ell=0 ℓ = 0 K K K N − U n N^{-U_n} N − U n
Fix p A p_A p A λ A 0 \lambda_A^0 λ A 0 π A \pi_A π A
Path scales and the two time conventions. For a descendant Q = Q s Q=Q_s Q = Q s a = N − s a=N^{-s} a = N − s h = N − k s h=N^{-ks} h = N − k s g = N − ℓ s g=N^{-\ell s} g = N − ℓ s
a s = N − s , q s ∼ log , N h , J s ∼ log , N h j g , ν Q ∼ log , N a d J s . (4.19) a_s=N^{-s},\qquad q_s\sim_{\log,N}h,\qquad J_s\sim_{\log,N}h^j g,\qquad\nu_Q\sim_{\log,N}a^dJ_s.\tag{4.19} a s = N − s , q s ∼ l o g , N h , J s ∼ l o g , N h j g , ν Q ∼ l o g , N a d J s . ( 4.19 ) In version 1 omit g g g P s P_s P s a Q a_Q a Q a a a
At an exact base, the true value and incidence-derivative bounds permit only K K K a Q a_Q a Q u u u Q s Q_s Q s q s J s q_sJ_s q s J s base Q \operatorname{base}_Q base Q
d ( base Q ) ∗ ( λ A 0 ∣ Q ) ( u ) ≤ K a Q d q s J s d u ≤ K q s ν Q d u . (4.20) \mathrm{d}(\operatorname{base}_Q)_*(\lambda_A^0|_Q)(u)\le K a_Q^d q_sJ_s\,\mathrm{d}u\le Kq_s\nu_Q\,\mathrm{d}u.\tag{4.20} d ( base Q ) ∗ ( λ A 0 ∣ Q ) ( u ) ≤ K a Q d q s J s d u ≤ K q s ν Q d u . ( 4.20 ) This is a cap in parent-time mass. Dividing by the block length q s q_s q s K ν Q K\nu_Q K ν Q Q s Q_s Q s Q Q Q K p d / ν Q Kp^d/\nu_Q K p d / ν Q Q Q Q K p d J s / ν Q Kp^dJ_s/\nu_Q K p d J s / ν Q p p p
∑ Q i n o n e b l o c k ν Q ≤ 1 , ∑ Q a t o n e n o d e q s ν Q ≤ 1. \sum_{Q\ \mathrm{in\ one\ block}}\nu_Q\le1,\qquad
\sum_{Q\ \mathrm{at\ one\ node}}q_s\nu_Q\le1. Q in one block ∑ ν Q ≤ 1 , Q at one node ∑ q s ν Q ≤ 1. They allow light labels to be removed whenever current incidence floors are needed, with thresholds chosen below the current selected mass. Spatial enlargements by subpower factors only change K K K
For positive finite narrowness, subtract a multiple of the Z Z Z D s D_s D s Y Y Y ∥ D s T e Y ∥ / ρ s \lVert D_s^T e_Y\rVert/\rho_s ∥ D s T e Y ∥ / ρ s ∣ det D s ∣ / ρ s |\det D_s|/\rho_s ∣ det D s ∣/ ρ s h h h h g hg h g
Y ′ = A s ⋅ ( 1 , v ) + O ( K g ) . Y'=A_s\cdot(1,v)+O(Kg). Y ′ = A s ⋅ ( 1 , v ) + O ( K g ) . The row A s A_s A s Q s Q_s Q s J s / q s 2 J_s/q_s^2 J s / q s 2
Stable graphs and point-state lists. Later arguments need velocity bounds after recording finite-mesh point states. We first construct the finite list of possible states; its size will be the input to the appended-entry floor below. Use normalized base coordinates u u u Q s Q_s Q s w w w B = 1 B = 1 B = 1 K g = N o ( 1 ) K_g = N^{o(1)} K g = N o ( 1 )
∣ P s ∣ ≤ K g a Q , K g − 1 ≤ ∣ P s , w ∣ ≤ K g , ∣ P s , w w ∣ ≤ K g / a Q |P_s| \le K_g a_Q,\qquad K_g^{-1} \le|P_{s,w}| \le K_g,\qquad|P_{s,ww}| \le K_g/a_Q ∣ P s ∣ ≤ K g a Q , K g − 1 ≤ ∣ P s , w ∣ ≤ K g , ∣ P s , w w ∣ ≤ K g / a Q on the counted incidences. First omit labels whose current mass is too small relative to q s ν Q q_s\nu_Q q s ν Q (4.20) then gives a base projection of inverse-subpower volume in every used label.
The polynomial discriminant
D ( u ) = P s , w 2 − 2 P s , w w P s \mathcal{D}(u) = P_{s,w}^{2} - 2P_{s,ww}P_s D ( u ) = P s , w 2 − 2 P s , w w P s is independent of w w w K K K D \mathcal{D} D K K K H H H H / K g 3 → ∞ H/K_g^3 \to\infty H / K g 3 → ∞ ∣ P s , w w ∣ = ( a Q H ) − 1 |P_{s,ww}| = (a_QH)^{-1} ∣ P s , w w ∣ = ( a Q H ) − 1
∣ w − F c t r ( u ) ∣ = ∣ P s , w ( u , w ) ∣ ∣ P s , w w ∣ ≤ K g H a Q ≤ K a . |w - F_{\mathrm{ctr}}(u)| = \frac{|P_{s,w}(u,w)|}{|P_{s,ww}|} \le K_gHa_Q \le Ka. ∣ w − F ctr ( u ) ∣ = ∣ P s , w w ∣ ∣ P s , w ( u , w ) ∣ ≤ K g H a Q ≤ K a . Below the cutoff,
2 ∣ P s , w w P s ∣ ≤ 2 K g H = o ( K g − 2 ) = o ( ∣ P s , w ∣ 2 ) , D ≥ 1 2 K g − 2 (4.21) 2|P_{s,ww}P_s| \le\frac{2K_g}{H} = o(K_g^{-2}) = o(|P_{s,w}|^2),\qquad\mathcal{D} \ge\frac{1}{2}K_g^{-2}\tag{4.21} 2∣ P s , w w P s ∣ ≤ H 2 K g = o ( K g − 2 ) = o ( ∣ P s , w ∣ 2 ) , D ≥ 2 1 K g − 2 ( 4.21 ) at those incidences. Partition the normalized box into core subboxes of inverse-subpower radius, each with a fixed-factor complex enlargement and a further margin. Choose the radius so that the variation of D \mathcal{D} D w w w K a Ka K a (3.24) . For a linear equation the same argument keeps its coefficient nonzero.
There are K K K q s ν Q q_s\nu_Q q s ν Q (4.20) supplies inverse-subpower real base volume for each retained entry. Compose its graph with the native test P ∗ P_* P ∗ K a Ka K a X X X ∣ X ∣ ≤ K |X| \le K ∣ X ∣ ≤ K K K K
An end history refining Q s Q_s Q s K q e n d / q s Kq_{\mathrm{end}}/q_s K q end / q s ( t , y , X ) (t,y,X) ( t , y , X ) K q e n d / q s Kq_{\mathrm{end}}/q_s K q end / q s K K K
These preparations use arbitrary fixed mesh exponents first, with sufficient window depth, and then slowly increasing grids. They may be made after the entropy-window choice with negligible total loss, without redefining λ A 0 \lambda_A^0 λ A 0
Original observation lists and full-layer witnesses. Finite-mesh point states in ( t , y , X ) (t,y,X) ( t , y , X ) p L p_L p L X X X K K K
For stored full-layer incidence witnesses, impose the same original pure-cap masks on the fixed density grids needed in the coordinate transfers. Their costs are power-small at every fixed tolerance in the exact problem. If a layer label loses at least half its incidence mass, saturation bounds its old block length times its full trajectory mass by an actual path factor times the lost incidence mass. Sum with the old world weights. The cost of dropping all such labels from the final path is therefore negligible when N 0 N_0 N 0
All these preparations use the already fixed diagonal. Their initial cap masks retain 1 − o ( 1 ) 1-o(1) 1 − o ( 1 )
Conditioning finite states and taking time slices
The prepared law has little information about the velocity in its end history. The next lemma converts that information bound to a pointwise joint ceiling. It also records how that ceiling passes to a point state and how much mass must be deleted before it becomes a conditional ceiling under a later restriction.
Lemma 4.8 (Likelihood truncation and finite-state conditioning). As N → ∞ N \to\infty N → ∞ p A ≥ 0 p_A \ge0 p A ≥ 0 ∑ A p A = 1 \sum_A p_A = 1 ∑ A p A = 1 λ A 0 \lambda_A^0 λ A 0 E , b E,b E , b
R A 0 ( E , b ) = λ A 0 { E , b } , λ A , p r e ( E ) = ∑ b R A 0 ( E , b ) , π A ( b ) = ∑ E R A 0 ( E , b ) . R_A^0(E,b)=\lambda_A^0\{E,b\},\qquad\lambda_{A,\mathrm{pre}}(E)=\sum_b R_A^0(E,b),\qquad\pi_A(b)=\sum_E R_A^0(E,b). R A 0 ( E , b ) = λ A 0 { E , b } , λ A , pre ( E ) = b ∑ R A 0 ( E , b ) , π A ( b ) = E ∑ R A 0 ( E , b ) . Suppose
∑ A p A ∑ E , b R A 0 ( E , b ) log R A 0 ( E , b ) λ A , p r e ( E ) π A ( b ) = o ( log N ) . \sum_A p_A\sum_{E,b}R_A^0(E,b)\log\frac{R_A^0(E,b)}{\lambda_{A,\mathrm{pre}}(E)\pi_A(b)}=o(\log N). A ∑ p A E , b ∑ R A 0 ( E , b ) log λ A , pre ( E ) π A ( b ) R A 0 ( E , b ) = o ( log N ) . Then deleting o ( 1 ) o(1) o ( 1 )
R A , g o o d ( E , b ) ≤ K λ A , p r e ( E ) π A ( b ) , K = N o ( 1 ) . (4.22) R_{A,\mathrm{good}}(E,b)\le K\lambda_{A,\mathrm{pre}}(E)\pi_A(b),\qquad K=N^{o(1)}.\tag{4.22} R A , good ( E , b ) ≤ K λ A , pre ( E ) π A ( b ) , K = N o ( 1 ) . ( 4.22 ) Let R A , 2 R_{A,2} R A , 2 d ′ d' d ′ D ≤ K D\le K D ≤ K E E E ( E , b ) (E,b) ( E , b ) R A , g o o d R_{A,\mathrm{good}} R A , good K 1 K_1 K 1 D / K 1 D/K_1 D / K 1 ( E , d ′ ) (E,d') ( E , d ′ ) K K 1 KK_1 K K 1 C = C ( E , d ′ ) C=C(E,d') C = C ( E , d ′ ) C C C w A ( C ) w_A(C) w A ( C ) T A T_A T A w A ( C ) / K 2 w_A(C)/K_2 w A ( C ) / K 2 1 / K 2 1/K_2 1/ K 2
T A ( b ∣ C ) ≤ K K 1 K 2 π A ( b ) . T_A(b\mid C)\le KK_1K_2\pi_A(b). T A ( b ∣ C ) ≤ K K 1 K 2 π A ( b ) . The subpowers K 1 , K 2 K_1,K_2 K 1 , K 2
Proof. Put R A , p r o d ( E , b ) = λ A , p r e ( E ) π A ( b ) R_{A,\mathrm{prod}}(E,b)=\lambda_{A,\mathrm{pre}}(E)\pi_A(b) R A , prod ( E , b ) = λ A , pre ( E ) π A ( b )
∑ R A 0 < R A , p r o d R A 0 log ( R A , p r o d / R A 0 ) ≤ 1 \sum_{R_A^0<R_{A,\mathrm{prod}}}R_A^0\log\left(R_{A,\mathrm{prod}}/R_A^0\right)\le1 R A 0 < R A , prod ∑ R A 0 log ( R A , prod / R A 0 ) ≤ 1 by x log ( 1 / x ) ≤ 1 / e x\log(1/x)\le1/e x log ( 1/ x ) ≤ 1/ e o ( log N ) o(\log N) o ( log N ) δ N ↓ 0 \delta_N\downarrow0 δ N ↓ 0 o ( δ N log N ) o(\delta_N\log N) o ( δ N log N ) δ N log N \delta_N\log N δ N log N o ( 1 ) o(1) o ( 1 )
R A , g o o d ( E , b ) ≤ N δ N λ A , p r e ( E ) π A ( b ) ≤ K λ A , p r e ( E ) π A ( b ) . (4.23) R_{A,\mathrm{good}}(E,b)\le N^{\delta_N}\lambda_{A,\mathrm{pre}}(E)\pi_A(b)\le K\lambda_{A,\mathrm{pre}}(E)\pi_A(b).\tag{4.23} R A , good ( E , b ) ≤ N δ N λ A , pre ( E ) π A ( b ) ≤ K λ A , pre ( E ) π A ( b ) . ( 4.23 ) The inequality holds in each retained parent; the small-loss assertion is only for their weighted mixture. Any further restrictions of this numerator leave its reference denominator unchanged. Let R A , 2 R_{A,2} R A , 2 d ′ d' d ′ D ≤ K D\le K D ≤ K ( E , b ) (E,b) ( E , b ) R A , g o o d R_{A,\mathrm{good}} R A , good
m A , p r e o m i t ( E , d ′ ) = ∑ b R A , 2 ( E , d ′ , b ) . m_{A,\mathrm{preomit}}(E,d')=\sum_b R_{A,2}(E,d',b). m A , preomit ( E , d ′ ) = b ∑ R A , 2 ( E , d ′ , b ) . Keep only entries satisfying the floor
m A , p r e o m i t ( E , d ′ ) ≥ λ A , p r e ( E ) / K 1 . (4.24) m_{A,\mathrm{preomit}}(E,d')\ge\lambda_{A,\mathrm{pre}}(E)/K_1.\tag{4.24} m A , preomit ( E , d ′ ) ≥ λ A , pre ( E ) / K 1 . ( 4.24 ) The omitted mass is at most D K 1 − 1 ∑ E λ A , p r e ( E ) = D / K 1 DK_1^{-1}\sum_E\lambda_{A,\mathrm{pre}}(E)=D/K_1 D K 1 − 1 ∑ E λ A , pre ( E ) = D / K 1 K 1 K_1 K 1 R A , 3 R_{A,3} R A , 3
R A , 3 ( E , d ′ , b ) ≤ R A , g o o d ( E , b ) ≤ K π A ( b ) λ A , p r e ( E ) ≤ K K 1 π A ( b ) m A , p r e o m i t ( E , d ′ ) . (4.25) \begin{aligned}
R_{A,3}(E,d',b)&\le R_{A,\mathrm{good}}(E,b)\le K\pi_A(b)\lambda_{A,\mathrm{pre}}(E)\\
&\le KK_1\pi_A(b)m_{A,\mathrm{preomit}}(E,d').
\end{aligned}\tag{4.25} R A , 3 ( E , d ′ , b ) ≤ R A , good ( E , b ) ≤ K π A ( b ) λ A , pre ( E ) ≤ K K 1 π A ( b ) m A , preomit ( E , d ′ ) . ( 4.25 ) Since whole entries were omitted, their retained conditional palette probabilities are bounded by K K 1 π A ( b ) KK_1\pi_A(b) K K 1 π A ( b )
For a coarser state C = C ( E , d ′ ) C=C(E,d') C = C ( E , d ′ ) (4.25) with the fixed reference weights
w A ( C ) = ∑ ( E , d ′ ) : C ( E , d ′ ) = C m A , p r e o m i t ( E , d ′ ) . w_A(C)=\sum_{(E,d'):C(E,d')=C}m_{A,\mathrm{preomit}}(E,d'). w A ( C ) = ( E , d ′ ) : C ( E , d ′ ) = C ∑ m A , preomit ( E , d ′ ) . Then R A , 3 ( C , b ) ≤ K K 1 π A ( b ) w A ( C ) R_{A,3}(C,b)\le KK_1\pi_A(b)w_A(C) R A , 3 ( C , b ) ≤ K K 1 π A ( b ) w A ( C )
A further sparse restriction T A ≤ R A , 3 T_A\le R_{A,3} T A ≤ R A , 3 T A T_A T A K 2 K_2 K 2
T A ( C ) < w A ( C ) / K 2 . T_A(C)<w_A(C)/K_2. T A ( C ) < w A ( C ) / K 2 . Their total mass is at most K 2 − 1 ∑ C w A ( C ) K_2^{-1}\sum_C w_A(C) K 2 − 1 ∑ C w A ( C ) K 2 − 1 K_2^{-1} K 2 − 1 T A ( b ∣ C ) ≤ K K 1 K 2 π A ( b ) T_A(b\mid C)\le KK_1K_2\pi_A(b) T A ( b ∣ C ) ≤ K K 1 K 2 π A ( b )
Apply the lemma with E E E b = [ v ] b=[v] b = [ v ] λ A 0 \lambda_A^0 λ A 0 K K K d ′ d' d ′ ( t , y , X ) (t,y,X) ( t , y , X ) w A ( C ) w_A(C) w A ( C ) v v v
The resulting bounds are integrated in time. The following elementary estimate gives fixed-time bounds for the trajectory prior while keeping that prior unchanged.
Lemma 4.9 (Retained duration and fixed-time prior caps). Let ν \nu ν d λ ( l , t ) = G ( l , t ) d ν ( l ) d t d\lambda(l,t)=G(l,t)\,d\nu(l)\,dt d λ ( l , t ) = G ( l , t ) d ν ( l ) d t 0 ≤ G ≤ K 0\leq G\leq K 0 ≤ G ≤ K [ 0 , 1 ] [0,1] [ 0 , 1 ] K 2 ≥ 1 K_2\geq1 K 2 ≥ 1 I I I
L I = { l : ∫ I G ( l , τ ) d τ ≥ ∣ I ∣ / K 2 } . L_I=\left\{l:\int_I G(l,\tau)\,d\tau\geq|I|/K_2\right\}. L I = { l : ∫ I G ( l , τ ) d τ ≥ ∣ I ∣/ K 2 } . Deleting the complementary line–interval pairs costs at most 1 / K 2 1/K_2 1/ K 2 t ∈ I t\in I t ∈ I C t C_t C t C τ + C_\tau^+ C τ + C t ( l ) ⇒ C τ + ( l ) C_t(l)\Rightarrow C_\tau^+(l) C t ( l ) ⇒ C τ + ( l ) τ ∈ I \tau\in I τ ∈ I
ν ( L I ∩ C t ) ≤ K 2 ∣ I ∣ ∫ I ∫ 1 C τ + ( l ) G ( l , τ ) d ν ( l ) d τ . (4.26) \nu(L_I\cap C_t)\leq\frac{K_2}{|I|}\int_I\int\mathbf{1}_{C_\tau^+}(l)G(l,\tau)\,d\nu(l)\,d\tau.\tag{4.26} ν ( L I ∩ C t ) ≤ ∣ I ∣ K 2 ∫ I ∫ 1 C τ + ( l ) G ( l , τ ) d ν ( l ) d τ . ( 4.26 ) For almost every t t t C t C_t C t K ν ( L I ∩ C t ) K\nu(L_I\cap C_t) K ν ( L I ∩ C t )
Proof. On a deleted line–interval pair the incidence duration is less than ∣ I ∣ / K 2 |I|/K_2 ∣ I ∣/ K 2 ν \nu ν ∑ I ∣ I ∣ = 1 \sum_I |I|=1 ∑ I ∣ I ∣ = 1 L I ∩ C t L_I\cap C_t L I ∩ C t
∫ I 1 C τ + ( l ) G ( l , τ ) d τ ≥ ∣ I ∣ / K 2 . \int_I \mathbf{1}_{C_\tau^+}(l)G(l,\tau)\,d\tau\geq|I|/K_2. ∫ I 1 C τ + ( l ) G ( l , τ ) d τ ≥ ∣ I ∣/ K 2 . Integration over these trajectories proves the displayed estimate. The last assertion follows from G ≤ K G\leq K G ≤ K □ \square □
In the prepared tangent problem, choose the time partition much finer than the finite collection of spatial meshes being used. Bounded motion of ( t , y , X ) (t,y,X) ( t , y , X ) C t ⇒ C τ + C_t\Rightarrow C_\tau^+ C t ⇒ C τ + (4.26) . This step requires no time regularity of w w w
After a selection, small conditional successes are omitted using the fixed mixture weights p A p_A p A K 2 K_2 K 2 log N 0 \log N_0 log N 0
From local candidates to actual improvements
The later geometric arguments will find a single improved fit inside any substantial residual of the reference path. Such a fit need not carry enough original mass, and its label need not yet be known from the original observation. We first specify the local fit and then prove a conversion criterion that supplies both properties.
Definition 4.10 (Improvement candidates). Work in a prepared tangent parent and write d = d ∗ d=d_* d = d ∗ Q = Q s Q=Q_s Q = Q s q s q_s q s a = N − s a=N^{-s} a = N − s ν Q \nu_Q ν Q s s s q s q_s q s
A time-improvement candidate of gain c > 0 c>0 c > 0 a ′ = a N − c a'=aN^{-c} a ′ = a N − c Q Q Q Q Q Q
∣ P ∣ ≤ K a ′ , ∣ P w ∣ ≤ K , ∣ P w w ∣ ≤ K / a ′ . |P|\leq Ka',\qquad|P_w|\leq K,\qquad|P_{ww}|\leq K/a'. ∣ P ∣ ≤ K a ′ , ∣ P w ∣ ≤ K , ∣ P w w ∣ ≤ K / a ′ . On counted incidences it also satisfies ∣ P w ∣ ≥ K − 1 |P_w|\geq K^{-1} ∣ P w ∣ ≥ K − 1
K − 1 N − d c ν Q . K^{-1}N^{-dc}\nu_Q. K − 1 N − d c ν Q . When 0 < ℓ < ∞ 0<\ell<\infty 0 < ℓ < ∞ g = N − ℓ s g=N^{-\ell s} g = N − ℓ s strip candidate of gain c > 0 c>0 c > 0 R R R
∣ Y ′ − R ⋅ ( 1 , v ) ∣ ≤ K g N − c |Y'-R\cdot(1,v)|\leq KgN^{-c} ∣ Y ′ − R ⋅ ( 1 , v ) ∣ ≤ K g N − c on trajectory pieces carrying incidence mass at least K − 1 ν Q K^{-1}\nu_Q K − 1 ν Q
These mass tests may use restrictions of λ A 0 \lambda_A^0 λ A 0 s A − 1 ≤ K s_A^{-1}\leq K s A − 1 ≤ K
For the whole-block upper estimates, extrapolating a polynomial along a trajectory from relative duration K − 1 K^{-1} K − 1 K K K
A time candidate improves the hidden-coordinate fit without changing the packet’s time block; a strip candidate improves its transverse velocity prediction. The mass in the definition is relative to the packet’s full trajectory assignment. The next proposition converts candidates in every substantial residual into one original known atlas.
Proposition 4.11 (Conversion of candidates to actual improvements). Consider the guarded tangent diagonal of Proposition 4.7. Suppose that, on a further subsequence of every subsequence carrying any substantial residual of the reference path, that residual supplies a time-improvement or strip candidate in the sense of Definition 4.10. Its depth remains in a bounded tangent range and its gain has a fixed positive finite limit, or is a fixed positive number.
Then one of the original-resolution outputs excluded by Lemma 4.6 exists. Its original success is at least p p a t h / log N 0 \mathfrak{p}_{\mathrm{path}}/\log N_0 p path / log N 0 K p a t h log N 0 K_{\mathrm{path}}\log N_0 K path log N 0 M p a t h M_{\mathrm{path}} M path
The proof has four steps. A finite greedy assignment first obtains substantial coverage by one fixed candidate scenario. Strip coverage is converted directly into narrower spatial charts. For time coverage, local graph comparison groups the candidate tests into short lists and then supplies the improved chart tests. Finally, the following recovery lemma transfers those lists from the analytical submeasure to the original law. The lemma is stated separately because it is also used in the unbounded-narrowness argument.
Lemma 4.12 (Original-law recovery of finite lists). Let μ \mu μ original unnormalized tempered path measure, and let ν ≤ μ \nu\le\mu ν ≤ μ A A A ν \nu ν O = ( x , o ) O=(x,o) O = ( x , o ) x = ( t , y ) x=(t,y) x = ( t , y ) o o o
Suppose a measurable rule L ( x , o ) L(x,o) L ( x , o ) ∣ L ( x , o ) ∣ ≤ D l i s t |L(x,o)| \le D_{\mathrm{list}} ∣ L ( x , o ) ∣ ≤ D list
ν { A ∈ L ( O ) } ≥ m . (4.27) \nu\{A \in L(O)\} \ge m.\tag{4.27} ν { A ∈ L ( O )} ≥ m . ( 4.27 ) Retain all original observation tags, path tags and output labels in the joint pushforward law λ \lambda λ μ \mu μ F ρ λ F_{\rho}\lambda F ρ λ ρ \rho ρ ∥ λ − F ρ λ ∥ T V ≤ e \|\lambda-F_{\rho}\lambda\|_{\mathrm{TV}} \le e ∥ λ − F ρ λ ∥ TV ≤ e L C ( o ) L_C(o) L C ( o )
μ { A ∈ L C ( x ) ( o ) } ≥ m − 2 e . (4.28) \mu\{A \in L_{C(x)}(o)\} \ge m-2e.\tag{4.28} μ { A ∈ L C ( x ) ( o )} ≥ m − 2 e . ( 4.28 ) If the density, ranges, copies and geometric predicates of this original joint law have complexity bounded by a fixed polynomial in M p a t h M_{\mathrm{path}} M path e = O ( N 0 − 1 ) e=O(N_0^{-1}) e = O ( N 0 − 1 ) ν \nu ν L L L
Proof. For the fixed geometric map A A A
μ { A ∈ L ( O ) } ≥ ν { A ∈ L ( O ) } ≥ m . (4.29) \mu\{A \in L(O)\} \ge\nu\{A \in L(O)\} \ge m.\tag{4.29} μ { A ∈ L ( O )} ≥ ν { A ∈ L ( O )} ≥ m . ( 4.29 ) Use total variation in the convention of supremum over measurable events. In each base cell C C C o o o D l i s t D_{\mathrm{list}} D list λ ( C , o , ⋅ ) \lambda(C,o,\cdot) λ ( C , o , ⋅ ) F ρ λ F_{\rho}\lambda F ρ λ C C C E L E_L E L
λ ( E L C ) ≥ ( F ρ λ ) ( E L C ) − e ≥ ( F ρ λ ) ( E L ) − e ≥ λ ( E L ) − 2 e ≥ m − 2 e . \begin{aligned}
\lambda(E_{L_C}) &\ge(F_{\rho}\lambda)(E_{L_C})-e \\
&\ge(F_{\rho}\lambda)(E_L)-e \\
&\ge\lambda(E_L)-2e \ge m-2e.
\end{aligned} λ ( E L C ) ≥ ( F ρ λ ) ( E L C ) − e ≥ ( F ρ λ ) ( E L ) − e ≥ λ ( E L ) − 2 e ≥ m − 2 e . All auxiliary tags remain joint data in this calculation. The maximization conditions only on C , o C,o C , o C C C
Here is the quantitative total-variation bound, including the effect of marginalizing hidden trajectory coefficients. In original coefficient coordinates replace the spatial intercept by y ( t ) y(t) y ( t ) ( x , ξ ) (x,\xi) ( x , ξ ) x = ( t , y ( t ) ) x=(t,y(t)) x = ( t , y ( t )) ξ \xi ξ V V V ξ \xi ξ
The original tempered density is piecewise constant on its defining semialgebraic pieces. Fix the other variables and translate one coordinate of x x x h h h D D D D D D H H H T T T D D D R R R J J J
∥ τ h f − f ∥ L 1 ≤ C H ( T D + 1 ) J ( 1 + R ) q ∣ h ∣ , (4.30) \lVert\tau_h f-f\rVert_{L^1}\leq CH(TD+1)J(1+R)^q|h|,\tag{4.30} ∥ τ h f − f ∥ L 1 ≤ C H ( T D + 1 ) J ( 1 + R ) q ∣ h ∣ , ( 4.30 ) where q q q ξ \xi ξ L 1 L^1 L 1
∥ λ − F ρ λ ∥ T V ≤ N 0 B S ( M p a t h ) ρ , (4.31) \lVert\lambda-F_\rho\lambda\rVert_{\mathrm{TV}}\leq N_0^{B_\mathcal{S}(M_{\mathrm{path}})}\rho,\tag{4.31} ∥ λ − F ρ λ ∥ TV ≤ N 0 B S ( M path ) ρ , ( 4.31 ) after absorbing fixed constants. Here B S B_\mathcal{S} B S S \mathcal{S} S
Choose ρ = N 0 − D m e s h \rho=N_0^{-D_{\mathrm{mesh}}} ρ = N 0 − D mesh D m e s h ≥ B S ( M p a t h ) + 2 D_{\mathrm{mesh}}\geq B_\mathcal{S}(M_{\mathrm{path}})+2 D mesh ≥ B S ( M path ) + 2 O ( N 0 − 1 ) O(N_0^{-1}) O ( N 0 − 1 ) N 0 P 0 ( M p a t h ) N_0^{P_0(M_{\mathrm{path}})} N 0 P 0 ( M path )
N 0 ( j + 1 ) ( D m e s h + P 0 ( M p a t h ) ) + O ( 1 ) . (4.32) N_0^{(j+1)(D_{\mathrm{mesh}}+P_0(M_{\mathrm{path}}))+O(1)}.\tag{4.32} N 0 ( j + 1 ) ( D mesh + P 0 ( M path )) + O ( 1 ) . ( 4.32 ) The number of observed bin entries, worlds and possible labels is also polynomial. Multiplying these counts gives the asserted polynomial table exponent. Arbitrary changes of a maximizing posterior only change the contents of this table. No posterior derivative or reciprocal probability denominator occurs.
Proof of Proposition 4.11. Candidate existence means existence of a parent, node, chart and test; the auxiliary construction proving it may use much sparser configurations. Only the residual is used to certify the candidate’s mass floor.
Use the preparations of Proposition 4.7 and Lemma 4.8, retaining 1 − o ( 1 ) 1-o(1) 1 − o ( 1 ) π A \pi_A π A K K K p A p_A p A λ A 0 \lambda_A^0 λ A 0
Coverage by one fixed candidate scenario. First fix a gain c > 0 c>0 c > 0 N σ N^\sigma N σ σ > 0 \sigma>0 σ > 0 Q Q Q
Assign to each chosen test all still available Q Q Q
Disjointness makes this priority assignment stop after at most N d c + σ N^{dc+\sigma} N d c + σ Q Q Q N σ N^\sigma N σ K K K
Choose a countable menu of scenarios. Each fixes a type, a bounded tangent-depth range, a positive gain c c c σ \sigma σ c n → c ∗ > 0 c_n\to c_*>0 c n → c ∗ > 0 c ∗ c_* c ∗ σ \sigma σ σ / 4 \sigma/4 σ /4 N − d c N^{-dc} N − d c
For each available scenario and node, make the finite greedy assignment maximal. The selections for different scenarios are made separately on the same prepared path. Suppose that the union of every fixed finite menu has reference coverage tending to zero. Choose J n → ∞ J_n \to\infty J n → ∞ J n J_n J n all path incidences lying on their assigned trajectory pieces throughout the respective blocks. The remaining path has substantial mass. The candidate hypothesis gives, on a further subsequence, a candidate of positive limiting gain in a bounded depth range. The preceding menu choice supplies a fixed compatible scenario, eventually among the first J n J_n J n
Hence a fixed finite menu, of size J J J η 0 > 0 \eta_0 > 0 η 0 > 0 G n G_n G n
β n ≥ η 0 J G n . (4.33) \beta_n \ge\frac{\eta_0}{J G_n}.\tag{4.33} β n ≥ J G n η 0 . ( 4.33 ) The node can vary with the sample; its depth remains in the fixed scenario’s bounded range. By (4.15) , β n log N 0 → ∞ \beta_n \log N_0 \to\infty β n log N 0 → ∞ β n − 1 = N o ( 1 ) \beta_n^{-1} = N^{o(1)} β n − 1 = N o ( 1 )
The following comparisons lose only negligible fractions of this coverage or bounded factors. In the time case retain the derivative lower bound N − 2 σ N^{-2\sigma} N − 2 σ N − σ N^{-\sigma} N − σ O ( N − σ ) O(N^{-\sigma}) O ( N − σ ) (4.18) bound their total mass by a fixed negative power, hence by o ( β n ) o(\beta_n) o ( β n )
Conversion of strip coverage. Take a true label deeper within the same parent, at a fixed target depth large enough that its velocity-strip thickness is ≪ g N − c \ll gN^{-c} ≪ g N − c ℓ > 0 \ell> 0 ℓ > 0 A d e e p A_{\mathrm{deep}} A deep D = R − A d e e p = ( D 0 , D 1 ) D = R - A_{\mathrm{deep}} = (D_0,D_1) D = R − A deep = ( D 0 , D 1 )
∣ D 0 + D 1 v ∣ ≤ C g N − c + σ . \lvert D_0 + D_1 v\rvert\le CgN^{-c+\sigma}. ∣ D 0 + D 1 v ∣ ≤ C g N − c + σ . If ∥ D ∥ > g N − c / 2 \lVert D\rVert> gN^{-c/2} ∥ D ∥ > g N − c /2 ∣ v ∣ ≤ K \lvert v\rvert\le K ∣ v ∣ ≤ K ∣ D 1 ∣ ≥ g N − c / 2 / K \lvert D_1\rvert\ge gN^{-c/2}/K ∣ D 1 ∣ ≥ g N − c /2 / K K K K
2 C g N − c + σ ∣ D 1 ∣ ≤ K N − c / 2 + σ ≤ N − c / 3 . (4.34) \frac{2CgN^{-c+\sigma}}{\lvert D_1\rvert} \le KN^{-c/2+\sigma} \le N^{-c/3}.\tag{4.34} ∣ D 1 ∣ 2 C g N − c + σ ≤ K N − c /2 + σ ≤ N − c /3 . ( 4.34 ) For each deep history E E E Q s Q_s Q s N σ N^\sigma N σ (4.23) and the angular ceiling therefore charge all these misaligned rows by at most
K N σ − S c / 3 λ A , p r e ( E ) . KN^{\sigma-Sc/3}\lambda_{A,\mathrm{pre}}(E). K N σ − S c /3 λ A , pre ( E ) . Use fine velocity bins and bounded enlargements to cover the intervals. Summing over histories, labels, and parents with weights p A p_A p A K N σ − S c / 3 KN^{\sigma-Sc/3} K N σ − S c /3 σ < min ( c / 12 , S c / 12 ) \sigma< \min(c/12,Sc/12) σ < min ( c /12 , S c /12 ) o ( β n ) o(\beta_n) o ( β n ) β n = N − o ( 1 ) \beta_n = N^{-o(1)} β n = N − o ( 1 )
∥ R − A d e e p ∥ ≤ g N − c / 2 . (4.35) \lVert R-A_{\mathrm{deep}}\rVert\le gN^{-c/2}.\tag{4.35} ∥ R − A deep ∥ ≤ g N − c /2 . ( 4.35 ) The unbinned rows stored in the priority assignment also have coefficients bounded by a fixed power of N N N Q Q Q K q s ν Q Kq_s\nu_Q K q s ν Q K q s ν Q Kq_s\nu_Q K q s ν Q (4.18) . For a sufficiently large fixed power this contradicts the greedy eligibility threshold.
Set τ = ( J s / q s 2 ) N − c / 4 \tau=(J_s/q_s^2)N^{-c/4} τ = ( J s / q s 2 ) N − c /4 J s , q s J_s,q_s J s , q s Q Q Q R R R τ \tau τ R 0 b , R 1 b R_0^b,R_1^b R 0 b , R 1 b p L p_L p L g N − c / 2 ≪ τ gN^{-c/2}\ll\tau g N − c /2 ≪ τ R R R K p a t h O ( 1 ) K_{\mathrm{path}}^{O(1)} K path O ( 1 ) Z Z Z
∣ Y ′ − R 0 b − R 1 b Z ′ ∣ ≤ K p a t h O ( 1 ) τ (4.36) \lvert Y'-R_0^b-R_1^bZ'\rvert\le K_{\mathrm{path}}^{O(1)}\tau\tag{4.36} ∣ Y ′ − R 0 b − R 1 b Z ′ ∣ ≤ K path O ( 1 ) τ ( 4.36 ) by the actual bound on v = Z ′ v=Z' v = Z ′ Y − R 1 b Z Y-R_1^bZ Y − R 1 b Z q s τ q_s\tau q s τ Q Q Q
Keep the old Z Z Z Y − R 1 b Z Y-R_1^bZ Y − R 1 b Z R 0 b R_0^b R 0 b q s ( 1 , R 1 b ) q_s(1,R_1^b) q s ( 1 , R 1 b ) q s ( 0 , τ ) q_s(0,\tau) q s ( 0 , τ ) ( Z , Y ) (Z,Y) ( Z , Y ) q s 2 τ = J s N − c / 4 q_s^2\tau=J_sN^{-c/4} q s 2 τ = J s N − c /4 Q Q Q N − c / 4 N^{-c/4} N − c /4
Comparison of the time candidates. We compare candidate graphs compatible with the same original observation. They agree at its base point to the candidate fitting accuracy. We will prove that, except on a negligible fraction of compatible pairs, they are also close on a whole short comparison box. Their finite jets can then replace the candidate indices as listable chart labels.
Work in each assigned Q s Q_s Q s u u u w w w a Q a_Q a Q a Q / a ∼ log , N 1 a_Q/a\sim_{\log,N}1 a Q / a ∼ l o g , N 1 N σ N^\sigma N σ σ \sigma σ O ( σ ) O(\sigma) O ( σ ) σ \sigma σ m Q = p A q s ν Q m_Q=p_Aq_s\nu_Q m Q = p A q s ν Q ∑ Q m Q ≤ 1 \sum_Qm_Q\le1 ∑ Q m Q ≤ 1
Each index (assigned test)’s base marginal on the counted occurrences costs at most
m Q N − d c + O ( σ ) d u . (4.37) m_QN^{-dc+O(\sigma)}\,du.\tag{4.37} m Q N − d c + O ( σ ) d u . ( 4.37 ) Indeed the candidate value and lower derivative tests require only O ( N O ( σ ) ) O(N^{O(\sigma)}) O ( N O ( σ ) ) a ′ a' a ′ K ( a ′ ) d K(a')^d K ( a ′ ) d q s J s q_sJ_s q s J s ν Q ∼ log , N a d J s \nu_Q \sim_{\log,N} a^d J_s ν Q ∼ l o g , N a d J s u u u ≥ N − O ( σ ) \ge N^{-O(\sigma)} ≥ N − O ( σ ) (4.37) (likewise if using the unnormalized parent prior for eligibility).
Convert each assigned test and the true Q s Q_s Q s w w w P P P a f i t = a ′ a_{\mathrm{fit}}=a' a fit = a ′ a f i t = a a_{\mathrm{fit}}=a a fit = a a f i t N σ a_{\mathrm{fit}}N^\sigma a fit N σ N σ N^\sigma N σ N − 2 σ N^{-2\sigma} N − 2 σ ∣ P w w ∣ ≥ N − C σ / a f i t \lvert P_{ww}\rvert\ge N^{-C\sigma}/a_{\mathrm{fit}} ∣ P w w ∣ ≥ N − C σ / a fit C C C a f i t N O ( σ ) a_{\mathrm{fit}}N^{O(\sigma)} a fit N O ( σ ) w w w
For the remaining tests the discriminant
D = P w 2 − 2 P w w P (4.38) \mathcal{D}=P_w^2-2P_{ww}P\tag{4.38} D = P w 2 − 2 P w w P ( 4.38 ) is independent of w w w P w 2 P_w^2 P w 2 K p a t h K_{\mathrm{path}} K path u u u N O ( σ ) N^{O(\sigma)} N O ( σ )
Subdivide the time of Q s Q_s Q s ρ \rho ρ N − C ′ σ N^{-C'\sigma} N − C ′ σ C ′ C' C ′ ρ \rho ρ N O ( σ ) N^{O(\sigma)} N O ( σ ) Q Q Q Q Q Q p L p_L p L K p a t h O ( 1 ) ρ K_{\mathrm{path}}^{O(1)}\rho K path O ( 1 ) ρ
Around each subbox center, take comparison balls or polydiscs with radius a fixed multiple of K p a t h O ( 1 ) ρ K_{\mathrm{path}}^{O(1)}\rho K path O ( 1 ) ρ D \mathcal{D} D C ′ C' C ′ N − 4 σ N^{-4\sigma} N − 4 σ w w w
For such nonreplaced tests, on any assigned line for these choices the low-curvature cutoff guarantees 2 ∣ P w w P ∣ ≪ D 2\lvert P_{ww}P\rvert\ll\mathcal{D} 2 ∣ P w w P ∣ ≪ D P w P_w P w f f f w w w O ( a f i t N O ( σ ) ) O(a_{\mathrm{fit}}N^{O(\sigma)}) O ( a fit N O ( σ ) ) P w P_w P w w w w D \sqrt{\mathcal{D}} D f 0 f_0 f 0
Let μ c o v \mu_{\mathrm{cov}} μ cov O = ( p L , Q s , subbox , true branch ) O=(p_L,Q_s,\text{subbox},\text{true branch}) O = ( p L , Q s , subbox , true branch ) Q s Q_s Q s κ O \kappa_O κ O O O O
μ p a i r = ∫ κ O ⊗ κ O d ( O ∗ μ c o v ) ( O ) . (4.39) \mu_{\mathrm{pair}}=\int\kappa_O\otimes\kappa_O\,d(O_* \mu_{\mathrm{cov}})(O).\tag{4.39} μ pair = ∫ κ O ⊗ κ O d ( O ∗ μ cov ) ( O ) . ( 4.39 ) Each marginal is μ c o v \mu_{\mathrm{cov}} μ cov N − O ( 1 ) N^{-O(1)} N − O ( 1 ) a ′ N O ( σ ) a' N^{O(\sigma)} a ′ N O ( σ ) p L p_L p L w w w
Fix d 0 = ( 1 + d ) / 2 < 1 d_0 = (1+d)/2 < 1 d 0 = ( 1 + d ) /2 < 1 ε ∗ \varepsilon_* ε ∗
ε = ε ∗ / 2 , θ = ε ∗ / 8. (4.40) \varepsilon= \varepsilon_*/2,\qquad\theta= \varepsilon_*/8.\tag{4.40} ε = ε ∗ /2 , θ = ε ∗ /8. ( 4.40 ) We will choose σ \sigma σ o ( 1 ) o(1) o ( 1 ) a N − 2 θ c aN^{-2\theta c} a N − 2 θ c
First discard entries (index, subbox, candidate and true branch signs) with marginal mass below m Q N − d c − C ′ ′ σ m_QN^{-dc-C''\sigma} m Q N − d c − C ′′ σ C ′ ′ C'' C ′′ f 0 f_0 f 0 a N O ( σ ) aN^{O(\sigma)} a N O ( σ ) (4.37) , this projection has relative measure at least N − O ( σ ) N^{-O(\sigma)} N − O ( σ ) O ( σ ) O(\sigma) O ( σ )
Suppose a nonvanishing relative fraction of the pair law were bad. The light-entry omissions leave a group ( Q , subbox , true branch ) (Q,\text{subbox},\text{true branch}) ( Q , subbox , true branch ) m Q N − O ( σ ) m_QN^{-O(\sigma)} m Q N − O ( σ ) f 0 f_0 f 0 M = a M=a M = a Δ = a ′ \Delta=a' Δ = a ′ (4.37) gives index-base ceilings with weights N − d c N^{-dc} N − d c N O ( σ ) N^{O(\sigma)} N O ( σ ) W c W_c W c N − d c N^{-dc} N − d c B B B
f i B ≤ N O ( σ ) m Q W c B N − d c W c . (4.41) \frac{f_i}{B} \le N^{O(\sigma)}\frac{m_QW_c}{B}\frac{N^{-dc}}{W_c}.\tag{4.41} B f i ≤ N O ( σ ) B m Q W c W c N − d c . ( 4.41 ) The list count gives W c ≤ N O ( σ ) W_c \le N^{O(\sigma)} W c ≤ N O ( σ ) B ≥ m Q N − O ( σ ) B \ge m_QN^{-O(\sigma)} B ≥ m Q N − O ( σ ) N − d c / W c N^{-dc}/W_c N − d c / W c N O ( σ ) N^{O(\sigma)} N O ( σ ) ∑ Q m Q ≤ 1 \sum_Q m_Q \le1 ∑ Q m Q ≤ 1 N σ N^\sigma N σ β n = N − o ( 1 ) \beta_n=N^{-o(1)} β n = N − o ( 1 ) o ( β n ) o(\beta_n) o ( β n ) β n − 1 \beta_n^{-1} β n − 1
Here is the tolerance check. Put R = M / Δ = N c R=M/\Delta=N^c R = M /Δ = N c M = a M=a M = a Δ = a ′ \Delta=a' Δ = a ′
a N − 2 θ c = M R − 2 θ > M R − ε . aN^{-2\theta c}=MR^{-2\theta}>MR^{-\varepsilon}. a N − 2 θ c = M R − 2 θ > M R − ε . The list counts are at most R d + C σ / c + o ( 1 ) R^{d+C\sigma/c+o(1)} R d + C σ / c + o ( 1 ) f 0 f_0 f 0 R C σ / c + o ( 1 ) R^{C\sigma/c+o(1)} R C σ / c + o ( 1 ) C C C σ \sigma σ C σ / c < ε ∗ / 8 C\sigma/c<\varepsilon_*/8 C σ / c < ε ∗ /8 o ( 1 ) o(1) o ( 1 ) β n − 1 \beta_n^{-1} β n − 1 ε ∗ / 8 \varepsilon_*/8 ε ∗ /8 d < d 0 d<d_0 d < d 0 ε = ε ∗ / 2 \varepsilon=\varepsilon_*/2 ε = ε ∗ /2
The improved time charts. Bin sufficiently many candidate branch jets at the subbox center (in comparison-ball scaled coordinates) to steps a N − θ c aN^{-\theta c} a N − θ c π O \pi_O π O O O O N ( b ) \mathcal{N}(b) N ( b ) b b b
∑ b π O ( b ) π O ( N ( b ) ) = P { B 2 ∈ N ( B 1 ) ∣ O } . (4.42) \sum_b \pi_O(b)\pi_O(\mathcal{N}(b)) = \mathbb{P}\{B_2 \in\mathcal{N}(B_1) \mid O\}.\tag{4.42} b ∑ π O ( b ) π O ( N ( b )) = P { B 2 ∈ N ( B 1 ) ∣ O } . ( 4.42 ) Choose a maximizing b b b O O O 1 − o ( 1 ) 1-o(1) 1 − o ( 1 ) o ( 1 ) o(1) o ( 1 )
Define the preliminary geometric label
c g e o m = ( Q s , short block , spatial subbox , true branch sign , candidate jet bin ) . c_{\mathrm{geom}} = (Q_s,\text{short block},\text{spatial subbox},\text{true branch sign},\text{candidate jet bin}). c geom = ( Q s , short block , spatial subbox , true branch sign , candidate jet bin ) . Keep the earlier priority assignment and its invariant branch signs, and merge assignments with the same c g e o m c_{\mathrm{geom}} c geom c g e o m c_{\mathrm{geom}} c geom p L p_L p L
For each nonempty jet bin in its group, choose one representative equation P ^ \widehat{P} P f ^ \hat{f} f ^ c g e o m c_{\mathrm{geom}} c geom f ^ \hat{f} f ^ O ( a N − θ c ) O(aN^{-\theta c}) O ( a N − θ c )
∣ w − f ^ ∣ ≲ a N − θ c |w-\hat{f}| \lesssim aN^{-\theta c} ∣ w − f ^ ∣ ≲ a N − θ c on each member’s entire short line piece, after choosing σ \sigma σ
If ∣ P ^ w w ∣ ≤ ( a N − θ c / 2 ) − 1 |\widehat{P}_{ww}| \le(aN^{-\theta c/2})^{-1} ∣ P w w ∣ ≤ ( a N − θ c /2 ) − 1 w w w 1 1 1 ≥ N − O ( σ ) \ge N^{-O(\sigma)} ≥ N − O ( σ ) w w w O ( N − θ c / 2 ) O(N^{-\theta c/2}) O ( N − θ c /2 ) ≲ a N − θ c \lesssim aN^{-\theta c} ≲ a N − θ c ≤ a − 1 N θ c / 2 + O ( σ ) \le a^{-1}N^{\theta c/2+O(\sigma)} ≤ a − 1 N θ c /2 + O ( σ ) a Q N − 2 θ c / 3 a_QN^{-2\theta c/3} a Q N − 2 θ c /3
For larger curvature, the representative branch is within a N − θ c / 2 + O ( σ ) aN^{-\theta c/2+O(\sigma)} a N − θ c /2 + O ( σ ) w w w a Q N − θ c / 3 a_QN^{-\theta c/3} a Q N − θ c /3
Choose σ \sigma σ θ \theta θ C ′ σ < k θ c / 12 C'\sigma< k\theta c/12 C ′ σ < k θ c /12 a Q / a a_Q/a a Q / a
Use the true spatial frame scaled by ρ \rho ρ w w w B B B
Keep the preliminary label c g e o m c_{\mathrm{geom}} c geom
The depth gain is at least ( θ c / 3 ) ϵ n (\theta c/3)\epsilon_n ( θ c /3 ) ϵ n C ′ σ ϵ n + o ( ϵ n ) C'\sigma\epsilon_n + o(\epsilon_n) C ′ σ ϵ n + o ( ϵ n ) σ \sigma σ k k k
Finite encoding and return to the original law. Both constructors have now specified their chart geometry, tests, full labels, and block-invariant trajectory assignments. We verify that these objects have bounded original-coordinate complexity and recover their lists on the original tempered path.
Store the ordered quadratic tests or row predictors at the single chosen node. The assignments use membership in the old chart’s trajectory set, whole-block line inequalities, and finite priority rules. Subbox certification flags, branch signs, merger tables, and representative tests are fixed data for each chart and test; a simple branch is selected by its midpoint derivative sign. Nonempty jet bins may be reindexed by discrete identifiers, so numerical jets need not enter a predicate. The analytical masks and the posterior search used to choose the lists are not part of these geometric assignments.
Coefficient bounds. Here is an explicit coefficient bound for every time candidate. In normalized chart coordinates write P ( u , w ) = A w 2 + B ( u ) w + C ( u ) P(u,w)=Aw^2+B(u)w+C(u) P ( u , w ) = A w 2 + B ( u ) w + C ( u ) B B B C C C ∣ A ∣ ≤ N s + c + O ( σ ) |A|\le N^{s+c+O(\sigma)} ∣ A ∣ ≤ N s + c + O ( σ ) δ ≥ N − C S σ \delta\ge N^{-C_{\mathcal{S}}\sigma} δ ≥ N − C S σ (4.37) ; its chart weight cancels when the floor is divided by the ceiling. On that witness ∣ w ∣ ≤ W ≤ N 0 P 0 ( M p a t h ) |w|\le W\le N_0^{P_0(M_{\mathrm{path}})} ∣ w ∣ ≤ W ≤ N 0 P 0 ( M path )
∣ B ( u ) ∣ ≤ N σ + 2 ∣ A ∣ W , ∣ C ( u ) ∣ ≤ a ′ N σ + ∣ A ∣ W 2 + ∣ B ( u ) ∣ W . (4.43) |B(u)| \le N^\sigma+ 2|A|W,\qquad|C(u)| \le a'N^\sigma+ |A|W^2 + |B(u)|W.\tag{4.43} ∣ B ( u ) ∣ ≤ N σ + 2∣ A ∣ W , ∣ C ( u ) ∣ ≤ a ′ N σ + ∣ A ∣ W 2 + ∣ B ( u ) ∣ W . ( 4.43 ) Fixed-degree polynomial Remez from this projection costs at most δ − C \delta^{-C} δ − C B , C B,C B , C N 0 P S ( M p a t h ) N_0^{P_\mathcal{S}(M_{\mathrm{path}})} N 0 P S ( M path ) w w w
Actual lists. On the masked successes the full output labels have lists at the original p L p_L p L ( K p a t h log N 0 ) r S (K_{\mathrm{path}}\log N_0)^{r_\mathcal{S}} ( K path log N 0 ) r S r S r_\mathcal{S} r S O ( N 0 − 1 ) O(N_0^{-1}) O ( N 0 − 1 )
Complexity and mass. The coefficient and predicate bounds above, followed by Lemma 4.12, bound every original-coordinate complexity exponent by a polynomial in M p a t h M_{\mathrm{path}} M path M # M^\# M # (4.13) ; its fixed power of actual path factors and logarithms likewise fits the allowance n n n
The comparison and representative choice retain some fixed fraction c 0 > 0 c_0 > 0 c 0 > 0 β n \beta_n β n
c 0 η 0 J G n p p a t h − O ( N 0 − 1 ) > p p a t h log N 0 (4.44) \frac{c_0\eta_0}{J G_n}\,\mathfrak{p}_{\mathrm{path}} - O(N_0^{-1}) > \frac{\mathfrak{p}_{\mathrm{path}}}{\log N_0}\tag{4.44} J G n c 0 η 0 p path − O ( N 0 − 1 ) > log N 0 p path ( 4.44 ) for large stages, by (4.15) . The original mass loss is negligible since N 0 − 1 log N 0 / p p a t h ≤ N 0 − 1 K p a t h log N 0 → 0 N_0^{-1}\log N_0/\mathfrak{p}_{\mathrm{path}} \le N_0^{-1}K_{\mathrm{path}}\log N_0 \to0 N 0 − 1 log N 0 / p path ≤ N 0 − 1 K path log N 0 → 0 γ n \gamma_n γ n k k k
Velocity breadth in isotropic parents
The isotropic argument will use finite point states to control gradients of packet graphs. A large gradient would predict the velocity in a power-thin affine strip. We now rule out such concentration on a substantial family by constructing a narrower parent atlas. The original-law list recovery and finite encoding just established realize this width improvement while keeping the parent depth and horizon fixed.
Corollary 4.13 (Breadth before sparse selections). In version 2 with ℓ = 0 \ell= 0 ℓ = 0 p L p_L p L K K K κ , u > 0 \kappa, u > 0 κ , u > 0 κ \kappa κ N − u N^{-u} N − u
In version 1 the corresponding assertion for intervals follows from the angular cap. These statements can be prepared at reciprocal-subpower separation thresholds before arbitrary sparse selections. They persist under selections of whole conditioning states, or controlled fixed-fraction selections within them; no version-2 breadth assertion is made for every velocity-dependent inverse-subpower subfamily.
Proof. If the version-2 assertion fails, on a subsequence there are state-predicted strips of the indicated width carrying a fixed positive amount of residual mass, still on substantial events of the prepared good law.
A finite family of heavy strips. Use unit normal and offset pairs ( n v , b v ′ ) (n_v,b'_v) ( n v , b v ′ ) T ≥ 1 T \ge1 T ≥ 1 O ( T ) O(T) O ( T ) 2 N − u 2N^{-u} 2 N − u π \pi π κ N = 1 / K \kappa_N = 1/K κ N = 1/ K κ N \kappa_N κ N
Per parent take maximal parameter-separated representatives at spacing r 1 = N − u / 2 r_1 = N^{-u/2} r 1 = N − u /2 M 1 M_1 M 1 ≤ K \le K ≤ K ∣ v ∣ ≤ T |v| \le T ∣ v ∣ ≤ T r 1 / ( 1 + T ) r_1/(1+T) r 1 / ( 1 + T ) 4 N − u + T ∣ n v , 1 − n v , 2 ∣ 4N^{-u} + T|n_{v,1} - n_{v,2}| 4 N − u + T ∣ n v , 1 − n v , 2 ∣ O ( ( 1 + T ) N − u / r 1 ) ≤ N − u / 3 O((1+T)N^{-u}/r_1) \le N^{-u/3} O (( 1 + T ) N − u / r 1 ) ≤ N − u /3 π \pi π
( M 1 κ N ) 2 ≤ M 1 + K N − S u / 3 M 1 2 , (4.45) (M_1\kappa_N)^2 \le M_1 + K N^{-Su/3}M_1^2,\tag{4.45} ( M 1 κ N ) 2 ≤ M 1 + K N − S u /3 M 1 2 , ( 4.45 ) giving the claim. Moreover for representatives separated by > r 2 = N − u / 4 >r_2=N^{-u/4} > r 2 = N − u /4 enlarged strips of widths 10 ( 1 + T ) r 1 10(1+T)r_1 10 ( 1 + T ) r 1 ≲ ( 1 + T ) 2 r 1 / r 2 \lesssim(1+T)^2r_1/r_2 ≲ ( 1 + T ) 2 r 1 / r 2
Trajectory assignments and the narrower atlas. On narrow events the velocity therefore lies in an enlarged representative strip, and any choice of such containing strip agrees to O ( r 2 ) O(r_2) O ( r 2 ) r 1 r_1 r 1 velocity alone on parent trajectories, invariantly over the block. Bin its parameters (with one orientation each) to rectangular r 2 r_2 r 2 ( n ^ , b ^ ) (\hat n,\hat b) ( n ^ , b ^ )
∣ n ^ ⋅ y ′ − b ^ ∣ ≲ ( 1 + T ) r 2 (4.46) |\hat n\cdot y' - \hat b| \lesssim(1+T)r_2\tag{4.46} ∣ n ^ ⋅ y ′ − b ^ ∣ ≲ ( 1 + T ) r 2 ( 4.46 ) on assigned pieces. Bin also the corresponding n ^ \hat n n ^ r 2 r_2 r 2 r 2 r_2 r 2 n ^ \hat n n ^ b ^ \hat b b ^
On the successful incidences, the assumed state knowledge supplies actual lists for the full output labels at the original p L p_L p L O ( N 0 − 1 ) O(N_0^{-1}) O ( N 0 − 1 )
In version 1 the corresponding avoidance of narrow velocity intervals on substantial families already follows from the angular cap and the deep-history joint bound (summing the subpower state ambiguities there). These almost-all breadth/separation conclusions can use reciprocal-subpower accuracies by taking fixed-power thresholds to zero sufficiently slowly. In applications requiring general position, one can prepare them before sparse selections and keep them through selections of whole conditioning states (or with small fixed-fraction losses controlled there). No breadth in version 2 on every K − 1 K^{-1} K − 1
Scalar concentration and polynomial fitting
We prove two consequences of scalar concentration. First, a small spacetime support forces a heavy bin of quadratic trajectory coefficients. Second, when a common row subtracts the tangential coordinates from a quadratic signal, concentration of the residual scalar forces polynomial fits to that row along labeled trajectories. The second argument begins with entropy estimates under simultaneous time tests. A planar dot-product theorem then gives affine row fits and first-jet information; divided differences raise the jet order, and a separate time test produces the polynomial fits.
The arguments in this section take place in tangent units. Thus K = N o ( 1 ) K = N^{o(1)} K = N o ( 1 ) dense if its mass is at least K − 1 K^{-1} K − 1 K K K [ 0 , 1 ] [0,1] [ 0 , 1 ]
We use measurable time labels E l ⊂ [ 0 , 1 ] E_l \subset[0,1] E l ⊂ [ 0 , 1 ]
ν { l : t ∈ E l , the indicated conditions hold } ; \nu\{l : t \in E_l,\ \text{the indicated conditions hold}\}; ν { l : t ∈ E l , the indicated conditions hold } ; it is not divided by the success probability at time t t t w ( l , t ) ≤ K p r e w(l,t) \le K_{\mathrm{pre}} w ( l , t ) ≤ K pre K p r e = N o ( 1 ) K_{\mathrm{pre}} = N^{o(1)} K pre = N o ( 1 ) ν ⊗ d t \nu\otimes dt ν ⊗ d t δ ≥ K p r e − 1 \delta\ge K_{\mathrm{pre}}^{-1} δ ≥ K pre − 1 w < δ / 2 w < \delta/2 w < δ /2 δ / 2 \delta/2 δ /2 δ / ( 2 K p r e ) \delta/(2K_{\mathrm{pre}}) δ / ( 2 K pre )
Concentration of quadratic coefficients
Lemma 5.1 (Scalar coefficient clustering). Let
x l ( t ) = x 0 , l + t x 1 , l + t 2 x 2 , l , x_l(t) = x_{0,l} + t x_{1,l} + t^2 x_{2,l}, x l ( t ) = x 0 , l + t x 1 , l + t 2 x 2 , l , and let the trajectory prior ν \nu ν K K K 0 ≤ d < 1 0 \le d < 1 0 ≤ d < 1 p = N − c + o ( 1 ) p = N^{-c+o(1)} p = N − c + o ( 1 ) c > 0 c > 0 c > 0 K d ν d t K\,d\nu\,dt K d ν d t K p − 1 − d Kp^{-1-d} K p − 1 − d p p p ( t , x ) (t,x) ( t , x )
ν { l : ( x 0 , l , x 1 , l , x 2 , l ) ∈ B } ≤ K r d (5.1) \nu\{l : (x_{0,l},x_{1,l},x_{2,l}) \in B\} \le K r^d\tag{5.1} ν { l : ( x 0 , l , x 1 , l , x 2 , l ) ∈ B } ≤ K r d ( 5.1 ) for every coefficient ball B B B r ≥ p r \ge p r ≥ p p p p p d / K p^d/K p d / K K − 1 K^{-1} K − 1
Proof. First discard trajectories of labeled length below a sufficiently small reciprocal subpower. The remaining trajectory mass is at least K − 1 K^{-1} K − 1 p − 1 / K p^{-1}/K p − 1 / K D ∈ [ 0 , 1 ] D \in[0,1] D ∈ [ 0 , 1 ]
z l , D ( t ) = ( x 0 , l + t ( x 1 , l − D ) , D + t x 2 , l ) . (5.2) z_{l,D}(t) = (x_{0,l} + t(x_{1,l} - D), D + t x_{2,l}).\tag{5.2} z l , D ( t ) = ( x 0 , l + t ( x 1 , l − D ) , D + t x 2 , l ) . ( 5.2 ) The identity
( 1 , t ) ⋅ z l , D ( t ) = x l ( t ) (1,t) \cdot z_{l,D}(t) = x_l(t) ( 1 , t ) ⋅ z l , D ( t ) = x l ( t ) shows that the marked lifted motions occupy at most K p − 2 − d Kp^{-2-d} K p − 2 − d ( t , x ) (t,x) ( t , x ) K / p K/p K / p K K K
Apply Lemma 2.15, namely the weighted full-trace form of OpenAI [24 ] (Lemma 2.3). Rescaling a subpower bounded spatial chart into a fixed bounded chart and rounding line parameters much more finely than p p p p − 1 / K p^{-1}/K p − 1 / K K p − 2 − d Kp^{-2-d} K p − 2 − d p 1 + d / K p^{1+d}/K p 1 + d / K
p / K ≤ u ≤ v ≤ K p/K \le u \le v \le K p / K ≤ u ≤ v ≤ K whose lifted trajectory mass is at least
K − 1 p d − 1 u v . (5.3) K^{-1}p^{d-1}uv.\tag{5.3} K − 1 p d − 1 uv . ( 5.3 ) Here the conversion from widths in cell units contributes p − 2 p^{-2} p − 2
All scalar quadratics arising from this plank lie in one coefficient ball of radius K v Kv K v ( 1 , t ) (1,t) ( 1 , t ) O ( v ) O(v) O ( v ) O ( v ) O(v) O ( v )
For fixed scalar coefficients, varying D D D D ( − t , 1 ) D(-t,1) D ( − t , 1 ) n = ( n 1 , n 2 ) n=(n_1,n_2) n = ( n 1 , n 2 )
max { ∣ n 2 ∣ , ∣ n 2 − n 1 ∣ } ≥ c \max\{|n_2|,|n_2-n_1|\}\ge c max { ∣ n 2 ∣ , ∣ n 2 − n 1 ∣ } ≥ c for an absolute c > 0 c>0 c > 0 t = 0 , 1 t=0,1 t = 0 , 1 D D D C u Cu C u K u v d Kuv^d K u v d v < p v<p v < p p p p v ≥ p / K v\ge p/K v ≥ p / K K K K (5.3) gives
v 1 − d ≤ K p 1 − d , v ≤ K p . v^{1-d}\le Kp^{1-d},\qquad v\le Kp. v 1 − d ≤ K p 1 − d , v ≤ K p . Dividing (5.3) by the dummy-coordinate acceptance bound gives coefficient mass at least K − 1 p d − 1 v ≥ p d / K K^{-1}p^{d-1}v\ge p^d/K K − 1 p d − 1 v ≥ p d / K K p Kp K p K K K p p p
Scalar-normal data and conditioning
The next definition separates the fixed trajectory prior from the time labels that we will repeatedly restrict. Its three inputs control scalar supports, comparisons of the row and residual slope, and spatial mass conditional on a coefficient bin.
Definition 5.2 (Scalar-normal data). Fix j ∈ { 1 , 2 } j\in\{1,2\} j ∈ { 1 , 2 } 0 ≤ d < 1 0\le d<1 0 ≤ d < 1 ζ \zeta ζ ν \nu ν x l x_l x l y l : [ 0 , 1 ] → R j y_l:[0,1]\to\mathbb{R}^j y l : [ 0 , 1 ] → R j K K K E l E_l E l ν ⊗ d t \nu\otimes dt ν ⊗ d t F ( t ) ∈ ( R j ) ∗ F(t)\in(\mathbb{R}^j)^* F ( t ) ∈ ( R j ) ∗ K K K
S l ( t ) = x l ( t ) − F ( t ) y l ( t ) . S_l(t)=x_l(t)-F(t)y_l(t). S l ( t ) = x l ( t ) − F ( t ) y l ( t ) . For a bounded set A A A N q ( A ) N_q(A) N q ( A ) q q q A A A
For every active time, every interval J J J r r r ζ ≤ q ≤ r ≤ 1 \zeta\le q\le r\le1 ζ ≤ q ≤ r ≤ 1
N q ( J ∩ { S l ( t ) : l with t ∈ E l } ) ≤ K ( r / q ) d . (5.4) N_q\bigl(J\cap\{S_l(t):l\text{ with }t\in E_l\}\bigr)\le K(r/q)^d.\tag{5.4} N q ( J ∩ { S l ( t ) : l with t ∈ E l } ) ≤ K ( r / q ) d . ( 5.4 ) For every spatial cube Q ⊂ R 1 + j Q\subset\mathbb{R}^{1+j} Q ⊂ R 1 + j q q q
ν { l : t ∈ E l , ( x l ( t ) , y l ( t ) ) ∈ Q } ≤ K q j + d . (5.5) \nu\{l:t\in E_l,\ (x_l(t),y_l(t))\in Q\}\le Kq^{j+d}.\tag{5.5} ν { l : t ∈ E l , ( x l ( t ) , y l ( t )) ∈ Q } ≤ K q j + d . ( 5.5 ) At active times,
∣ F ( t ) − F ( u ) ∣ ≤ K ( ∣ t − u ∣ + ζ ) . (5.6) |F(t)-F(u)|\le K(|t-u|+\zeta).\tag{5.6} ∣ F ( t ) − F ( u ) ∣ ≤ K ( ∣ t − u ∣ + ζ ) . ( 5.6 ) Writing R l ( t ) = x l ′ ( t ) − F ( t ) y l ′ R_l(t)=x_l'(t)-F(t)y_l' R l ( t ) = x l ′ ( t ) − F ( t ) y l ′
∣ R l ( t ) − R l ~ ( u ) ∣ ≤ K ( dist ( ( t , x l ( t ) , y l ( t ) ) , ( u , x l ~ ( u ) , y l ~ ( u ) ) ) + ζ ) . (5.7) |R_l(t)-R_{\tilde l}(u)|\le K\bigl(\operatorname{dist}((t,x_l(t),y_l(t)),(u,x_{\tilde l}(u),y_{\tilde l}(u)))+\zeta\bigr).\tag{5.7} ∣ R l ( t ) − R l ~ ( u ) ∣ ≤ K ( dist (( t , x l ( t ) , y l ( t )) , ( u , x l ~ ( u ) , y l ~ ( u ))) + ζ ) . ( 5.7 ) Let l r l_r l r r r r B B B q q q ζ ≤ q ≤ r \zeta\le q \le r ζ ≤ q ≤ r
ν { l ∈ B : t ∈ E l , ( x l ( t ) , y l ( t ) ) ∈ Q } ν ( B ) ≤ K ( q / r ) j + d . (5.8) \frac{\nu\{l \in B : t \in E_l,\ (x_l(t),y_l(t)) \in Q\}}{\nu(B)} \le K(q/r)^{j+d}.\tag{5.8} ν ( B ) ν { l ∈ B : t ∈ E l , ( x l ( t ) , y l ( t )) ∈ Q } ≤ K ( q / r ) j + d . ( 5.8 ) These are absolute labeled masses relative to the stated prior. A common null set of times may be removed. A finite world parameter is permitted, with the hypotheses holding in each retained conditional world; every conditioning variable below then includes that world.
We use fixed dyadic grids, allowing bounded enlargements. All exponents of resolutions used in one application range over a fixed finite interval. Slowly densifying exponent grids give intermediate covering bounds by subdivision, since consecutive scale ratios are subpower. The conditional cap (5.8) is needed on compatible parameter grids at the entropy scales. A restriction of labels preserves every upper hypothesis in Definition 5.2 with the same prior.
For these data we will approximate F F F d < 1 d < 1 d < 1
For clarity, write H \mathsf{H} H I \mathsf{I} I o ∗ = o ( log N ) o_* = o(\log N) o ∗ = o ( log N )
Lemma 5.3 (Entropy on a dense tested event). Presample a finite time tuple independently of the trajectory in its world, with absolutely continuous marginal laws for its tested times. Let T T T A \mathcal{A} A Q Q Q t t t
H Q ( S ( t ) r ∣ T ) = d log ( 1 / r ) + o ∗ , (5.9) \mathsf{H}_Q(S(t)_r \mid T) = d\log(1/r) + o_*,\tag{5.9} H Q ( S ( t ) r ∣ T ) = d log ( 1/ r ) + o ∗ , ( 5.9 ) H Q ( y ( t ) r ∣ T , S ( t ) r ) = j log ( 1 / r ) + o ∗ , (5.10) \mathsf{H}_Q(y(t)_r \mid T,S(t)_r) = j\log(1/r) + o_*,\tag{5.10} H Q ( y ( t ) r ∣ T , S ( t ) r ) = j log ( 1/ r ) + o ∗ , ( 5.10 ) I Q ( l r ; S ( t ) q ∣ T , S ( t ) r ) = o ∗ , q < r . (5.11) \mathsf{I}_Q(l_r;S(t)_q \mid T,S(t)_r) = o_*, \qquad q < r.\tag{5.11} I Q ( l r ; S ( t ) q ∣ T , S ( t ) r ) = o ∗ , q < r . ( 5.11 ) Moreover, for every T T T e e e r ≥ q r \ge q r ≥ q
H Q ( ( e y ( t ) ) r ∣ T , S ( t ) q ) ≥ log ( 1 / r ) − o ∗ . (5.12) \mathsf{H}_Q((e\,y(t))_r \mid T,S(t)_q) \ge\log(1/r) - o_*.\tag{5.12} H Q (( e y ( t ) ) r ∣ T , S ( t ) q ) ≥ log ( 1/ r ) − o ∗ . ( 5.12 ) The same assertions hold anew on every further dense event; the event may depend on all the times and on the trajectory.
Proof. Let P P P p A = P ( A ) p_\mathcal{A} = P(\mathcal{A}) p A = P ( A ) s ( T ) = P ( A ∣ T ) s(T) = P(\mathcal{A} \mid T) s ( T ) = P ( A ∣ T )
E Q 1 s ( T ) ≤ 1 p A , E Q log 1 s ( T ) ≤ log 1 p A = o ∗ . (5.13) \mathbb{E}_Q \frac{1}{s(T)} \le\frac{1}{p_\mathcal{A}}, \qquad\mathbb{E}_Q \log\frac{1}{s(T)} \le\log\frac{1}{p_\mathcal{A}} = o_*.\tag{5.13} E Q s ( T ) 1 ≤ p A 1 , E Q log s ( T ) 1 ≤ log p A 1 = o ∗ . ( 5.13 ) The first inequality follows by integrating s ( T ) / p A s(T)/p_\mathcal{A} s ( T ) / p A T T T { s > 0 } \{s > 0\} { s > 0 }
The shear ( x , y ) ↦ ( S , y ) (x,y) \mapsto(S,y) ( x , y ) ↦ ( S , y ) T T T Z = ( S l ( t ) r , y l ( t ) r ) Z = (S_l(t)_r,y_l(t)_r) Z = ( S l ( t ) r , y l ( t ) r ) A \mathcal{A} A t t t z z z
P ( A , Z = z ∣ T ) ≤ P ( t ∈ E l , Z = z ∣ T ) ≤ K r j + d . P(\mathcal{A},Z=z \mid T) \le P(t \in E_l,Z=z \mid T) \le Kr^{j+d}. P ( A , Z = z ∣ T ) ≤ P ( t ∈ E l , Z = z ∣ T ) ≤ K r j + d . For the last inequality, the trajectory law at fixed T T T K K K r r r (5.5) applies. Thus, whenever s ( T ) > 0 s(T)>0 s ( T ) > 0
Q ( Z = z ∣ T ) = P ( A , Z = z ∣ T ) s ( T ) ≤ K r j + d s ( T ) . Q(Z=z\mid T)=\frac{P(\mathcal{A},Z=z\mid T)}{s(T)}\leq\frac{Kr^{j+d}}{s(T)}. Q ( Z = z ∣ T ) = s ( T ) P ( A , Z = z ∣ T ) ≤ s ( T ) K r j + d . The entropy is therefore at least
( j + d ) log ( 1 / r ) − O ( log K ) − E Q log ( 1 / s ( T ) ) . (j+d)\log(1/r)-O(\log K)-\mathbb{E}_{Q}\log(1/s(T)). ( j + d ) log ( 1/ r ) − O ( log K ) − E Q log ( 1/ s ( T )) . The support bounds are K r − d Kr^{-d} K r − d S r S_r S r K r − j Kr^{-j} K r − j y r y_r y r
For (5.11) , put B = l r B=l_r B = l r s ( T , B ) = P ( A ∣ T , B ) s(T,B)=P(\mathcal{A}\mid T,B) s ( T , B ) = P ( A ∣ T , B ) (5.8) , because the tuple was trajectory-independent. In B B B K ( r / q ) j K(r/q)^j K ( r / q ) j y q y_q y q S q S_q S q
P ( A , S q = z ∣ T , B ) ≤ K ( q / r ) d . P(\mathcal{A},S_q=z\mid T,B)\leq K(q/r)^d. P ( A , S q = z ∣ T , B ) ≤ K ( q / r ) d . The calculation in (5.13) gives
E Q log ( 1 / s ( T , B ) ) ≤ log ( 1 / p A ) , \mathbb{E}_{Q}\log(1/s(T,B))\leq\log(1/p_\mathcal{A}), E Q log ( 1/ s ( T , B )) ≤ log ( 1/ p A ) , with no dependence on the number or individual prior masses of B B B
H Q ( S q ∣ T , B ) ≥ d log ( r / q ) − o ∗ . \mathsf{H}_Q(S_q\mid T,B)\geq d\log(r/q)-o_*. H Q ( S q ∣ T , B ) ≥ d log ( r / q ) − o ∗ . Since the coefficients vary by O ( r ) O(r) O ( r ) B B B K K K S r S_r S r
H Q ( S q ∣ T , B , S r ) ≥ H Q ( S q ∣ T , B ) − H Q ( S r ∣ T , B ) ≥ d log ( r / q ) − o ∗ . \mathsf{H}_Q(S_q\mid T,B,S_r)\geq \mathsf{H}_Q(S_q\mid T,B)-\mathsf{H}_Q(S_r\mid T,B)\geq d\log(r/q)-o_*. H Q ( S q ∣ T , B , S r ) ≥ H Q ( S q ∣ T , B ) − H Q ( S r ∣ T , B ) ≥ d log ( r / q ) − o ∗ . On the other hand, (5.4) bounds H Q ( S q ∣ T , S r ) \mathsf{H}_Q(S_q\mid T,S_r) H Q ( S q ∣ T , S r ) d log ( r / q ) + o ∗ d\log(r/q)+o_* d log ( r / q ) + o ∗ (5.11) .
For (5.12) , fix T T T r r r y y y K r q − j Krq^{-j} K r q − j q q q (5.10) at scale q q q p A p_A p A
We shall repeatedly use the following exact entropy inequality, valid for any discrete Z Z Z A A A B B B C C C T T T
H ( Z ∣ T , C ) ≤ H ( Z ∣ T , B ) + H ( Z ∣ T , A ) + I ( A ; B ∣ T , C ) . (5.14) \mathsf{H}(Z\mid T,C)\leq \mathsf{H}(Z\mid T,B)+\mathsf{H}(Z\mid T,A)+\mathsf{I}(A;B\mid T,C).\tag{5.14} H ( Z ∣ T , C ) ≤ H ( Z ∣ T , B ) + H ( Z ∣ T , A ) + I ( A ; B ∣ T , C ) . ( 5.14 ) Indeed, expand the left side as H ( Z ∣ T , B , C ) + I ( Z ; B ∣ T , C ) \mathsf{H}(Z\mid T,B,C)+\mathsf{I}(Z;B\mid T,C) H ( Z ∣ T , B , C ) + I ( Z ; B ∣ T , C ) Z Z Z ( A , Z ) (A,Z) ( A , Z ) Z Z Z ( T , S q ) (T,S_q) ( T , S q ) ( T , l r ) (T,l_r) ( T , l r ) A = S q A=S_q A = S q B = l r B=l_r B = l r C = S r C=S_r C = S r (5.11) then gives negligible entropy conditional on ( T , S r ) (T,S_r) ( T , S r ) O ( log K ) O(\log K) O ( log K )
The first jet
Proposition 5.4 (Scalar-normal first jet). Assume Definition 5.2. Fix a finite nested hierarchy of dyadic time lengths of fixed positive power, with finest length H ∗ H_* H ∗ ζ ≤ H ∗ 3 / K \zeta\le H_*^3/K ζ ≤ H ∗ 3 / K K K K A ( t ) A(t) A ( t ) K K K
∣ F ( v ) − F ( u ) − ( v − u ) A ( u ) ∣ ≤ K H 11 / 10 (5.15) \lvert F(v)-F(u)-(v-u)A(u)\rvert\le K H^{11/10}\tag{5.15} ∣ F ( v ) − F ( u ) − ( v − u ) A ( u )∣ ≤ K H 11/10 ( 5.15 ) whenever the retained active times u , v u,v u , v H H H
J 1 ( l , t ) = x l ′ ( t ) − F ( t ) y l ′ − A ( t ) y l ( t ) . J_1(l,t)=x_l'(t)-F(t)y_l'-A(t)y_l(t). J 1 ( l , t ) = x l ′ ( t ) − F ( t ) y l ′ − A ( t ) y l ( t ) . For every finite presampled tuple and every dense tested event as in Lemma 5.3, on which all tested times carry the retained labels,
H ( J 1 ( l , t ) r ∣ T , S l ( t ) r ) = o ∗ (5.16) \mathsf{H}(J_1(l,t)_r\mid T,S_l(t)_r)=o_*\tag{5.16} H ( J 1 ( l , t ) r ∣ T , S l ( t ) r ) = o ∗ ( 5.16 ) if r ≥ H ∗ ( 1 − d ) / 64 r\ge H_*^{(1-d)/64} r ≥ H ∗ ( 1 − d ) /64 A A A
The first step is geometric: we fit each component of F F F F F F d < 1 d<1 d < 1
Comparing these fits across lengths will produce the row A A A J 1 J_1 J 1
Theorem 5.5 (Planar dot-product alternative). For every ε > 0 \varepsilon>0 ε > 0 η , δ 0 > 0 \eta,\delta_0>0 η , δ 0 > 0 0 < p ≤ δ 0 0<p\le\delta_0 0 < p ≤ δ 0 Let A , B ⊂ [ 0 , 1 ] 2 \mathcal{A},\mathcal{B}\subset[0,1]^2 A , B ⊂ [ 0 , 1 ] 2
N s ( E ∩ B ( z , w ) ) ≤ p − η w / s for E ∈ { A , B } , p ≤ s ≤ w . (5.17) N_s(E\cap B(z,w))\le p^{-\eta}w/s\qquad\text{for }E\in\{\mathcal{A},\mathcal{B}\},\quad p\le s\le w.\tag{5.17} N s ( E ∩ B ( z , w )) ≤ p − η w / s for E ∈ { A , B } , p ≤ s ≤ w . ( 5.17 ) At least one of the following alternatives holds:
There are affine lines ℓ , ℓ ⊥ \ell,\ell^\perp ℓ , ℓ ⊥
N p ( A ∩ { z : dist ( z , ℓ ) ≤ p } ) , N p ( B ∩ { z : dist ( z , ℓ ⊥ ) ≤ p } ) N_p\bigl(\mathcal{A}\cap\{z:\operatorname{dist}(z,\ell)\le p\}\bigr),\qquad
N_p\bigl(\mathcal{B}\cap\{z:\operatorname{dist}(z,\ell^\perp)\le p\}\bigr) N p ( A ∩ { z : dist ( z , ℓ ) ≤ p } ) , N p ( B ∩ { z : dist ( z , ℓ ⊥ ) ≤ p } ) are at least p ε − 1 p^{\varepsilon-1} p ε − 1
Every restricted triple set H ⊂ A × B × B \mathcal{H}\subset \mathcal{A}\times \mathcal{B}\times \mathcal{B} H ⊂ A × B × B N p ( H ) ≥ p η − 3 N_p(\mathcal{H})\ge p^{\eta-3} N p ( H ) ≥ p η − 3 s ≥ p s\ge p s ≥ p J J J ∣ J ∣ ≥ p − η s \lvert J\rvert\ge p^{-\eta}s ∣ J ∣ ≥ p − η s
N s ( J ∩ { a ⋅ ( b − b ′ ) : ( a , b , b ′ ) ∈ H } ) ≥ ( ∣ J ∣ / s ) 1 − ε . N_s\bigl(J\cap\{a\cdot(b-b'):(a,b,b')\in\mathcal{H}\}\bigr)\ge(\lvert J\rvert/s)^{1-\varepsilon}. N s ( J ∩ { a ⋅ ( b − b ′ ) : ( a , b , b ′ ) ∈ H } ) ≥ (∣ J ∣ / s ) 1 − ε . We use the restatement of Theorem 5.2 at the start of Section 8 of Wang and Zahl [31 ] , together with Definition 4.2, with theorem numbering fixed to arXiv:2210.09581v2 (2025). In particular, (5.17) is only an upper covering condition, and the second alternative concerns a restricted triple set. We use the first alternative only for A \mathcal{A} A
Proof of Proposition 5.4. Before any label selection, insert finitely many intermediate dyadic lengths so that adjacent lengths H ′ ≤ H H' \le H H ′ ≤ H H ′ ≥ H 33 / 32 / 2 H' \ge H^{33/32}/2 H ′ ≥ H 33/32 /2
We construct those fits one row component and one hierarchy length at a time. The row F F F
Preparing the current labels. We perform the following preparation before fitting each row component at each hierarchy length H H H E l c u r E_l^{\mathrm{cur}} E l cur
m ( t ) = ν { l : t ∈ E l c u r } , δ = ∫ 0 1 m ( t ) d t ≥ K − 1 . m(t)=\nu\{l:t\in E_l^{\mathrm{cur}}\},\qquad\delta=\int_0^1 m(t)\,dt\ge K^{-1}. m ( t ) = ν { l : t ∈ E l cur } , δ = ∫ 0 1 m ( t ) d t ≥ K − 1 . Choose a reciprocal-subpower threshold τ = o ( δ ) \tau=o(\delta) τ = o ( δ ) m ( t ) < τ m(t)<\tau m ( t ) < τ H H H τ H \tau H τ H τ \tau τ E l p r e E_l^{\mathrm{pre}} E l pre δ / 2 \delta/2 δ /2 τ H ≥ H / K \tau H\ge H/K τ H ≥ H / K m ( t ) ≥ τ m(t)\ge\tau m ( t ) ≥ τ ν \nu ν H H H δ / ( 2 H ) \delta/(2H) δ / ( 2 H )
Local normalization. Fix a surviving block I = [ t I , t I + H ] I=[t_I,t_I+H] I = [ t I , t I + H ] u = ( t − t I ) / H u=(t-t_I)/H u = ( t − t I ) / H
U l = { u ∈ [ 0 , 1 ] : t I + H u ∈ E l p r e ∩ I } . U_l=\{u\in[0,1]:t_I+Hu\in E_l^{\mathrm{pre}}\cap I\}. U l = { u ∈ [ 0 , 1 ] : t I + H u ∈ E l pre ∩ I } . Averaging and Cauchy–Schwarz provide u ∗ u_* u ∗
∫ 1 { u ∗ ∈ U l } ∣ U l ∣ d ν ( l ) ≥ K − 1 . \int\mathbf{1}_{\{u_*\in U_l\}}|U_l|\,d\nu(l)\ge K^{-1}. ∫ 1 { u ∗ ∈ U l } ∣ U l ∣ d ν ( l ) ≥ K − 1 . During the local construction retain trajectories labeled at t ∗ = t I + H u ∗ t_*=t_I+Hu_* t ∗ = t I + H u ∗
F ∗ = F ( t ∗ ) , B l = x l ( t I ) − F ∗ y l ( t I ) . F_*=F(t_*),\qquad B_l=x_l(t_I)-F_*y_l(t_I). F ∗ = F ( t ∗ ) , B l = x l ( t I ) − F ∗ y l ( t I ) . The difference B l − S l ( t ∗ ) B_l-S_l(t_*) B l − S l ( t ∗ ) O ( K H ) O(KH) O ( K H ) H H H C C C B l B_l B l K H − d KH^{-d} K H − d K H d KH^d K H d (5.5) at time t ∗ t_* t ∗ K H − j KH^{-j} K H − j H H H ∣ B l − S l ( t ∗ ) ∣ ≤ K H |B_l-S_l(t_*)|\le KH ∣ B l − S l ( t ∗ ) ∣ ≤ K H C C C H d H^d H d ν C \nu_C ν C ∫ ∣ U l ∣ d ν C < K 1 − 1 \int|U_l|\,d\nu_C<K_1^{-1} ∫ ∣ U l ∣ d ν C < K 1 − 1 K / K 1 K/K_1 K / K 1 K 1 K_1 K 1
For a lower endpoint b C b_C b C
Y l = y l ( t I ) , z l = ( B l − b C ) / H , q l = x l ′ ( t I ) − F ∗ y l ′ , Y_l=y_l(t_I),\qquad z_l=(B_l-b_C)/H,\qquad q_l=x_l'(t_I)-F_*y_l', Y l = y l ( t I ) , z l = ( B l − b C ) / H , q l = x l ′ ( t I ) − F ∗ y l ′ , ϕ ( u ) = ( F ( t I + H u ) − F ∗ ) / H , L l ( u ) = z l + u q l − ϕ ( u ) Y l . \phi(u)=(F(t_I+Hu)-F_*)/H,\qquad L_l(u)=z_l+u q_l-\phi(u)Y_l. ϕ ( u ) = ( F ( t I + H u ) − F ∗ ) / H , L l ( u ) = z l + u q l − ϕ ( u ) Y l . The normalized row ϕ \phi ϕ L l L_l L l Y l Y_l Y l q l q_l q l K K K ϕ \phi ϕ ζ / H \zeta/H ζ / H x 2 , l x_{2,l} x 2 , l
S l ( t I + H u ) − b C H = L l ( u ) + H ( u 2 x 2 , l − u ϕ ( u ) y l ′ ) . (5.18) \frac{S_l(t_I+Hu)-b_C}{H}=L_l(u)+H\bigl(u^2x_{2,l}-u\phi(u)y_l'\bigr).\tag{5.18} H S l ( t I + H u ) − b C = L l ( u ) + H ( u 2 x 2 , l − u ϕ ( u ) y l ′ ) . ( 5.18 ) The final term is O ( K H ) O(KH) O ( K H ) p = H 1 / 4 p = H^{1/4} p = H 1/4 p 0 = H 1 / 8 p_0 = H^{1/8} p 0 = H 1/8 L l ( u ) L_l(u) L l ( u ) d d d
ν C { l : u ∈ U l , ( Y l , L l ( u ) ) in given p -bins } ≤ K p j + d . (5.19) \nu_C\{l : u \in U_l,\ (Y_l,L_l(u))\text{ in given }p\text{-bins}\} \le Kp^{j+d}.\tag{5.19} ν C { l : u ∈ U l , ( Y l , L l ( u )) in given p -bins } ≤ K p j + d . ( 5.19 ) Indeed, in the original variables the corresponding region has tangential widths K p Kp K p K H p KHp K H p K H − j KH^{-j} K H − j H p Hp H p H d H^d H d K p j + d Kp^{j+d} K p j + d (5.19) with p 0 p_0 p 0 t ∗ t_* t ∗
∣ q l − q l ~ ∣ ≤ K ( ∣ Y l − Y l ~ ∣ + H ) . (5.20) |q_l-q_{\widetilde{l}}| \le K(|Y_l-Y_{\widetilde{l}}|+H).\tag{5.20} ∣ q l − q l ∣ ≤ K ( ∣ Y l − Y l ∣ + H ) . ( 5.20 ) Here y l ( t ∗ ) = Y l + O ( K H ) y_l(t_*)=Y_l+O(KH) y l ( t ∗ ) = Y l + O ( K H ) O ( K H ) O(KH) O ( K H ) x l ′ ( t ∗ ) x'_l(t_*) x l ′ ( t ∗ ) x l ′ ( t I ) x'_l(t_I) x l ′ ( t I ) O ( K H ) O(KH) O ( K H ) x l x_l x l O ( K H ) O(KH) O ( K H ) p = H 1 / 4 p=H^{1/4} p = H 1/4 Y l Y_l Y l q l q_l q l ϕ \phi ϕ
Fitting one row component. Slice the other j − 1 j-1 j − 1 Y l Y_l Y l p p p j = 1 j=1 j = 1 C C C K H d p j − 1 KH^dp^{j-1} K H d p j − 1 t ∗ t_* t ∗ H d p j − 1 H^dp^{j-1} H d p j − 1 ν s \nu_s ν s
ν s ( all trajectories ) ≤ K , ∫ ∣ U l ∣ d ν s ( l ) ≥ K − 1 . \nu_s(\text{all trajectories}) \le K,\qquad\int|U_l|\,d\nu_s(l) \ge K^{-1}. ν s ( all trajectories ) ≤ K , ∫ ∣ U l ∣ d ν s ( l ) ≥ K − 1 . Let v l v_l v l Y l Y_l Y l q l q_l q l v v v (5.20) gives a function G G G
∣ G ( v ) − G ( v ′ ) ∣ ≤ K ( ∣ v − v ′ ∣ + p ) . |G(v)-G(v')| \le K(|v-v'|+p). ∣ G ( v ) − G ( v ′ ) ∣ ≤ K ( ∣ v − v ′ ∣ + p ) . Translate L l ( u ) L_l(u) L l ( u ) ϕ ( u ) \phi(u) ϕ ( u )
L l a ( u ) = z l + u G ( v l ) − ϕ 1 ( u ) v l + O ( K p ) . (5.21) L_l^a(u)=z_l+uG(v_l)-\phi_1(u)v_l+O(Kp).\tag{5.21} L l a ( u ) = z l + u G ( v l ) − ϕ 1 ( u ) v l + O ( K p ) . ( 5.21 ) They have the local upper d d d
ν s { l : u ∈ U l , ( v l , L l a ( u ) ) in given p -bins } ≤ K p 1 + d . (5.22) \nu_s\{l:u\in U_l,\ (v_l,L_l^a(u))\text{ in given }p\text{-bins}\}\le Kp^{1+d}.\tag{5.22} ν s { l : u ∈ U l , ( v l , L l a ( u )) in given p -bins } ≤ K p 1 + d . ( 5.22 ) Choose u 0 u_0 u 0 ∫ 1 { u 0 ∈ U l } ∣ U l ∣ d ν s ≥ K − 1 \int1_{\{u_0\in U_l\}}|U_l|\,d\nu_s\ge K^{-1} ∫ 1 { u 0 ∈ U l } ∣ U l ∣ d ν s ≥ K − 1 u 0 u_0 u 0 u u u L l a ( u ) L_l^a(u) L l a ( u ) d ν s ( l ) d ν s ( l ′ ) d u d\nu_s(l)d\nu_s(l')\,du d ν s ( l ) d ν s ( l ′ ) d u K p − d Kp^{-d} K p − d p d / K p^d/K p d / K u 0 u_0 u 0 p 2 d / K p^{2d}/K p 2 d / K (5.21) implies
( u − u 0 ) ( G ( v l ) − G ( v l ′ ) ) − ( ϕ 1 ( u ) − ϕ 1 ( u 0 ) ) ( v l − v l ′ ) = L l ′ a ( u 0 ) − L l a ( u 0 ) + O ( K p ) . (5.23) (u-u_0)(G(v_l)-G(v_{l'}))-(\phi_1(u)-\phi_1(u_0))(v_l-v_{l'})
= L_{l'}^a(u_0)-L_l^a(u_0)+O(Kp).\tag{5.23} ( u − u 0 ) ( G ( v l ) − G ( v l ′ )) − ( ϕ 1 ( u ) − ϕ 1 ( u 0 )) ( v l − v l ′ ) = L l ′ a ( u 0 ) − L l a ( u 0 ) + O ( K p ) . ( 5.23 ) At fixed u u u p p p v l , v l ′ v_l,v_{l'} v l , v l ′ K p 1 + d Kp^{1+d} K p 1 + d K K K ( u , v l , v l ′ ) (u,v_l,v_{l'}) ( u , v l , v l ′ ) K p 3 + 2 d Kp^{3+2d} K p 3 + 2 d u u u K p 2 d Kp^{2d} K p 2 d (5.22) over the tangential bins for the first starting value and the second matching value.
For each time p p p J J J T J ⊂ J T_J\subset J T J ⊂ J ∣ T J ∣ < p / K 1 |T_J|<p/K_1 ∣ T J ∣ < p / K 1 T J T_J T J O ( p − 1 ) O(p^{-1}) O ( p − 1 ) K p 2 d / K 1 Kp^{2d}/K_{1} K p 2 d / K 1 K 1 K_{1} K 1
a = ( u − u 0 , − ϕ 1 ( u ) + ϕ 1 ( u 0 ) ) , b = ( G ( v l ) , v l ) , b ′ = ( G ( v l ′ ) , v l ′ ) a=(u-u_{0},-\phi_{1}(u)+\phi_{1}(u_{0})),\qquad b=(G(v_{l}),v_{l}),\qquad b'=(G(v_{l'}),v_{l'}) a = ( u − u 0 , − ϕ 1 ( u ) + ϕ 1 ( u 0 )) , b = ( G ( v l ) , v l ) , b ′ = ( G ( v l ′ ) , v l ′ ) then have p p p p − 3 / K p^{-3}/K p − 3 / K
Retain a quadrant for a a a H \mathcal{H} H A \mathcal{A} A B \mathcal{B} B H ⊂ A × B × B \mathcal{H}\subset\mathcal{A}\times\mathcal{B}\times\mathcal{B} H ⊂ A × B × B K − 1 K^{-1} K − 1 1 1 1 [ 0 , 1 ] 2 [0,1]^2 [ 0 , 1 ] 2 b − b ′ b-b' b − b ′ (5.17) with a subpower constant: covering a horizontal interval for the first graph, or a vertical interval for the second, by s s s K w / s Kw/s K w / s
Fix 0 < ε < 1 − d 0<\varepsilon<1-d 0 < ε < 1 − d N N N p ± η p^{\pm\eta} p ± η (5.23) , all dot products are within K p Kp K p K ( ∣ J ∣ / s ) d s K(|J|/s)^d s K ( ∣ J ∣/ s ) d s J J J
( ∣ J ∣ / s ) 1 − ε − d ≤ K , ∣ J ∣ / s ≥ p − η , (|J|/s)^{1-\varepsilon-d}\le K,\qquad|J|/s\ge p^{-\eta}, ( ∣ J ∣/ s ) 1 − ε − d ≤ K , ∣ J ∣/ s ≥ p − η , a contradiction. The additive error and inverse dilations cost only subpowers at s ≥ p s\ge p s ≥ p
We therefore obtain at least p ε − 1 / K p^{\varepsilon-1}/K p ε − 1 / K ϕ 1 \phi_{1} ϕ 1 K K K p ε / K p^{\varepsilon}/K p ε / K K p Kp K p J I \mathcal{J}_{I} J I J ∈ J I J\in\mathcal{J}_{I} J ∈ J I T J T_J T J K p Kp K p p p p K K K
T I = t I + H ⋃ J ∈ J I T J . T_I=t_I+H\bigcup_{J\in\mathcal{J}_I}T_J. T I = t I + H J ∈ J I ⋃ T J . Every T J T_J T J p / K 1 p/K_1 p / K 1
∣ T I ∣ ≥ H p ε / K . |T_I|\ge Hp^\varepsilon/K. ∣ T I ∣ ≥ H p ε / K . In particular, the fitting set consists of prepared active times, not the whole selected bins. Let ε \varepsilon ε η , δ \eta,\delta η , δ N N N ∣ T I ∣ ≥ H / K |T_I|\ge H/K ∣ T I ∣ ≥ H / K K p Kp K p
In the original variables this is an affine component fit to F F F K H 5 / 4 KH^{5/4} K H 5/4 T I T_I T I all prepared labels E l p r e E_l^{\mathrm{pre}} E l pre
∑ I s u r v i v i n g ∫ T I m ( t ) d t ≥ τ ∑ I ∣ T I ∣ ≥ τ δ 2 H H K = δ τ 2 K . (5.24) \sum_{I\ \mathrm{surviving}}\int_{T_I}m(t)\,\mathrm{d}t\ge\tau\sum_I|T_I|\ge\tau\frac{\delta}{2H}\frac{H}{K}=\frac{\delta\tau}{2K}.\tag{5.24} I surviving ∑ ∫ T I m ( t ) d t ≥ τ I ∑ ∣ T I ∣ ≥ τ 2 H δ K H = 2 K δ τ . ( 5.24 ) This is inverse-subpower. It uses both the instantaneous floor and the number of surviving blocks, each established before the local search. For j = 2 j=2 j = 2 m , δ , τ m,\delta,\tau m , δ , τ
Subsequent label restrictions preserve the earlier fits and all upper inputs. We obtain affine rows P I ( t ) P_I(t) P I ( t ) A I A_I A I K K K
∣ F ( t ) − P I ( t ) ∣ ≤ K H 5 / 4 (5.25) \lvert F(t)-P_I(t)\rvert\le K H^{5/4}\tag{5.25} ∣ F ( t ) − P I ( t )∣ ≤ K H 5/4 ( 5.25 ) on every retained active time in the corresponding interval. Let I ∗ ( t ) I_*(t) I ∗ ( t ) t t t
A ( t ) = A I ∗ ( t ) A(t)=A_{I_*(t)} A ( t ) = A I ∗ ( t ) on the finest blocks with retained labels, and set A ( t ) = 0 A(t)=0 A ( t ) = 0 K K K
Comparison across lengths. We now compare the affine fits through the intermediate lengths fixed before any selections. Let μ f i t \mu_{\mathrm{fit}} μ fit H H H I I I μ f i t ( I ) < H / K 1 \mu_{\mathrm{fit}}(I)<H/K_1 μ fit ( I ) < H / K 1 1 / K 1 1/K_1 1/ K 1 K 1 K_1 K 1 μ f i t \mu_{\mathrm{fit}} μ fit H ′ / K 1 H'/K_1 H ′ / K 1 ν \nu ν H ′ / K 1 H'/K_1 H ′ / K 1 H ′ / K H'/K H ′ / K
∣ A I ′ − A I ∣ ≤ K H 5 / 4 H ′ ≤ K H 7 / 32 . \lvert A_{I'}-A_I\rvert\le K\frac{H^{5/4}}{H'} \le K H^{7/32}. ∣ A I ′ − A I ∣ ≤ K H ′ H 5/4 ≤ K H 7/32 . Summing through the fixed finite chain compares the finest slope A ( u ) A(u) A ( u ) A I A_I A I K H 7 / 32 K H^{7/32} K H 7/32 (5.25) , this gives
∣ F ( v ) − F ( u ) − ( v − u ) A ( u ) ∣ ≤ K ( H 5 / 4 + H 1 + 7 / 32 ) ≤ K H 11 / 10 . \lvert F(v)-F(u)-(v-u)A(u)\rvert\le K\left(H^{5/4}+H^{1+7/32}\right)\le K H^{11/10}. ∣ F ( v ) − F ( u ) − ( v − u ) A ( u )∣ ≤ K ( H 5/4 + H 1 + 7/32 ) ≤ K H 11/10 . This proves Equation (5.15), which persists under the label restrictions below.
Concentrating the residual slopes. The affine fits now hold on a dense set of incidences. We next restrict the trajectory labels so that their residual slopes admit short lists once the time and scalar value are known. At the finest length H = H ∗ H=H_* H = H ∗ C C C p 0 = H 1 / 8 p_0=H^{1/8} p 0 = H 1/8 ϕ \phi ϕ L l L_l L l
k l = q l − A I Y l k_l=q_l-A_IY_l k l = q l − A I Y l and error at most K p 0 Kp_0 K p 0 K p 0 Kp_0 K p 0 Y Y Y p 0 p_0 p 0 γ = ( 1 − d ) / 4 \gamma=(1-d)/4 γ = ( 1 − d ) /4 p 0 γ p_0^\gamma p 0 γ K − 1 K^{-1} K − 1
If this failed, there would be a sequence of groups and a fixed c > 0 c>0 c > 0 N − c N^{-c} N − c τ = N − c / 4 \tau=N^{-c/4} τ = N − c /4 τ \tau τ Y Y Y p 0 j τ p_0^j\tau p 0 j τ K τ K\tau K τ p 0 d / K p_0^d/K p 0 d / K
For a fixed first trajectory, second trajectories with slope within p 0 j p_0^j p 0 j
K N − c τ − 2 p 0 − j K N^{-c}\tau^{-2}p_0^{-j} K N − c τ − 2 p 0 − j retained Y Y Y K p 0 Kp_0 K p 0 τ \tau τ K N − c / τ KN^{-c}/\tau K N − c / τ p 0 j τ p_0^j\tau p 0 j τ K p 0 j + d Kp_0^{j+d} K p 0 j + d (5.19) . Thus the close-slope collision contribution is at most
K p 0 d N − c τ − 2 = K p 0 d N − c / 2 . Kp_0^dN^{-c}\tau^{-2}=Kp_0^dN^{-c/2}. K p 0 d N − c τ − 2 = K p 0 d N − c /2 . For the other pairs, the two affine approximations can agree to K p 0 Kp_0 K p 0 K p 0 1 − γ Kp_0^{1-\gamma} K p 0 1 − γ o ( p 0 d / K ) o(p_0^d/K) o ( p 0 d / K ) 1 − γ > d 1-\gamma>d 1 − γ > d
Keep one such slope interval in each good group. Since
J 1 ( l , t ) − k l = H ( 2 u x 2 , l − ϕ ( u ) y l ′ − u A I y l ′ ) = O ( K H ) , (5.26) J_1(l,t)-k_l=H(2ux_{2,l}-\phi(u)y'_l-uA_Iy'_l)=O(KH),\tag{5.26} J 1 ( l , t ) − k l = H ( 2 u x 2 , l − ϕ ( u ) y l ′ − u A I y l ′ ) = O ( K H ) , ( 5.26 ) this gives a deterministic interval cover for J 1 J_1 J 1 t t t S l ( t ) H S_l(t)_H S l ( t ) H K K K C C C ∣ B l − S l ( t ) ∣ ≤ K H |B_l-S_l(t)|\le KH ∣ B l − S l ( t ) ∣ ≤ K H J 1 J_1 J 1 K K K
K p 0 γ = K H ( 1 − d ) / 32 . Kp_0^\gamma=KH^{(1-d)/32}. K p 0 γ = K H ( 1 − d ) /32 . At any reading scale r ≥ H ( 1 − d ) / 64 r\ge H^{(1-d)/64} r ≥ H ( 1 − d ) /64 K K K r r r T T T J 1 J_1 J 1 K K K r r r T , l r T,l_r T , l r (5.14) and (5.11) anew on each dense tested event, with fine scalar S H S_H S H S r S_r S r (5.16) . The argument rederives the entropy estimate on the specified event, using covers that persist under restriction. Together with the Taylor estimate already proved, this completes the proposition.
Higher jets and polynomial fits
Fix 0 ≤ d < 1 0\le d<1 0 ≤ d < 1 M ≥ 2 M\ge2 M ≥ 2 h = N − c + o ( 1 ) h=N^{-c+o(1)} h = N − c + o ( 1 ) c > 0 c>0 c > 0 M M M h / K h/K h / K H H H H 1.01 H^{1.01} H 1.01
Define
c 0 = .1 , r i = min ( .009 , c i − 1 / 2 ) , c i = r i / 2 ( 1 ≤ i ≤ M − 1 ) , c_0=.1,\qquad r_i=\min(.009,c_{i-1}/2),\qquad c_i=r_i/2\quad(1\le i\le M-1), c 0 = .1 , r i = min ( .009 , c i − 1 /2 ) , c i = r i /2 ( 1 ≤ i ≤ M − 1 ) , and put
r min = min 1 ≤ i ≤ M − 1 r i = .009 4 − ( M − 2 ) , α 0 = ( 1 − d ) / 64. r_{\min}=\min_{1\le i\le M-1}r_i=.009\,4^{-(M-2)},\qquad\alpha_0=(1-d)/64. r m i n = 1 ≤ i ≤ M − 1 min r i = .009 4 − ( M − 2 ) , α 0 = ( 1 − d ) /64. Both numbers are positive and depend only on M , d M,d M , d h h h
h ≫ H 1 ≫ H 2 ≫ ⋯ ≫ H M − 1 ≫ H ∗ , h\gg H_1\gg H_2\gg\cdots\gg H_{M-1}\gg H_*, h ≫ H 1 ≫ H 2 ≫ ⋯ ≫ H M − 1 ≫ H ∗ , by taking at each step the largest dyadic length satisfying the corresponding inequality
H 1 r min ≤ h M + 11 , H_1^{r_{\min}}\le h^{M+11}, H 1 r m i n ≤ h M + 11 , H i r min ≤ H i − 1 1.02 ( 2 ≤ i ≤ M − 1 ) , H_i^{r_{\min}}\le H_{i-1}^{1.02}\qquad(2\le i\le M-1), H i r m i n ≤ H i − 1 1.02 ( 2 ≤ i ≤ M − 1 ) , H ∗ α 0 ≤ H M − 1 1.02 . (5.27) H_*^{\alpha_0}\le H_{M-1}^{1.02}.\tag{5.27} H ∗ α 0 ≤ H M − 1 1.02 . ( 5.27 ) Choose the available floor to satisfy
ζ ≤ K − 1 min { H ∗ 3 , h M + 1 / 2 } . (5.28) \zeta\le K^{-1}\min\{H_*^3,h^{M+1/2}\}.\tag{5.28} ζ ≤ K − 1 min { H ∗ 3 , h M + 1/2 } . ( 5.28 ) These choices use finite fixed power ratios. We will verify the required entropy readings in the proof. The strict exponent margins absorb dyadic rounding and subpower factors. When specifying the floor by a fixed power, one may place it a further fixed power of h h h K K K
Proposition 5.6 (Higher-jet polynomial fitting). For the parameters M , d , h M,d,h M , d , h (5.28) . Then a dense set of labeled incidences ( l , t ) (l,t) ( l , t ) P P P M M M K K K h h h t t t
∣ F ( v ) − P ( v ) ∣ ≤ K h M + 1 / 4 |F(v)-P(v)|\le Kh^{M+1/4} ∣ F ( v ) − P ( v ) ∣ ≤ K h M + 1/4 on labeled times v ∈ E l v\in E_l v ∈ E l h / K h/K h / K v = t v=t v = t
Proof. Use the hierarchy and floor fixed in Equations (5.27) and (5.28) . Apply Proposition 5.4, inserting its finite intermediate Taylor scales before making any label selections. We retain the resulting single-time labels and the row A ( t ) A(t) A ( t )
An independent time tree. Choose u u u [ 0 , 1 ] [0,1] [ 0 , 1 ] H 1 H_1 H 1 H 2 H_2 H 2 M − 1 M-1 M − 1 t ′ t' t ′ h h h u u u T T T t ′ t' t ′
All tested times are labeled with dense probability. Here is a direct verification. For a fixed trajectory and a fixed containing interval, write a a a a 2 M − 1 a^{2^{M-1}} a 2 M − 1 h h h t ′ t' t ′ a a a ( 2 M − 1 + 1 ) (2^{M-1}+1) ( 2 M − 1 + 1 )
We may also require that every pair gap at scale H i H_i H i H i / K H_i/K H i / K 2 / K 2/K 2/ K K K K
Inductive information statement. Put F 0 = F F_0=F F 0 = F F 1 = A F_1=A F 1 = A t t t F 1 ( t ) , … , F m ( t ) F_1(t),\ldots,F_m(t) F 1 ( t ) , … , F m ( t )
J i ( t ) = x l ( i ) ( t ) − F i ( t ) y l ( t ) − i F i − 1 ( t ) y l ′ , 1 ≤ i ≤ m . (5.29) J_i(t)=x_l^{(i)}(t)-F_i(t)y_l(t)-iF_{i-1}(t)y_l',
\qquad1\le i\le m.\tag{5.29} J i ( t ) = x l ( i ) ( t ) − F i ( t ) y l ( t ) − i F i − 1 ( t ) y l ′ , 1 ≤ i ≤ m . ( 5.29 ) Every row is a function of its subtree times and the world only. In particular it is independent of the trajectory and does not use the separate time t ′ t' t ′ K K K
H ( J i ( t ) r ∣ T , S l ( t ) r ) = o ∗ (5.30) \mathsf{H}(J_i(t)r \mid T,S_l(t)r)=o_*\tag{5.30} H ( J i ( t ) r ∣ T , S l ( t ) r ) = o ∗ ( 5.30 ) on each further dense event in the original trajectory–tuple experiment. At the leaves this is (5.16) for the fixed row A A A t ′ t' t ′ T T T
Consider a pair t , w t,w t , w H H H m m m q = H 1.01 q=H^{1.01} q = H 1.01 q q q x l , y l x_l,y_l x l , y l (5.15) , give
S l ( w ) = S l ( t ) + ( w − t ) J 1 ( t ) + O ( K H 1.1 ) . (5.31) S_l(w)=S_l(t)+(w-t)J_1(t)+O(KH^{1.1}).\tag{5.31} S l ( w ) = S l ( t ) + ( w − t ) J 1 ( t ) + O ( K H 1.1 ) . ( 5.31 ) The error is smaller than q q q S l ( w ) q S_l(w)_q S l ( w ) q q q q T , S l ( t ) q T,S_l(t)_q T , S l ( t ) q
∇ Z = Z ( w ) − Z ( t ) w − t , \nabla Z=\frac{Z(w)-Z(t)}{w-t}, ∇ Z = w − t Z ( w ) − Z ( t ) , where endpoint rows use their own subtree data. Since ∣ w − t ∣ ≥ H / K \lvert w-t\rvert\ge H/K ∣ w − t ∣ ≥ H / K ∇ J i \nabla J_i ∇ J i K H .01 KH^{.01} K H .01 o ∗ o_* o ∗ T , S l ( t ) q T,S_l(t)_q T , S l ( t ) q
For i ≥ 1 i\ge1 i ≥ 1 x l x_l x l y l y_l y l
∇ J i = x l ( i + 1 ) ( t ) − ( ∇ F i ) y l ( t ) − ( F i ( w ) + i ∇ F i − 1 ) y l ′ . (5.32) \nabla J_i=x_l^{(i+1)}(t)-(\nabla F_i)y_l(t)-(F_i(w)+i\nabla F_{i-1})y_l'.\tag{5.32} ∇ J i = x l ( i + 1 ) ( t ) − ( ∇ F i ) y l ( t ) − ( F i ( w ) + i ∇ F i − 1 ) y l ′ . ( 5.32 ) For i = 1 i=1 i = 1 x l ′ x_l' x l ′ x l ′ ′ x_l'' x l ′′ i ≥ 2 i\ge2 i ≥ 2 (5.32) are simply the exact product difference for an affine y l y_l y l
Proceed through i = 1 , … , m i=1,\ldots,m i = 1 , … , m
∇ F i − 1 = F i ( t ) + O ( K H c i − 1 ) , c i − 1 > 0. (5.33) \nabla F_{i-1}=F_i(t)+O(KH^{c_{i-1}}),\qquad c_{i-1}>0.\tag{5.33} ∇ F i − 1 = F i ( t ) + O ( K H c i − 1 ) , c i − 1 > 0. ( 5.33 ) Initially (5.15) , divided by the pair gap, supplies the chosen exponent c 0 = .1 c_0=.1 c 0 = .1
First, ∇ F i \nabla F_i ∇ F i ∣ ∇ F i ∣ ≥ N α \lvert\nabla F_i\rvert\ge N^\alpha ∣ ∇ F i ∣ ≥ N α α > 0 \alpha>0 α > 0 (5.32) by that norm. The other displayed coefficients are bounded by K K K (5.33) , and x l ( i + 1 ) , y l ′ x_l^{(i+1)},y_l' x l ( i + 1 ) , y l ′ K K K y l ( t ) y_l(t) y l ( t ) T T T ( ∇ F i ) / ∣ ∇ F i ∣ (\nabla F_i)/\lvert\nabla F_i\rvert ( ∇ F i ) / ∣ ∇ F i ∣ q q q H = N − b + o ( 1 ) H=N^{-b+o(1)} H = N − b + o ( 1 ) N − β N^{-\beta} N − β 0 < β < min ( α , b ) 0<\beta<\min(\alpha,b) 0 < β < min ( α , b ) q q q (5.12) on that very event. It follows that every fixed-power tail is negligible relative to the current dense tuple mass, and a slow choice of cutoffs yields ∣ ∇ F i ∣ ≤ K \lvert\nabla F_i\rvert\le K ∣ ∇ F i ∣ ≤ K
We now also have F i ( w ) − F i ( t ) = O ( K H ) F_i(w)-F_i(t)=O(KH) F i ( w ) − F i ( t ) = O ( K H )
r i = min ( .009 , c i − 1 / 2 ) > 0. r_i=\min(.009,c_{i-1}/2)>0. r i = min ( .009 , c i − 1 /2 ) > 0. Equations (5.32) and (5.33) encode
U i + 1 = x l ( i + 1 ) ( t ) − ( ∇ F i ) y l ( t ) − ( i + 1 ) F i ( t ) y l ′ (5.34) U_{i+1}=x_l^{(i+1)}(t)-(\nabla F_i)y_l(t)-(i+1)F_i(t)y_l'\tag{5.34} U i + 1 = x l ( i + 1 ) ( t ) − ( ∇ F i ) y l ( t ) − ( i + 1 ) F i ( t ) y l ′ ( 5.34 ) to width H r i H^{r_i} H r i o ∗ o_* o ∗ T T T S l ( t ) q S_l(t)_q S l ( t ) q K H .01 KH^{.01} K H .01 O ( K H ) O(KH) O ( K H ) O ( K H c i − 1 ) O(KH^{c_i-1}) O ( K H c i − 1 )
If i < m i < m i < m J i + 1 ( t ) J_{i+1}(t) J i + 1 ( t ) ( ∇ F i − F i + 1 ( t ) ) y l ( t ) (\nabla F_i-F_{i+1}(t))y_l(t) ( ∇ F i − F i + 1 ( t )) y l ( t ) H r i / 2 H^{r_i/2} H r i /2 H r i / 3 H^{r_i/3} H r i /3 (5.12) on any dense such event. After a negligible deletion,
∇ F i − F i + 1 ( t ) = O ( K H r i / 2 ) . \nabla F_i-F_{i+1}(t)=O(KH^{r_i/2}). ∇ F i − F i + 1 ( t ) = O ( K H r i /2 ) . This proves the compatibility assertion with the chosen c i = r i / 2 c_i=r_i/2 c i = r i /2 i i i
For i = m i=m i = m F m + 1 ( t ) = ∇ F m F_{m+1}(t)=\nabla F_m F m + 1 ( t ) = ∇ F m t t t t ′ t' t ′ U m + 1 U_{m+1} U m + 1 r = H r m r=H^{r_m} r = H r m T T T S l ( t ) q S_l(t)_q S l ( t ) q T T T l r l_r l r K K K (5.14) with
Z = J m + 1 ( t ) r , A = S l ( t ) q , B = l r , C = S l ( t ) r Z=J_{m+1}(t)_r,\qquad A=S_l(t)_q,\qquad B=l_r,\qquad C=S_l(t)_r Z = J m + 1 ( t ) r , A = S l ( t ) q , B = l r , C = S l ( t ) r and invoking Equation (5.11) on the current event gives Equation (5.30) for the new jet. The older jets retain their hereditary assertions.
There are finitely many rows and discrepancy tests for fixed M M M (5.14) , and Lemma 5.3. We never infer hereditary entropy by conditioning an earlier average. These cutoff tests concern rows already defined from the time tree. They do not introduce a dependence on the trajectory or on t ′ t' t ′
We can now verify the schedule. A jet formed at a pair scale H i H_i H i H i r m H_i^{r_m} H i r m m m m r m ≥ r min r_m\ge r_{\min} r m ≥ r m i n
K H 1 r m ≤ K H 1 r min ≤ K h M + 11 ≪ p : = h M + 10 . KH_1^{r_m}\le KH_1^{r_{\min}}\le Kh^{M+11}\ll p:=h^{M+10}. K H 1 r m ≤ K H 1 r m i n ≤ K h M + 11 ≪ p := h M + 10 . At a child scale H i H_i H i i ≥ 2 i\ge2 i ≥ 2
K H i r m ≤ K H i − 1 1.02 ≪ H i − 1 1.01 , KH_i^{r_m}\le KH_{i-1}^{1.02}\ll H_{i-1}^{1.01}, K H i r m ≤ K H i − 1 1.02 ≪ H i − 1 1.01 , which supplies the reading q q q (5.16) applies at every reading at least H ∗ α 0 H_*^{\alpha_0} H ∗ α 0 (5.27) , this is finer than H M − 1 1.01 H_{M-1}^{1.01} H M − 1 1.01 ζ ≤ H ∗ 3 / K \zeta\le H_*^3/K ζ ≤ H ∗ 3 / K H ∗ H_* H ∗ q q q
Testing the polynomial at a separate time. At the root u u u
P F ( v ) = ∑ i = 0 M F i ( u ) i ! ( v − u ) i , E = F ( t ′ ) − P F ( t ′ ) , J 0 ( u ) = S l ( u ) . P_F(v)=\sum_{i=0}^{M}\frac{F_i(u)}{i!}(v-u)^i,\qquad E=F(t')-P_F(t'),\qquad J_0(u)=S_l(u). P F ( v ) = i = 0 ∑ M i ! F i ( u ) ( v − u ) i , E = F ( t ′ ) − P F ( t ′ ) , J 0 ( u ) = S l ( u ) . The coefficients of P F P_F P F K K K v v v M M M u ∈ [ 0 , 1 ] u \in[0,1] u ∈ [ 0 , 1 ] t ′ t' t ′ x l x_l x l y l y_l y l
S l ( t ′ ) = ∑ i = 0 M J i ( u ) i ! ( t ′ − u ) i − E y l ( t ′ ) + O ( K h M + 1 ) . (5.35) S_l(t') = \sum_{i=0}^{M} \frac{J_i(u)}{i!}(t'-u)^i - E y_l(t') + O(Kh^{M+1}).\tag{5.35} S l ( t ′ ) = i = 0 ∑ M i ! J i ( u ) ( t ′ − u ) i − E y l ( t ′ ) + O ( K h M + 1 ) . ( 5.35 ) The last term is the single extra degree produced when the degree M M M P F P_F P F y l y_l y l
Suppose ∣ E ∣ ≥ h M + 1 / 4 |E| \ge h^{M+1/4} ∣ E ∣ ≥ h M + 1/4 T , S l ( u ) p T,S_l(u)_p T , S l ( u ) p p p p o ∗ o_* o ∗ (5.35) places S l ( t ′ ) S_l(t') S l ( t ′ ) K ∣ E ∣ K|E| K ∣ E ∣ r = h 1 / 4 r = h^{1/4} r = h 1/4 (5.4) allows at most K r − d Kr^{-d} K r − d ∣ E ∣ r |E|r ∣ E ∣ r T T T
∣ E ∣ r ≥ h M + 1 / 4 h 1 / 4 = h M + 1 / 2 ≥ K ζ , |E|r \ge h^{M+1/4}h^{1/4} = h^{M+1/2} \ge K\zeta, ∣ E ∣ r ≥ h M + 1/4 h 1/4 = h M + 1/2 ≥ K ζ , by (5.28) ; its containing interval is larger still. Thus both are within the available scale range; dyadic enlargement or the unit cutoff costs only K K K
After division by ∣ E ∣ |E| ∣ E ∣ O ( K h 3 / 4 ) O(Kh^{3/4}) O ( K h 3/4 ) O ( r ) O(r) O ( r ) y l ( t ′ ) = y l ( u ) + O ( K h ) y_l(t') = y_l(u) + O(Kh) y l ( t ′ ) = y l ( u ) + O ( K h ) J \mathcal{J} J p p p T , S l ( u ) p , J T,S_l(u)_p,\mathcal{J} T , S l ( u ) p , J r r r ( E / ∣ E ∣ ) y l ( u ) (E/|E|)y_l(u) ( E /∣ E ∣ ) y l ( u ) K r − d Kr^{-d} K r − d H ( J ∣ T , S l ( u ) p ) = o ∗ \mathsf{H}(\mathcal{J} \mid T,S_l(u)_p) = o_* H ( J ∣ T , S l ( u ) p ) = o ∗
H ( ( ( E / ∣ E ∣ ) y l ( u ) ) r ∣ T , S l ( u ) p ) ≤ d log ( 1 / r ) + o ∗ . \mathsf{H}(((E/|E|)y_l(u))_r \mid T,S_l(u)_p) \le d\log(1/r) + o_*. H ((( E /∣ E ∣ ) y l ( u ) ) r ∣ T , S l ( u ) p ) ≤ d log ( 1/ r ) + o ∗ . (5.12) , applied on this same event, gives the lower bound log ( 1 / r ) − o ∗ \log(1/r)-o_* log ( 1/ r ) − o ∗ d < 1 d < 1 d < 1 h h h ∣ F ( t ′ ) − P F ( t ′ ) ∣ ≤ K h M + 1 / 4 |F(t')-P_F(t')| \le Kh^{M+1/4} ∣ F ( t ′ ) − P F ( t ′ ) ∣ ≤ K h M + 1/4
Recovering labeled time fibers. Let z z z t ′ t' t ′ P F P_F P F z z z t ′ t' t ′ h h h p g ≥ K − 1 p_g \ge K^{-1} p g ≥ K − 1 g ( z ) g(z) g ( z ) t ′ t' t ′ ∫ g ( z ) d P ( z ) = p g \int g(z)\,dP(z) = p_g ∫ g ( z ) d P ( z ) = p g δ \delta δ
∫ { g < p g δ } g ( z ) d P ( z ) ≤ p g δ . \int_{\{g<p_g\delta\}} g(z)\,dP(z) \le p_g\delta. ∫ { g < p g δ } g ( z ) d P ( z ) ≤ p g δ . Choose δ → 0 \delta\to0 δ → 0 h p g δ ≥ h / K hp_g\delta\ge h/K h p g δ ≥ h / K
The pre-restriction marginal law of ( l , t ′ ) (l,t') ( l , t ′ ) u u u h h h t ′ t' t ′ ( l , t ′ ) (l,t') ( l , t ′ ) h / K h/K h / K h M + 1 / 4 h^{M+1/4} h M + 1/4
Every construction began with upper inputs and dense mass, so it may be performed anew on a dense residual. This proves the proposition.
The isotropic improvement
This section treats the one-spatial-coordinate model and the isotropic two-spatial-coordinate model. Its purpose is to turn incidence information along trajectories into a quadratic test with a smaller thickness and the same time interval. We first construct a velocity graph, prove that its coefficients have polynomial fits along many trajectories, and interpolate these fits to a polynomial field. A switched-line argument makes that field approximately conservative. The final step restores the degree allowed for a chart. In two spatial coordinates, broad directions can still lie on a conic; the higher-order terms they fail to detect are expressed through the hidden coordinate to obtain a quadratic test.
Theorem 6.1 (Isotropic improvement). Consider the tangent data of Proposition 4.7, with 0 < k ≤ 1 0 < k \le1 0 < k ≤ 1 0 ≤ d < 1 0 \le d < 1 0 ≤ d < 1 j = 1 j = 1 j = 1 j = 2 j = 2 j = 2 ℓ = 0 \ell= 0 ℓ = 0 Q = Q s Q = Q_s Q = Q s a = N − s a = N^{-s} a = N − s h ∼ log , N a k h \sim_{\log,N} a^k h ∼ l o g , N a k
a 2 = a h e 2 , e 2 > 0. a_2 = a h^{e_2}, \qquad e_2 > 0. a 2 = a h e 2 , e 2 > 0. The candidate has incidence mass at least
h ν Q K ( a 2 a ) d , K = N o ( 1 ) . \frac{h\nu_Q}{K}\left(\frac{a_2}{a}\right)^d, \qquad K = N^{o(1)}. K h ν Q ( a a 2 ) d , K = N o ( 1 ) . Writing the candidate as P n e w P_{\mathrm{new}} P new ∣ P n e w ∣ ≤ K a 2 |P_{\mathrm{new}}| \le K a_2 ∣ P new ∣ ≤ K a 2 ∣ ∂ w P n e w ∣ ≤ K |\partial_w P_{\mathrm{new}}| \le K ∣ ∂ w P new ∣ ≤ K h h h K − 1 K^{-1} K − 1 K / a 2 K/a_2 K / a 2 w − F ( t , y ) w - F(t,y) w − F ( t , y ) F F F ( t , y , w ) (t,y,w) ( t , y , w )
Here substantial refers to positive probability in the original reference path, whereas a dense discovery family may have only inverse-subpower probability. The distinction is needed when Proposition 4.11 converts candidates into a known atlas. We do not use an arbitrary tangent subpower as an actual N 0 N_0 N 0
Measures, scales, and inherited estimates
Write
z = ( t , y ) , m = 1 + j , v = ( 1 , V ) , W = ( z , X ) . z = (t,y), \qquad m = 1 + j, \qquad\mathbf{v} = (1,V), \qquad W = (z,X). z = ( t , y ) , m = 1 + j , v = ( 1 , V ) , W = ( z , X ) . A parent world and its label are included in every state below. The parent prior is ν \nu ν λ \lambda λ
d λ ( l , t ) = G w t ( l , t ) d ν ( l ) d t , 0 ≤ G w t ≤ K . (6.1) d\lambda(l,t) = G_{\mathrm{wt}}(l,t)\,d\nu(l)\,dt, \qquad0 \le G_{\mathrm{wt}} \le K.\tag{6.1} d λ ( l , t ) = G wt ( l , t ) d ν ( l ) d t , 0 ≤ G wt ≤ K . ( 6.1 ) All initial losses are summed in the path mixture of parents. If a line portion has weighted duration at least H / K 1 H/K_1 H / K 1 H / K 1 H/K_1 H / K 1 H / ( K K 1 ) H/(K K_1) H / ( K K 1 )
All exponent ranges are fixed first, with room for finer comparisons. A finite list of pure and joint caps is prepared before sparse selection. Lists of exponents subsequently increase slowly enough that products of their subpower losses remain subpower. Deletion budgets are assigned for the whole finite list before it is enlarged. We use successively slower grids for physical counts, parameter estimates, field comparisons, strip filling, and residual comparisons. Interpolation between neighboring dyadic widths whose ratio is at most K K K
We use the following consequences of Proposition 4.7 and Lemma 4.8. For every required fixed power width p p p X X X K p d Kp^d K p d z z z π \pi π K π ( J ) K\pi(J) K π ( J ) J J J
The true packets Q u Q_u Q u p = N − u p=N^{-u} p = N − u H ∼ log , N p k H\sim_{\log,N}p^k H ∼ l o g , N p k
ν Q ∼ log , N p d H j . (6.2) \nu_Q\sim_{\log,N}p^dH^j.\tag{6.2} ν Q ∼ l o g , N p d H j . ( 6.2 ) Their assigned trajectory sets are disjoint in each time block, and assignments are invariant throughout that block. All spatial axes have length H H H K K K K K K K H ν Q KH\nu_Q K H ν Q K K K
Physical support and parameter masses
The physical support will have exponent m + d m+d m + d m + 1 m+1 m + 1 ( v , X ′ ) (v,X') ( v , X ′ ) m m m
We first prove, after negligible deletion on the required grids,
# W p ≤ K p − ( m + d ) , λ ( W p ) ≤ K p m + d , λ ( z p ) ≤ K p m . (6.3) \#W_p\le Kp^{-(m+d)},\qquad\lambda(W_p)\le Kp^{m+d},\qquad\lambda(z_p)\le Kp^m.\tag{6.3} # W p ≤ K p − ( m + d ) , λ ( W p ) ≤ K p m + d , λ ( z p ) ≤ K p m . ( 6.3 ) Here # W p \#W_p # W p
At scale p p p K K K
X = f ( z ) + O ( K p ) . X=f(z)+O(Kp). X = f ( z ) + O ( K p ) . In version 2 these are obtained by composing the native quadratic test with a simple root of the packet equation, or with its affine quadratic center in the large-curvature case. The root error is O ( K p ) O(Kp) O ( K p ) X X X O ( K p ) O(Kp) O ( K p ) H H H H ν Q / K 1 H\nu_Q/K_1 H ν Q / K 1 K / K 1 K/K_1 K / K 1 ∑ Q H ν Q ≤ 1 \sum_QH\nu_Q\le1 ∑ Q H ν Q ≤ 1 K − 1 K^{-1} K − 1 K K K
Prune line/packet/ball/sign portions of weighted duration below H / K 1 H/K_1 H / K 1 K / K 1 K/K_1 K / K 1 f ( z ( t ) ) − X l ( t ) f(z(t))-X_l(t) f ( z ( t )) − X l ( t ) K p Kp K p H / K H/K H / K
∣ D v f − X ′ ∣ ≤ K p / H . (6.4) \lvert D_{\mathbf v}f-X'\rvert\le Kp/H.\tag{6.4} ∣ D v f − X ′ ∣ ≤ K p / H . ( 6.4 ) As k ≤ 1 k \le1 k ≤ 1 ∣ X ′ ∣ ≤ K \lvert X' \rvert\le K ∣ X ′ ∣ ≤ K K K K ∣ ∇ z f ∣ \lvert\nabla_z f \rvert ∣ ∇ z f ∣ (6.4) and ∣ V ∣ ≤ K \lvert V \rvert\le K ∣ V ∣ ≤ K
∂ t f + ∇ y f ⋅ V = O ( K ) \partial_t f+\nabla_y f\cdot V=O(K) ∂ t f + ∇ y f ⋅ V = O ( K ) would predict V V V j = 1 j=1 j = 1 p L p_L p L K K K K / H 2 K/H^2 K / H 2 ∣ ∇ z f ∣ ≤ K \lvert\nabla_z f \rvert\le K ∣ ∇ z f ∣ ≤ K
A packet now uses at most K ( 1 + H / p ) m K(1+H/p)^m K ( 1 + H / p ) m W p W_p W p H ≥ p / K H \ge p/K H ≥ p / K (6.2) over packets and H − 1 H^{-1} H − 1
K H 1 p d H j ( H p ) m ≤ K p − ( m + d ) . \frac{K}{H}\frac{1}{p^dH^j}\left(\frac{H}{p}\right)^m \le Kp^{-(m+d)}. H K p d H j 1 ( p H ) m ≤ K p − ( m + d ) . This proves (6.3) . Remove globally light W p W_p W p p m + d / K 1 p^{m+d}/K_1 p m + d / K 1 K 1 K_1 K 1
# { W p ⊂ W w 0 } ≤ K ( w 0 / p ) m + d ( w 0 ≥ p ) , # { W p over a fixed z p } ≤ K p − d , (6.5) \#\{W_p\subset W_{w_0}\}\le K(w_0/p)^{m+d}\quad(w_0\ge p),\qquad\#\{W_p\text{ over a fixed }z_p\}\le Kp^{-d},\tag{6.5} # { W p ⊂ W w 0 } ≤ K ( w 0 / p ) m + d ( w 0 ≥ p ) , # { W p over a fixed z p } ≤ K p − d , ( 6.5 ) with bounded enlargements at boundaries.
Let B p B_p B p
X = x 0 + t x 1 + t 2 x 2 , y = y 0 + t V . X=x_0+tx_1+t^2x_2,\qquad y=y_0+tV. X = x 0 + t x 1 + t 2 x 2 , y = y 0 + t V . Additional trajectory tags stay in the prior but are not part of this coefficient bin. There is one fixed, unnormalized, preceding long-line prior ν ∘ \nu^\circ ν ∘
K − 1 p j + d π ( [ V ] p ) ≤ ν ∘ ( B p ) ≤ K p j + d π ( [ V ] p ) . (6.6) K^{-1}p^{j+d}\pi([V]_p)\le\nu^\circ(B_p)\le Kp^{j+d}\pi([V]_p).\tag{6.6} K − 1 p j + d π ([ V ] p ) ≤ ν ∘ ( B p ) ≤ K p j + d π ([ V ] p ) . ( 6.6 ) A physical cell W p W_p W p [ V ] p [V]_p [ V ] p K K K B p B_p B p
The joint caps imply, for a time interval I I I y y y J y J_y J y X X X w 0 w_0 w 0 J J J
λ ( these constraints ) ≤ K ∣ I ∣ J y w 0 d π ( J ) . (6.7) \lambda(\text{these constraints})\le K\lvert I\rvert J_yw_0^d\pi(J).\tag{6.7} λ ( these constraints ) ≤ K ∣ I ∣ J y w 0 d π ( J ) . ( 6.7 ) For the current finite grid of spatial meshes, take one common time partition much finer than every mesh on that grid. Deleting line/bin pairs of weighted duration below ∣ I ∣ / K 1 \lvert I\rvert/K_1 ∣ I ∣ / K 1 1 / K 1 1/K_1 1/ K 1 I I I active at t ∈ I t\in I t ∈ I I I I t t t (6.7) by the retained duration, as in (4.26) , gives the fixed-time active-prior estimate
ν { l : active at t , ( y l ( t ) , X l ( t ) ) ∈ B , [ V l ] β = J } ≤ K p j + d π ( J ) (6.8) \nu\{l:\text{ active at }t,\ (y_l(t),X_l(t))\in B,\ [V_l]_\beta=J\}\le Kp^{j+d}\pi(J)\tag{6.8} ν { l : active at t , ( y l ( t ) , X l ( t )) ∈ B , [ V l ] β = J } ≤ K p j + d π ( J ) ( 6.8 ) for a spatial cube B B B p p p ν \nu ν 1 / K 1/K 1/ K ν ∘ \nu^\circ ν ∘ λ ≤ K ν ∘ d t \lambda\le K\nu^\circ\,dt λ ≤ K ν ∘ d t ν ∘ \nu^\circ ν ∘
For an affine bin α = [ y 0 , V ] p \alpha=[y_0,V]_p α = [ y 0 , V ] p Π α = π ( [ V ] p ) \Pi_\alpha=\pi([V]_p) Π α = π ([ V ] p )
ν ∘ ( α ) ≤ K p j Π α , ν ∘ { α , ( x 0 , x 1 , x 2 ) ∈ B r } ≤ K r d p j Π α ( r ≥ p ) . \nu^\circ(\alpha)\le Kp^j\Pi_\alpha,\qquad\nu^\circ\{\alpha,\ (x_0,x_1,x_2)\in B_r\}\le Kr^dp^j\Pi_\alpha\quad(r\ge p). ν ∘ ( α ) ≤ K p j Π α , ν ∘ { α , ( x 0 , x 1 , x 2 ) ∈ B r } ≤ K r d p j Π α ( r ≥ p ) . At each time these coefficient restrictions put the position in the corresponding moving boxes, so (6.7) applies. This proves the upper half of (6.6) .
If the lower half failed by a fixed power on a substantial residual, a parent and an α \alpha α p j Π α / K p^j\Pi_\alpha/K p j Π α / K p j Π α p^j\Pi_\alpha p j Π α K K K ν ∘ \nu^\circ ν ∘ α \alpha α p j Π α p^j\Pi_\alpha p j Π α K r d Kr^d K r d ( t , X ) (t,X) ( t , X ) K p − 1 − d Kp^{-1-d} K p − 1 − d α \alpha α y y y K K K (6.5) counts scalar options. Lemma 5.1 gives a scalar coefficient p p p p d / K p^d/K p d / K B p B_p B p p − 1 ≤ K p^{-1}\le K p − 1 ≤ K
A block B p B_p B p K / p K/p K / p W p W_p W p p ν ∘ ( B p ) / K 1 p\nu^\circ(B_p)/K_1 p ν ∘ ( B p ) / K 1 K / K 1 K/K_1 K / K 1 K p m + d π ( [ V ] p ) Kp^{m+d}\pi([V]_p) K p m + d π ([ V ] p ) p m + d π ( [ V ] p ) / K p^{m+d}\pi([V]_p)/K p m + d π ([ V ] p ) / K K K K
X ′ to width p has at most K options given ( W p , [ V ] p ) . (6.9) X' \text{ to width }p\text{ has at most }K\text{ options given }(W_p,[V]_p).\tag{6.9} X ′ to width p has at most K options given ( W p , [ V ] p ) . ( 6.9 ) Between grid widths p ′ < p ′ ′ p'<p'' p ′ < p ′′ p ′ ′ / p ′ ≤ K p''/p'\le K p ′′ / p ′ ≤ K K K K K K K (6.6) concerns the same ν ∘ \nu^\circ ν ∘
A velocity graph and its coarse lists
Fix the scales of the intended improvement before constructing its graph. Choose a = N − s a=N^{-s} a = N − s s > 0 s>0 s > 0 h ∼ log , N a k h\sim_{\log,N}a^k h ∼ l o g , N a k M ≥ 2 M\ge2 M ≥ 2 h M ≪ a 2 h^M\ll a^2 h M ≪ a 2 ζ \zeta ζ h , M h,M h , M
ρ ≪ ζ , ρ ≪ h M + 1 / 4 . \rho\ll\zeta,\qquad\rho\ll h^{M+1/4}. ρ ≪ ζ , ρ ≪ h M + 1/4 . The exponent κ > 0 \kappa>0 κ > 0 m , d m,d m , d b b b b / ρ b/\rho b / ρ b κ / ρ b^\kappa/\rho b κ / ρ M M M
On any substantial residual, renew breadth at the fine physical states W b W_b W b j = 1 j=1 j = 1 m m m K − 1 K^{-1} K − 1 j = 2 j=2 j = 2 j = 1 j=1 j = 1 K π K\pi K π
We use the determinant multilinear Kakeya estimate in the following form. If T i \mathcal{T}_i T i 1 ≤ i ≤ n 1 \le i \le n 1 ≤ i ≤ n r r r R n \mathbb{R}^n R n e T e_T e T w T w_T w T M i = ∑ T ∈ T i w T M_i = \sum_{T \in\mathcal{T}_i} w_T M i = ∑ T ∈ T i w T
∫ R n ( ∑ T i ∈ T i 1 ≤ i ≤ n ∣ det ( e T 1 , … , e T n ) ∣ ∏ i = 1 n w T i 1 T i ( x ) ) 1 / ( n − 1 ) d x ≤ C n r n ∏ i = 1 n M i 1 / ( n − 1 ) . (6.10) \int_{\mathbb{R}^n} \left(\sum_{\substack{T_i \in\mathcal{T}_i\\1 \le i \le n}} \left\lvert\det(e_{T_1},\ldots,e_{T_n})\right\rvert\prod_{i=1}^{n} w_{T_i}\mathbf{1}_{T_i}(x)\right)^{1/(n-1)} dx \le C_n r^n \prod_{i=1}^{n} M_i^{1/(n-1)}.\tag{6.10} ∫ R n T i ∈ T i 1 ≤ i ≤ n ∑ ∣ det ( e T 1 , … , e T n ) ∣ i = 1 ∏ n w T i 1 T i ( x ) 1/ ( n − 1 ) d x ≤ C n r n i = 1 ∏ n M i 1/ ( n − 1 ) . ( 6.10 ) This is the equal ambient and wedge dimension case of Carbery and Valdimarsson [7 ] , Theorem 1, after scaling unit-radius tubes by r r r [2 ] ; Guth [13 ] proved the endpoint, and Bourgain and Guth [6 ] , Section 7, obtained a wedge-weighted formulation. Only 2 ≤ n ≤ 4 2 \le n \le4 2 ≤ n ≤ 4
Proposition 6.2 (Physical velocity graph). For a sufficiently fine fixed power b b b G = ( G 0 , G y ) G = (G_0,G_y) G = ( G 0 , G y ) W b W_b W b K K K
∣ X ′ ( t ) − G ( W b ) ⋅ v ∣ ≤ b κ , (6.11) \lvert X'(t) - G(W_b) \cdot v\rvert\le b^\kappa,\tag{6.11} ∣ X ′ ( t ) − G ( W b ) ⋅ v ∣ ≤ b κ , ( 6.11 ) where κ > 0 \kappa> 0 κ > 0 m m m 1 − d 1-d 1 − d s 0 ≥ b κ s_0 \ge b^\kappa s 0 ≥ b κ G G G s 0 s_0 s 0 K K K W s 0 W_{s_0} W s 0
Proof. Put n ′ = m + 1 n' = m+1 n ′ = m + 1
0 < θ < ( 1 − d ) / 3 , 0 < τ < θ / ( n ′ − 1 ) . 0 < \theta< (1-d)/3,\qquad0 < \tau< \theta/(n'-1). 0 < θ < ( 1 − d ) /3 , 0 < τ < θ / ( n ′ − 1 ) . If no m m m 0.9 0.9 0.9 ( v , X ′ ) (v,X') ( v , X ′ ) b τ b^\tau b τ b τ b^\tau b τ m m m n ′ n' n ′ b τ ( n ′ − 1 ) b^{\tau(n'-1)} b τ ( n ′ − 1 ) b θ b^\theta b θ N N N
Suppose these bad cells carry substantial mass. Group them into dyadic W W W w 0 ∼ b 2 / 3 w_0 \sim b^{2/3} w 0 ∼ b 2/3 w 0 / K 1 w_0/K_1 w 0 / K 1 K / w 0 K/w_0 K / w 0 K / K 1 K/K_1 K / K 1 M > 0 M > 0 M > 0 m E m_E m E M M M K M / w 0 KM/w_0 K M / w 0
Approximate each trajectory there by its affine tangent at the cube’s central time. The position error is O ( K w 0 2 ) ≪ b O(Kw_0^2) \ll b O ( K w 0 2 ) ≪ b O ( K w 0 ) ≪ b θ O(Kw_0) \ll b^\theta O ( K w 0 ) ≪ b θ O ( b ) O(b) O ( b ) K b Kb K b ( m E / b ) n ′ / K (m_E/b)^{n'}/K ( m E / b ) n ′ / K (6.10) with n = n ′ n=n' n = n ′ b n ′ b^{n'} b n ′
∑ E ( b θ ( m E / b ) n ′ / K ) 1 / ( n ′ − 1 ) ≤ K ( M / w 0 ) n ′ / ( n ′ − 1 ) . \sum_E \left(b^\theta(m_E/b)^{n'}/K\right)^{1/(n'-1)} \le K(M/w_0)^{n'/(n'-1)}. E ∑ ( b θ ( m E / b ) n ′ / K ) 1/ ( n ′ − 1 ) ≤ K ( M / w 0 ) n ′ / ( n ′ − 1 ) . The power-mean inequality forces at least K − 1 b θ ( w 0 / b ) n ′ K^{-1}b^{\theta}(w_0/b)^{n'} K − 1 b θ ( w 0 / b ) n ′ (6.5) there are at most K ( w 0 / b ) m + d K(w_0/b)^{m+d} K ( w 0 / b ) m + d
K − 1 b θ − ( 1 − d ) / 3 ⟶ ∞ , K^{-1}b^{\theta-(1-d)/3}\longrightarrow\infty, K − 1 b θ − ( 1 − d ) /3 ⟶ ∞ , a contradiction.
Thus the lifted directions concentrate near an m m m m m m K − 1 K^{-1} K − 1 X ′ X' X ′ K − 1 K^{-1} K − 1 K K K (6.11) , for instance with κ = τ / 2 \kappa=\tau/2 κ = τ /2
Keep G G G W b W_b W b γ \gamma γ λ r e f , γ \lambda_{\mathrm{ref},\gamma} λ ref , γ p γ p_\gamma p γ
Λ r e f = ∑ γ p γ λ r e f , γ , L r e f = Λ r e f ( a l l ) . (6.12) \Lambda_{\mathrm{ref}}=\sum_{\gamma}p_\gamma\lambda_{\mathrm{ref},\gamma},\qquad L_{\mathrm{ref}}=\Lambda_{\mathrm{ref}}(\mathrm{all}).\tag{6.12} Λ ref = γ ∑ p γ λ ref , γ , L ref = Λ ref ( all ) . ( 6.12 ) This reference includes trajectory indices and true packet labels. It is substantial: the bad-cell mass tends to zero, good cells retain their fixed captured fraction, and the renewal has vanishing loss. Hence L r e f L_{\mathrm{ref}} L ref G G G
For each fine cell, the tuples of width-s 0 s_0 s 0 K − 1 K^{-1} K − 1 (6.9) and (6.11) allow at most K K K s 0 s_0 s 0 ∣ G ∣ ≤ K |G|\le K ∣ G ∣ ≤ K W s 0 W_{s_0} W s 0 K K K K K K
Polynomial fits along reference trajectories
Use the scales a a a h h h M M M ζ \zeta ζ ρ \rho ρ b b b Λ r e f \Lambda_{\mathrm{ref}} Λ ref
Proposition 6.3 (Polynomial qualification). There are a subpower K q u a l = N o ( 1 ) K_{\mathrm{qual}}=N^{o(1)} K qual = N o ( 1 ) U N U_N U N
e N : = Λ r e f ( U N ) L r e f ⟶ 0. (6.13) e_N:=\frac{\Lambda_{\mathrm{ref}}(U_N)}{L_{\mathrm{ref}}}\longrightarrow0.\tag{6.13} e N := L ref Λ ref ( U N ) ⟶ 0. ( 6.13 ) Every reference incidence outside U N U_N U N t t t M M M K q u a l K_{\mathrm{qual}} K qual G G G h M h^M h M h / K q u a l h/K_{\mathrm{qual}} h / K qual Q s Q_s Q s
The coarse lists do not yet compare the rows at nearby points: two points in the same cell may use different members of its list. The following selection makes a common choice in every retained cell. We will use it first for the physical row and then for a scalar residual inside a parameter block.
Lemma 6.4 (A common choice from local lists). Let μ \mu μ ω \omega ω Z ( e ) ∈ R D Z(e) \in\mathbb{R}^{D} Z ( e ) ∈ R D Y ( e ) ∈ R q Y(e) \in\mathbb{R}^{q} Y ( e ) ∈ R q e e e ∣ Y ( e ) ∣ ≤ B \lvert Y(e)\rvert\le B ∣ Y ( e )∣ ≤ B L ≥ 2 L \ge2 L ≥ 2 K 1 , … , K L ≥ 1 K_1,\ldots,K_L \ge1 K 1 , … , K L ≥ 1
1 = s 1 > s 2 > ⋯ > s L = r ∗ , A = max i < L s i / s i + 1 . 1=s_1>s_2>\cdots>s_L=r_*,\qquad A=\max_{i<L}s_i/s_{i+1}. 1 = s 1 > s 2 > ⋯ > s L = r ∗ , A = i < L max s i / s i + 1 . At width s i s_i s i s i s_i s i Z Z Z Y Y Y K i K_i K i s i s_i s i μ ′ ≤ μ \mu' \le\mu μ ′ ≤ μ
μ ′ ( ω ) ≥ μ ( ω ) ∏ i = 1 L ( 4 D K i ) − 1 , \mu'(\omega)\ge\mu(\omega)\prod_{i=1}^{L}(4^D K_i)^{-1}, μ ′ ( ω ) ≥ μ ( ω ) i = 1 ∏ L ( 4 D K i ) − 1 , and any two retained incidences in the same world satisfy
∣ Y ( e ) − Y ( e ^ ) ∣ ≤ C D , q ( A + B ) ( ∣ Z ( e ) − Z ( e ^ ) ∣ + r ∗ ) . \lvert Y(e)-Y(\widehat e)\rvert\le C_{D,q}(A+B)\bigl(\lvert Z(e)-Z(\widehat e)\rvert+r_*\bigr). ∣ Y ( e ) − Y ( e )∣ ≤ C D , q ( A + B ) ( ∣ Z ( e ) − Z ( e )∣ + r ∗ ) . If, within each world, both Z Z Z Y Y Y
Proof. Order the scales as displayed. At scale s i s_i s i s i ( n + [ 0 , 1 ) D ) s_i(n+[0,1)^D) s i ( n + [ 0 , 1 ) D ) n m o d 4 n\bmod4 n mod 4 Y Y Y K i K_i K i
These two operations keep at least ( 4 D K i ) − 1 (4^D K_i)^{-1} ( 4 D K i ) − 1
Distinct coordinate cells of the same color are separated by at least 3 s i 3s_i 3 s i ∣ Z ( e ) − Z ( e ^ ) ∣ ∞ ≤ s i \lvert Z(e)-Z(\widehat e)\rvert_\infty\le s_i ∣ Z ( e ) − Z ( e ) ∣ ∞ ≤ s i ∣ Y ( e ) − Y ( e ^ ) ∣ ≤ q s i \lvert Y(e)-Y(\widehat e)\rvert\le\sqrt{q}\,s_i ∣ Y ( e ) − Y ( e )∣ ≤ q s i r ∗ r_* r ∗ A A A ∣ Y ( e ) − Y ( e ^ ) ∣ ≤ 2 B \lvert Y(e)-Y(\widehat e)\rvert\le2B ∣ Y ( e ) − Y ( e )∣ ≤ 2 B □ \square □
We apply this lemma only on fixed finite grids first. When all entering masses are inverse-subpower and B B B K i = N o ( 1 ) K_i=N^{o(1)} K i = N o ( 1 )
D L log 4 + ∑ i log K i = o ( log N ) , A = N o ( 1 ) . DL\log4+\sum_i\log K_i=o(\log N),\qquad A=N^{o(1)}. D L log 4 + i ∑ log K i = o ( log N ) , A = N o ( 1 ) . The selected mass is then inverse-subpower and the comparison costs only K K K r r r ( log N ) / r (\log N)/r ( log N ) / r
Proof of Proposition 6.3. Fix ϵ > 0 \epsilon>0 ϵ > 0 U N , ϵ U_{N,\epsilon} U N , ϵ N ϵ N^\epsilon N ϵ h M h^M h M h N − ϵ hN^{-\epsilon} h N − ϵ h M / 4 h^M/4 h M /4 h 1 / 4 h^{1/4} h 1/4
If Λ r e f ( U N , ϵ ) / L r e f \Lambda_{\mathrm{ref}}(U_N,\epsilon)/L_{\mathrm{ref}} Λ ref ( U N , ϵ ) / L ref L r e f L_{\mathrm{ref}} L ref G G G W b W_b W b W b W_b W b
We will apply Proposition 5.6 inside a full coefficient block B ρ B_\rho B ρ ρ \rho ρ G y G_y G y ν ∘ ( ⋅ ∣ B ρ ) \nu^\circ(\cdot\mid B_\rho) ν ∘ ( ⋅ ∣ B ρ )
Apply Lemma 6.4 on the grid between ρ \rho ρ Z Z Z W b W_b W b Y = G ( W b ) Y=G(W_b) Y = G ( W b ) O ( K b ) O(Kb) O ( K b ) ρ \rho ρ
∣ G ( W b ) − G ( W ^ b ) ∣ ≤ K ( ∣ W − W ^ ∣ + ρ ) . (6.14) |G(W_b)-G(\widehat{W}_b)|\le K(|W-\widehat{W}|+\rho).\tag{6.14} ∣ G ( W b ) − G ( W b ) ∣ ≤ K ( ∣ W − W ∣ + ρ ) . ( 6.14 ) In a physical cube C = W ρ C=W_\rho C = W ρ G C G_C G C K ρ K\rho K ρ 1 ≥ s ′ ≥ ζ 1\ge s'\ge\zeta 1 ≥ s ′ ≥ ζ p = ρ s ′ p=\rho s' p = ρ s ′ W b W_b W b p p p X − G C ⋅ z X-G_C\cdot z X − G C ⋅ z O ( K ρ ) O(K\rho) O ( K ρ ) C C C K ρ 2 + K b ≪ p K\rho^2+Kb\ll p K ρ 2 + K b ≪ p K K K K / ρ K/\rho K / ρ
Delete line/strip portions of duration below ρ / K 1 \rho/K_1 ρ / K 1 W b W_b W b M M M K M / ρ KM/\rho K M / ρ z p z_p z p K − 1 K^{-1} K − 1
Apply (6.10) in R m \mathbb{R}^m R m m E m_E m E K p Kp K p
∑ E ( ( m E / p ) m / K ) 1 / ( m − 1 ) ≤ K ( M / ρ ) m / ( m − 1 ) . \sum_E \left((m_E/p)^m/K\right)^{1/(m-1)}\le K(M/\rho)^{m/(m-1)}. E ∑ ( ( m E / p ) m / K ) 1/ ( m − 1 ) ≤ K ( M / ρ ) m / ( m − 1 ) . At least K − 1 ( ρ / p ) m K^{-1}(\rho/p)^m K − 1 ( ρ / p ) m W p W_p W p K K K ∣ G C ∣ ≤ K |G_C|\le K ∣ G C ∣ ≤ K C C C O ( K r ) O(Kr) O ( K r ) p ≤ r ≤ ρ p\le r\le\rho p ≤ r ≤ ρ K ( ρ / r ) m K(\rho/r)^m K ( ρ / r ) m r r r K ( ρ / r ) m ( r / p ) m + d K(\rho/r)^m(r/p)^{m+d} K ( ρ / r ) m ( r / p ) m + d
# { occupied p -strips meeting a normal r -interval } ≤ K ( r / p ) d . (6.15) \#\{\text{occupied }p\text{-strips meeting a normal }r\text{-interval}\}\le K(r/p)^d.\tag{6.15} # { occupied p -strips meeting a normal r -interval } ≤ K ( r / p ) d . ( 6.15 ) Use each filling only to certify its upper support list. Those lists persist on further restrictions, and intermediate resolutions follow by subdivision and enlargement.
The parameter zoom. The physical cubes C C C B ρ B_\rho B ρ q = ( γ , B ρ ) \mathfrak{q}=(\gamma,B_\rho) q = ( γ , B ρ ) γ \gamma γ B ρ B_\rho B ρ ν γ \nu^\gamma ν γ q \mathfrak{q} q X c q ( t ) , y c q ( t ) X_c^\mathfrak{q}(t),y_c^\mathfrak{q}(t) X c q ( t ) , y c q ( t ) ρ \rho ρ
x = X − X c q ρ , y ~ = y − y c q ρ . x=\frac{X-X_c^\mathfrak{q}}{\rho},\qquad\widetilde{y}=\frac{y-y_c^\mathfrak{q}}{\rho}. x = ρ X − X c q , y = ρ y − y c q . Time is unchanged. All new trajectory coefficients are bounded; aligned width-r ′ r' r ′ B ρ r ′ B_{\rho r'} B ρ r ′ B ρ B_\rho B ρ F q ( t ) F_\mathfrak{q}(t) F q ( t ) G y G_y G y ρ \rho ρ F = F q F=F_\mathfrak{q} F = F q (6.14) ,
∣ F ( t ) − G y ( W b ) ∣ ≤ K ρ , ∣ F ( t ) − F ( u ) ∣ ≤ K ( ∣ t − u ∣ + ρ ) (6.16) \lvert F(t)-G_y(W_b)\rvert\le K\rho,\qquad\lvert F(t)-F(u)\rvert\le K(\lvert t-u\rvert+\rho)\tag{6.16} ∣ F ( t ) − G y ( W b )∣ ≤ K ρ , ∣ F ( t ) − F ( u )∣ ≤ K (∣ t − u ∣ + ρ ) ( 6.16 ) at active occurrences. The physical positions lie within K ρ K\rho K ρ K K K ρ ( x − F ( t ) y ~ ) \rho(x-F(t)\widetilde{y}) ρ ( x − F ( t ) y ) K K K C C C K ( ρ 2 + b ) K(\rho^2+b) K ( ρ 2 + b ) ρ ζ \rho\zeta ρζ (6.15) supplies the scalar-normal local d d d ζ \zeta ζ K K K
Take the fixed prior ν ∘ ( ⋅ ∣ B ρ ) \nu^\circ(\cdot\mid B_\rho) ν ∘ ( ⋅ ∣ B ρ ) ζ ≤ q ′ ≤ r ′ ≤ 1 \zeta\le q'\le r'\le1 ζ ≤ q ′ ≤ r ′ ≤ 1 B ρ r ′ B_{\rho r'} B ρ r ′ q ′ q' q ′
K ( ρ q ′ ) j + d π ( [ V ] ρ r ′ ) . K(\rho q')^{j+d}\pi([V]_{\rho r'}). K ( ρ q ′ ) j + d π ([ V ] ρ r ′ ) . Its prior denominator is at least ( ρ r ′ ) j + d π ( [ V ] ρ r ′ ) / K (\rho r')^{j+d}\pi([V]_{\rho r'})/K ( ρ r ′ ) j + d π ([ V ] ρ r ′ ) / K (6.6) . Consequently the conditional spatial cap is
K ( q ′ / r ′ ) j + d . (6.17) K(q'/r')^{j+d}.\tag{6.17} K ( q ′ / r ′ ) j + d . ( 6.17 ) The case r ′ = 1 r'=1 r ′ = 1
The residual velocity. Put R = x ′ − F ( t ) y ~ ′ R=x'-F(t)\widetilde{y}' R = x ′ − F ( t ) y ′ W b W_b W b B ρ B_\rho B ρ ζ \zeta ζ v c = y c ′ v_c=y'_c v c = y c ′
ρ R = G 0 + G y v c − X c ′ ( t ) + ( G y − F ( t ) ) ( V − v c ) + O ( b κ ) . \rho R=G_0+G_yv_c-X'_c(t)+(G_y-F(t))(V-v_c)+O(b^\kappa). ρR = G 0 + G y v c − X c ′ ( t ) + ( G y − F ( t )) ( V − v c ) + O ( b κ ) . Here ∣ V − v c ∣ ≤ K ρ \lvert V-v_c\rvert\le K\rho ∣ V − v c ∣ ≤ K ρ ∣ X c ′ ′ ∣ ≤ K \lvert X''_c\rvert\le K ∣ X c ′′ ∣ ≤ K W b W_b W b b b b b , ρ , ζ b,\rho,\zeta b , ρ , ζ
Fix s ′ ≥ ζ s'\ge\zeta s ′ ≥ ζ β = ρ s ′ \beta=\rho s' β = ρ s ′ β \beta β K / s ′ K/s' K / s ′ s ′ s' s ′ ρ \rho ρ s ′ ν ∘ ( B β ) / K 1 s'\nu^\circ(B_\beta)/K_1 s ′ ν ∘ ( B β ) / K 1 K / K 1 K/K_1 K / K 1 B ρ B_\rho B ρ s ′ s' s ′ K β K\beta K β J = [ V ] β J=[V]_\beta J = [ V ] β (6.8) along that moving cross-section gives
λ ( zoomed cell , J ) ≤ K s ′ β j + d π ( J ) . \lambda(\text{zoomed cell},J)\le Ks'\beta^{j+d}\pi(J). λ ( zoomed cell , J ) ≤ K s ′ β j + d π ( J ) . Every surviving block pays at least
s ′ ν ∘ ( B β ) K 1 ≥ K − 1 s ′ β j + d π ( J ) . \frac{s'\nu^\circ(B_\beta)}{K_1}\ge K^{-1}s'\beta^{j+d}\pi(J). K 1 s ′ ν ∘ ( B β ) ≥ K − 1 s ′ β j + d π ( J ) . Both estimates contain s ′ s' s ′ K K K β \beta β O ( s ′ ) O(s') O ( s ′ ) K K K x ′ x' x ′ K s ′ Ks' K s ′ (6.16) varies by at most K ( s ′ + ρ ) ≤ K s ′ K(s' + \rho) \le Ks' K ( s ′ + ρ ) ≤ K s ′ K K K s ′ s' s ′ R R R
We next remove the velocity condition with a summed denominator argument. Write K p r e = N o ( 1 ) K_{\mathrm{pre}} = N^{o(1)} K pre = N o ( 1 ) m 0 ( W ) m_0(W) m 0 ( W )
λ ( W , J ) ≤ K p r e m 0 ( W ) π ( J ) , ∑ W m 0 ( W ) ≤ K p r e . \lambda(W,J) \le K_{\mathrm{pre}}m_0(W)\pi(J), \qquad\sum_W m_0(W) \le K_{\mathrm{pre}}. λ ( W , J ) ≤ K pre m 0 ( W ) π ( J ) , W ∑ m 0 ( W ) ≤ K pre . For a coarse velocity bin I = [ V ] ρ I = [V]_\rho I = [ V ] ρ K p r e K_{\mathrm{pre}} K pre B ρ B_\rho B ρ W , I W,I W , I K p r e K_{\mathrm{pre}} K pre W W W K b / ρ ≪ ζ Kb/\rho\ll\zeta K b / ρ ≪ ζ K p r e 2 K_{\mathrm{pre}}^2 K pre 2 E = ( W , B ρ , s ′ -cell ) E=(W,B_\rho,s'\text{-cell}) E = ( W , B ρ , s ′ -cell ) W , I W,I W , I m 0 ( W ) π ( I ) / K c u t m_0(W)\pi(I)/K_{\mathrm{cut}} m 0 ( W ) π ( I ) / K cut
K p r e 2 K c u t ∑ W m 0 ( W ) ∑ I π ( I ) ≤ K p r e 3 K c u t . \frac{K_{\mathrm{pre}}^2}{K_{\mathrm{cut}}}\sum_W m_0(W)\sum_I \pi(I) \le\frac{K_{\mathrm{pre}}^3}{K_{\mathrm{cut}}}. K cut K pre 2 W ∑ m 0 ( W ) I ∑ π ( I ) ≤ K cut K pre 3 . Choose the subpower K c u t K_{\mathrm{cut}} K cut
λ ( J ∣ E ) ≤ K π ( J ) π ( I ) . (6.18) \lambda(J \mid E) \le K\frac{\pi(J)}{\pi(I)}.\tag{6.18} λ ( J ∣ E ) ≤ K π ( I ) π ( J ) . ( 6.18 ) This uses neither a new prior nor a current incidence floor from (6.6) .
Every occurring residual bin has a retained entry witnessing it. Near constancy within that entry and (6.18) make its supporting velocity bins have total palette mass at least π ( I ) / K \pi(I)/K π ( I ) / K K K K I I I K K K
Residual comparison and the scalar-normal input. It remains to make a common residual choice in each zoomed cell. Carry out the preceding deletions on a fixed finite grid { s 1 , … , s L } ⊂ [ ζ , 1 ] \{s_1,\ldots,s_L\} \subset[\zeta,1] { s 1 , … , s L } ⊂ [ ζ , 1 ] K / K 1 K/K_1 K / K 1 K p r e 3 / K c u t K_{\mathrm{pre}}^3/K_{\mathrm{cut}} K pre 3 / K cut μ 0 \mu_0 μ 0 δ 0 \delta_0 δ 0 K − 1 K^{-1} K − 1
In world q = ( γ , B ρ ) \mathfrak{q} = (\gamma,B_\rho) q = ( γ , B ρ )
Z q ( e ) = ( t , y ~ ( t ) , x l ( t ) ) , R q ( e ) = x l ′ ( t ) − F q ( t ) y ~ l . Z_{\mathfrak{q}}(e) = (t,\widetilde{y}(t),x_l(t)), \qquad R_{\mathfrak{q}}(e) = x'_l(t)-F_{\mathfrak{q}}(t)\widetilde{y}_l. Z q ( e ) = ( t , y ( t ) , x l ( t )) , R q ( e ) = x l ′ ( t ) − F q ( t ) y l . The dimension of Z q Z_{\mathfrak{q}} Z q D = j + 2 D=j+2 D = j + 2 s i s_i s i s i s_i s i K i = N o ( 1 ) K_i=N^{o(1)} K i = N o ( 1 ) B ρ B_\rho B ρ I = [ V ] ρ I=[V]_\rho I = [ V ] ρ ( W , B ρ , C ) (W,B_\rho,C) ( W , B ρ , C )
Apply Lemma 6.4 with Y = R q Y=R_{\mathfrak{q}} Y = R q μ L \mu_L μ L δ ≥ K − 1 \delta\ge K^{-1} δ ≥ K − 1
∣ R q ( e ) − R q ( e ^ ) ∣ ≤ K ( ∣ Z q ( e ) − Z q ( e ^ ) ∣ + ζ ) . (6.19) \lvert R_{\mathfrak{q}}(e)-R_{\mathfrak{q}}(\widehat e)\rvert\le K\bigl(\lvert Z_{\mathfrak{q}}(e)-Z_{\mathfrak{q}}(\widehat e)\rvert+\zeta\bigr).\tag{6.19} ∣ R q ( e ) − R q ( e )∣ ≤ K ( ∣ Z q ( e ) − Z q ( e )∣ + ζ ) . ( 6.19 ) for every pair of retained incidences in the same world. Coordinates and rows need not agree across worlds. This time the selection may remove individual edges: the last use of breadth in this construction was strip filling, and its remaining steps use only the inherited upper bounds.
Put a q = p γ ν γ ∘ ( B ρ ) a_\mathfrak{q} = p_\gamma \nu^\circ_\gamma(B_\rho) a q = p γ ν γ ∘ ( B ρ ) μ L ( q ) / a q ≥ δ \mu_L(\mathfrak{q})/a_\mathfrak{q} \ge\delta μ L ( q ) / a q ≥ δ λ ≤ K ν γ ∘ d t \lambda\le K \nu^\circ_\gamma\,dt λ ≤ K ν γ ∘ d t K − 1 K^{-1} K − 1 ν γ ∘ ( ⋅ ∣ B ρ ) d t \nu^\circ_\gamma(\mathord\cdot\mid B_\rho)\,dt ν γ ∘ ( ⋅ ∣ B ρ ) d t
We can now apply the scalar-normal results of Section 5 in this world. The trajectories are ( x , y ~ ) (x,\widetilde y) ( x , y ) F q F_\mathfrak{q} F q ν γ ∘ ( ⋅ ∣ B ρ ) \nu^\circ_\gamma(\mathord\cdot\mid B_\rho) ν γ ∘ ( ⋅ ∣ B ρ ) (6.17) gives both spatial caps, including conditioning on a parameter bin; (6.16) gives row comparison; and (6.19) gives residual comparison. These are exactly the hypotheses of Definition 5.2, with the floor already chosen for Proposition 5.6. Apply Proposition 5.6. It produces a degree-M M M K K K F F F G y G_y G y K h M + 1 / 4 K h^{M+1/4} K h M + 1/4 h / K h/K h / K (6.11) and G 0 = X l ′ − G y V l + O ( b κ ) G_0 = X'_l - G_y V_l + O(b^\kappa) G 0 = X l ′ − G y V l + O ( b κ ) G 0 G_0 G 0 Q s Q_s Q s K ′ K' K ′ ϵ > 0 \epsilon> 0 ϵ > 0 N N N N ϵ N^\epsilon N ϵ h N − ϵ hN^{-\epsilon} h N − ϵ K ′ h M + 1 / 4 ≤ h M / 4 K'h^{M+1/4} \le h^M/4 K ′ h M + 1/4 ≤ h M /4 h M h^M h M N ϵ N^\epsilon N ϵ U N , ϵ U_{N,\epsilon} U N , ϵ ϵ > 0 \epsilon> 0 ϵ > 0
Λ r e f ( U N , ϵ ) L r e f ⟶ 0. \frac{\Lambda_{\mathrm{ref}}(U_{N,\epsilon})}{L_{\mathrm{ref}}} \longrightarrow0. L ref Λ ref ( U N , ϵ ) ⟶ 0. For ϵ r = 1 / r \epsilon_r = 1/r ϵ r = 1/ r 1 / r 1/r 1/ r r ( N ) ⟶ ∞ r(N) \longrightarrow\infty r ( N ) ⟶ ∞ K q u a l = N 1 / r ( N ) K_{\mathrm{qual}} = N^{1/r(N)} K qual = N 1/ r ( N ) U N = U N , 1 / r ( N ) U_N = U_{N,1/r(N)} U N = U N , 1/ r ( N ) K q u a l K_{\mathrm{qual}} K qual (6.13) follows. For any required fixed finite collection of applications, divide the failure budget among its members before enlarging the collection. This proves a relative o ( 1 ) o(1) o ( 1 ) N N N
Cartesian interpolation of the physical field
Put p ∗ = 1 / k p_* = 1/k p ∗ = 1/ k a ∼ log , N h p ∗ a \sim_{\log,N} h^{p_*} a ∼ l o g , N h p ∗
0 < α < 1 2 , ε T = h p ∗ − 1 + α , a 1 = a h e 1 , e 1 = α / 2 , c 0 = e 1 ( 1 − d ) / 2 , a 2 = a h e 2 , e 2 = c 0 / 3. (6.20) \begin{aligned}
0 &< \alpha< \frac{1}{2}, & \varepsilon_T &= h^{p_* - 1 + \alpha}, & a_1 &= a h^{e_1}, & e_1 &= \alpha/2, \\
c_0 &= e_1(1-d)/2, & a_2 &= a h^{e_2}, & e_2 &= c_0/3.
\end{aligned}\tag{6.20} 0 c 0 < α < 2 1 , = e 1 ( 1 − d ) /2 , ε T a 2 = h p ∗ − 1 + α , = a h e 2 , a 1 e 2 = a h e 1 , = c 0 /3. e 1 = α /2 , ( 6.20 ) In particular 0 < e 2 < 1 / 24 0 < e_2 < 1/24 0 < e 2 < 1/24 ξ \xi ξ K h ξ ≪ a 1 Kh\xi\ll a_1 K h ξ ≪ a 1 b b b a 1 , a 2 a_1,a_2 a 1 , a 2 G G G
Use the states
D = ( parent world , Q s , W b ) . \mathcal{D} = (\text{parent world}, Q_s, W_b). D = ( parent world , Q s , W b ) . They have actual list knowledge and the required deep-history ambiguity. Return to the substantial graph-reference measure Λ r e f \Lambda_{\mathrm{ref}} Λ ref Λ p r e p ≤ Λ r e f \Lambda_{\mathrm{prep}} \le\Lambda_{\mathrm{ref}} Λ prep ≤ Λ ref
m D = Λ p r e p ( D ) , q D = Λ p r e p ( U N ∩ D ) m D . m_{\mathcal{D}} = \Lambda_{\mathrm{prep}}(\mathcal{D}), \qquad q_{\mathcal{D}} = \frac{\Lambda_{\mathrm{prep}}(U_N \cap\mathcal{D})}{m_{\mathcal{D}}}. m D = Λ prep ( D ) , q D = m D Λ prep ( U N ∩ D ) . For any fixed η > 0 \eta> 0 η > 0 (6.13) gives
∑ q D > η m D ≤ η − 1 Λ p r e p ( U N ) ≤ ( e N / η ) L r e f = o ( 1 ) . (6.21) \sum_{q_{\mathcal{D}}>\eta} m_{\mathcal{D}} \le\eta^{-1}\Lambda_{\mathrm{prep}}(U_N) \le(e_N/\eta)L_{\mathrm{ref}} = o(1).\tag{6.21} q D > η ∑ m D ≤ η − 1 Λ prep ( U N ) ≤ ( e N / η ) L ref = o ( 1 ) . ( 6.21 ) Thus the states with q D ≤ η q_{\mathcal{D}} \le\eta q D ≤ η m m m p 0 > 0 p_0 > 0 p 0 > 0 η ≤ p 0 / ( 2 m ) \eta\le p_0/(2m) η ≤ p 0 / ( 2 m ) p 0 − m η ≥ p 0 / 2 p_0-m\eta\ge p_0/2 p 0 − m η ≥ p 0 /2
Choose a parent and Q = Q s Q = Q_s Q = Q s h ν Q / K h\nu_Q/K h ν Q / K h ν Q h\nu_Q h ν Q
u = ( z − z c ) / h u = (z-z_c)/h u = ( z − z c ) / h with a fixed origin such that ∣ u ∣ ≤ K |u| \le K ∣ u ∣ ≤ K h ν Q h\nu_Q h ν Q K K K u u u
Over a specified sufficiently fine base cell, G G G ε T \varepsilon_T ε T K K K Q Q Q X X X K a K a K a K K K ε T ≫ K a \varepsilon_T \gg K a ε T ≫ K a K K K ε T \varepsilon_T ε T
Conditional product sampling at a good state gives tuples of m m m K − 1 K^{-1} K − 1 K − 1 K^{-1} K − 1 h ν Q / K h\nu_Q/K h ν Q / K A 0 A_0 A 0 v i = ( 1 , v i ) \mathbf{v}_i = (1,v_i) v i = ( 1 , v i ) ∥ A 0 ∥ + ∥ A 0 − 1 ∥ ≤ K \|A_0\| + \|A_0^{-1}\| \le K ∥ A 0 ∥ + ∥ A 0 − 1 ∥ ≤ K ξ \xi ξ A 0 − 1 u A_0^{-1}u A 0 − 1 u
For each direction i i i ε T \varepsilon_T ε T K K K K ξ K\xi K ξ ξ \xi ξ K ξ K\xi K ξ M + 1 M+1 M + 1 h / K 1 h/K_1 h / K 1 K − 1 K^{-1} K − 1 K 1 K_1 K 1 K K K h M ≪ ε T h^M \ll\varepsilon_T h M ≪ ε T
For each state fix one qualifying witness, with its polynomial fit, in each of the m m m G ( W b ) G(W_b) G ( W b ) m m m m + 1 m+1 m + 1 K K K h ν Q / K h\nu_Q/K h ν Q / K K ε T K\varepsilon_T K ε T M M M K ξ K\xi K ξ
Delete cells of mass below h ν Q ξ m / K 1 h\nu_Q\xi^m/K_1 h ν Q ξ m / K 1 K ξ − m K\xi^{-m} K ξ − m η 0 ≥ K − 1 \eta_0 \ge K^{-1} η 0 ≥ K − 1 M + 1 M+1 M + 1 η 0 ( M + 2 ) m \eta_0^{(M+2)^m} η 0 ( M + 2 ) m K 1 − 1 K_1^{-1} K 1 − 1 K 1 K_1 K 1
Tensor Lagrange interpolation from the pure anchor vertices gives a polynomial row T ( u ) T(u) T ( u ) M M M K K K K K K K ε T K\varepsilon_T K ε T h ν Q / K h\nu_Q/K h ν Q / K
∣ G ( W b ) − T ( u ) ∣ ≤ K ε T . (6.22) \lvert G(W_b)-T(u)\rvert\le K\varepsilon_T.\tag{6.22} ∣ G ( W b ) − T ( u )∣ ≤ K ε T . ( 6.22 ) This statement depends only on the state. We now restore all prepared reference edges of the selected good states, preserving their breadth and palette bounds.
For a line of weighted duration at least h / K h/K h / K
∣ X l ′ ( t ) − T ( u l ( t ) ) ⋅ v l ∣ ≤ K ε T (6.23) \lvert X'_l(t)-T(u_l(t))\cdot v_l\rvert\le K\varepsilon_T\tag{6.23} ∣ X l ′ ( t ) − T ( u l ( t )) ⋅ v l ∣ ≤ K ε T ( 6.23 ) throughout its Q Q Q
The comparison is polynomial in time with a fixed degree.
We will also need the joint support count
# { ( ξ -bin of u , a i -bin of X ) } ≤ K ξ − m ( a / a i ) d , i = 1 , 2. (6.24) \#\{(\xi\text{-bin of }u,\ a_i\text{-bin of }X)\}\le K\xi^{-m}(a/a_i)^d,\qquad i=1,2.\tag{6.24} # {( ξ -bin of u , a i -bin of X )} ≤ K ξ − m ( a / a i ) d , i = 1 , 2. ( 6.24 ) Use true descendants at thickness a i a_i a i h i ∼ log N a i k h_i\sim\log_N a_i^k h i ∼ log N a i k K K K a i d h i j / K a_i^d h_i^j/K a i d h i j / K Q Q Q h i h_i h i K ( h i / ( h ξ ) ) m K(h_i/(h\xi))^m K ( h i / ( h ξ ) ) m ξ \xi ξ h / h i h/h_i h / h i
K h h i ν Q a i d h i j ( h i h ξ ) m = K ξ − m ν Q a i d h i j ≤ K ξ − m ( a / a i ) d . K\frac{h}{h_i}\frac{\nu_Q}{a_i^d h_i^j}\left(\frac{h_i}{h\xi}\right)^m
=K\xi^{-m}\frac{\nu_Q}{a_i^d h_i^j}\le K\xi^{-m}(a/a_i)^d. K h i h a i d h i j ν Q ( h ξ h i ) m = K ξ − m a i d h i j ν Q ≤ K ξ − m ( a / a i ) d . Switched lines force approximate conservativity
A conservative field has a simple geometric property: its integral along any broken base path depends only on the two endpoints. If the skew part of D T DT D T X / h X/h X / h Figure 2 illustrates this local obstruction. The proof below uses two bases of directions to turn it into a nonzero Jacobian minor. Its lower bound makes the probability of landing in the support from (6.24) tend to zero when d < 1 d<1 d < 1
Figure 2. Two orders of the same two base displacements. The difference between their line integrals has mixed term s t ( D T ( u 0 ) [ v i ] ⋅ v j − D T ( u 0 ) [ v j ] ⋅ v i ) st(DT(u_0)[v_i]\cdot v_j-DT(u_0)[v_j]\cdot v_i) s t ( D T ( u 0 ) [ v i ] ⋅ v j − D T ( u 0 ) [ v j ] ⋅ v i ) 2 m 2m 2 m m m m
Proposition 6.5 (Conservativity of the interpolated field). In coefficient norm, or equivalently in sup norm on the required enlarged box up to K K K
∥ skew D T ∥ ≤ K h p ∗ − 1 + c 0 . \lVert\operatorname{skew} DT\rVert\le Kh^{p_*-1+c_0}. ∥ skew D T ∥ ≤ K h p ∗ − 1 + c 0 . Proof. Otherwise restrict to a subsequence where the skew coefficient norm is at least h p ∗ − 1 + c 0 h^{p_*-1+c_0} h p ∗ − 1 + c 0 Q Q Q P \mathbb{P} P h ν Q / K h\nu_Q/K h ν Q / K K K K h p ∗ − 1 + c 0 / K h^{p_*-1+c_0}/K h p ∗ − 1 + c 0 / K
Starting from e 0 = ( l 0 , t 0 ) e_0=(l_0,t_0) e 0 = ( l 0 , t 0 ) 2 m 2m 2 m e i − 1 e_{i-1} e i − 1 e ⋄ = ( l i , t ⋄ ) e^\diamond=(l_i,t^\diamond) e ⋄ = ( l i , t ⋄ ) P ( ⋅ ∣ D ( e i − 1 ) ) \mathbb{P}(\mathord\cdot\mid \mathcal{D}(e_{i-1})) P ( ⋅ ∣ D ( e i − 1 )) t i t_i t i P \mathbb{P} P l i l_i l i e i = ( l i , t i ) e_i=(l_i,t_i) e i = ( l i , t i ) P \mathbb{P} P h / K 1 h/K_1 h / K 1 P \mathbb{P} P K / K 1 K/K_1 K / K 1 ν Q \nu_Q ν Q
Before these longness restrictions, the velocity at each switch, conditional on its whole past, is sampled from a prepared state law. Its probability in any specified thin pencil is as small as needed. Sequential avoidance therefore gives, with probability bounded below, a well-conditioned basis from the first m m m m m m
For every current edge and past history, the unnormalized next-step measure, restricted to long choices, obeys
d P { [ V l i ] = J , t i ∈ d t i , l i l o n g ∣ current edge and past } ≤ K π ( J ) d t i h . (6.25) \mathrm{d}\mathbb{P}\{[V_{l_i}]=J,\ t_i\in\mathrm{d}t_i,\ l_i\ \mathrm{long}\mid\text{current edge and past}\}\le K\pi(J)\frac{\mathrm{d}t_i}{h}.\tag{6.25} d P {[ V l i ] = J , t i ∈ d t i , l i long ∣ current edge and past } ≤ K π ( J ) h d t i . ( 6.25 ) The state law supplies K π ( J ) K\pi(J) K π ( J ) (6.1) divided by its duration bounds the endpoint-time density by K / h K/h K / h K K K P \mathbb{P} P d t i / h \mathrm{d}t_i/h d t i / h
δ i = ( t i − t i − 1 ) / h \delta_i=(t_i-t_{i-1})/h δ i = ( t i − t i − 1 ) / h is triangular with Jacobian one in normalized times; they range in [ − 1 , 1 ] 2 m [-1,1]^{2m} [ − 1 , 1 ] 2 m
For binned control directions v ~ i \widetilde{v}_i v i v ~ i δ i \widetilde{v}_i\delta_i v i δ i u u u X / h X/h X / h T ⋅ v ~ i T\cdot\widetilde{v}_i T ⋅ v i W b W_b W b K b Kb K b (6.23) and the bounded coefficients of T T T
K ( b / h + p v ) ≪ ξ in u , K ( b / h + p v + ε T ) ≪ a 1 / h in X / h , K(b/h+p_v)\ll\xi\text{ in }u,\qquad K(b/h+p_v+\varepsilon_T)\ll a_1/h \text{ in }X/h, K ( b / h + p v ) ≪ ξ in u , K ( b / h + p v + ε T ) ≪ a 1 / h in X / h , where p v p_v p v
ε T / ( a 1 / h ) ∼ log , N h α / 2 . \varepsilon_T/(a_1/h)\sim_{\log,N}h^{\alpha/2}. ε T / ( a 1 / h ) ∼ l o g , N h α /2 . Write the endpoint map as
δ ↦ ( u 0 + A δ , R ( δ ) ) , \delta\mapsto(u_0+A\delta,\mathcal{R}(\delta)), δ ↦ ( u 0 + A δ , R ( δ )) , where A A A v ~ i \widetilde{v}_i v i R \mathcal{R} R I , J I,J I , J m m m I I I j ′ ∈ J j'\in J j ′ ∈ J I I I j ′ ∈ J j'\in J j ′ ∈ J I ∪ { j ′ } I\cup\{j'\} I ∪ { j ′ } det A I \det A_I det A I
∂ j ′ R − ( ∇ I R ) T A I − 1 A j ′ . \partial_{j'}\mathcal{R}-(\nabla_I\mathcal{R})^{\mathsf T}A_I^{-1}A_{j'}. ∂ j ′ R − ( ∇ I R ) T A I − 1 A j ′ . Differentiate this identity at zero in each coordinate of I I I η \eta η K η K\eta K η
A I T H s y m A J , H s y m = A I − T ( D 2 R ( 0 ) ) I , J A J − 1 , A_I^{\mathsf T}H_{\mathrm{sym}}A_J,\qquad H_{\mathrm{sym}}=A_I^{-\mathsf T}(D^2\mathcal{R}(0))_{I,J}A_J^{-1}, A I T H sym A J , H sym = A I − T ( D 2 R ( 0 ) ) I , J A J − 1 , a symmetric matrix transported between the two bases. Here fixed-degree polynomial derivative bounds control a derivative at zero by the minor norm. On the other hand, directly differentiating the successive integrals gives, for i ∈ I , j ′ ∈ J i\in I,j'\in J i ∈ I , j ′ ∈ J
( D 2 R ( 0 ) ) i , j ′ = D T ( u 0 ) [ v ~ i ] ⋅ v ~ j ′ . (D^2\mathcal{R}(0))_{i,j'}=DT(u_0)[\widetilde{v}_i]\cdot\widetilde{v}_{j'}. ( D 2 R ( 0 ) ) i , j ′ = D T ( u 0 ) [ v i ] ⋅ v j ′ . Later legs have zero duration at zero, so there are no additional cross terms. Inverting both bases bounds the skew of D T ( u 0 ) DT(u_0) D T ( u 0 ) K η K\eta K η h p ∗ − 1 + c 0 / K h^{p_*-1+c_0}/K h p ∗ − 1 + c 0 / K
For every start and control tuple, choose such a minor and remove the region where its absolute value is below h p ∗ − 1 + c 0 / K 1 h^{p_*-1+c_0}/K_1 h p ∗ − 1 + c 0 / K 1 (6.25) , make the removed path mass arbitrarily small. On the passing region, fix the complementary m − 1 m-1 m − 1
K ξ m ( a 1 / h ) h − ( p ∗ − 1 + c 0 ) . K\xi^m(a_1/h)h^{-(p_*-1+c_0)}. K ξ m ( a 1 / h ) h − ( p ∗ − 1 + c 0 ) . The complementary coordinates have bounded volume. The resulting cell estimate has the full m + 1 m+1 m + 1 ξ m ( a 1 / h ) \xi^m(a_1/h) ξ m ( a 1 / h ) (6.24) . Summing and integrating the start and palette weights bounds the successful-path probability by
K ( a / a 1 ) d ( a 1 / h ) h − ( p ∗ − 1 + c 0 ) ≤ K h e 1 ( 1 − d ) − c 0 = K h e 1 ( 1 − d ) / 2 ⟶ 0. \begin{aligned}
K(a/a_1)^d(a_1/h)h^{-(p_*-1+c_0)}&\le Kh^{e_1(1-d)-c_0} \\
&=Kh^{e_1(1-d)/2}\longrightarrow0.
\end{aligned} K ( a / a 1 ) d ( a 1 / h ) h − ( p ∗ − 1 + c 0 ) ≤ K h e 1 ( 1 − d ) − c 0 = K h e 1 ( 1 − d ) /2 ⟶ 0. This contradicts the positive lower probability and proves the proposition.
A potential and concentrated intercept groups
Define the polynomial
P ( u ) = h ∫ 0 1 T ( t u ) ⋅ u d t . P(u)=h\int_{0}^{1}T(tu)\cdot u\,\mathrm{d}t. P ( u ) = h ∫ 0 1 T ( t u ) ⋅ u d t . Differentiating under the integral and comparing with h T ( u ) hT(u) h T ( u ) u u u
∣ ∇ P − h T ∣ ≤ K a h c 0 . (6.26) \lvert\nabla P-hT\rvert\le Kah^{c_0}.\tag{6.26} ∣ ∇ P − h T ∣ ≤ K a h c 0 . ( 6.26 ) The norms of all fixed positive derivatives of P P P K h Kh K h (6.23) , the variation of X − P ( u ) X-P(u) X − P ( u ) K a h c 0 Kah^{c_0} K a h c 0 h h h h ε T ≤ K a h α h\varepsilon_T\le Kah^\alpha h ε T ≤ K a h α c 0 < α c_0<\alpha c 0 < α
Bin the midpoint value of X − P ( u ) X-P(u) X − P ( u ) a 2 a_2 a 2 C C C D \mathcal{D} D C C C K a h c 0 Kah^{c_0} K a h c 0 a 2 a_2 a 2 X , u X,u X , u P P P h ν Q h\nu_Q h ν Q
K ( a 2 / a ) d . (6.27) K(a_2/a)^d.\tag{6.27} K ( a 2 / a ) d . ( 6.27 ) Indeed apply the exact-base X X X a 2 a_2 a 2 P ( u ) + C P(u)+C P ( u ) + C ν Q ∼ log , N a d h j \nu_Q\sim_{\log,N}a^dh^j ν Q ∼ l o g , N a d h j
We next bound the number of groups without asserting that an arbitrary restriction preserves breadth. First remove short lines globally in Q Q Q C C C C C C C C C ( D , C ) (\mathcal{D},C) ( D , C ) h ν Q / K h\nu_Q/K h ν Q / K M M M K M / h KM/h K M / h ( D , C ) (\mathcal{D},C) ( D , C )
Apply (6.10) in dimension m m m u u u ξ \xi ξ K h ξ Kh\xi K h ξ O ( ξ ) O(\xi) O ( ξ ) m E m_E m E
∑ E ( m E / ( h ξ ) m K ) 1 / ( m − 1 ) ≤ K ( M / h ) m / ( m − 1 ) . \sum_E\left(\frac{m_E/(h\xi)^m}{K}\right)^{1/(m-1)}
\le K(M/h)^{m/(m-1)}. E ∑ ( K m E / ( h ξ ) m ) 1/ ( m − 1 ) ≤ K ( M / h ) m / ( m − 1 ) . Power mean gives at least K − 1 ξ − m K^{-1}\xi^{-m} K − 1 ξ − m X X X a 2 a_2 a 2 K K K P P P a 2 a_2 a 2 (6.24) with i = 2 i=2 i = 2 K ( a / a 2 ) d K(a/a_2)^d K ( a / a 2 ) d
Delete groups below
K − 1 h ν Q ( a 2 / a ) d (6.28) K^{-1}h\nu_Q(a_2/a)^d\tag{6.28} K − 1 h ν Q ( a 2 / a ) d ( 6.28 ) with a sufficiently large denominator. The summed loss is small. On every retained long line in any group,
∣ X l ( t ) − P ( u l ( t ) ) − C ∣ ≤ K a 2 (6.29) \lvert X_l(t)-P(u_l(t))-C\rvert\le Ka_2\tag{6.29} ∣ X l ( t ) − P ( u l ( t )) − C ∣ ≤ K a 2 ( 6.29 ) throughout the entire Q Q Q
All subsequent counts use these heavy groups and their immediately preceding long witnesses.
Seven directions and the exceptional conic
The remaining task is to replace the potential fit by an allowed quadratic test. In three base variables, even directions with every triple independent can all lie on a nonsingular conic. Third directional tests along those directions therefore need not force every higher-order term to vanish. We must identify the cubic and quartic terms that survive, and later express them using the hidden coordinate so that the final test is quadratic.
For a symmetric rank-r r r T T T R 3 \mathbb{R}^3 R 3 F T ( x ) = T ( x , … , x ) F_T(x)=T(x,\ldots,x) F T ( x ) = T ( x , … , x ) T ( p i , p i , p i , ⋅ , … , ⋅ ) = 0 T(p_i,p_i,p_i,\cdot,\ldots,\cdot)=0 T ( p i , p i , p i , ⋅ , … , ⋅ ) = 0 p i p_i p i F T F_T F T r − 3 r-3 r − 3 r r r q = 0 q=0 q = 0 q q q q 2 q^2 q 2 T ( p i , p i , ⋅ ) T(p_i,p_i,\cdot) T ( p i , p i , ⋅ )
Lemma 6.6 (Seven-direction tensor tests). Let p i = ( 1 , v i ) ∈ R 3 p_i=(1,v_i)\in\mathbb{R}^3 p i = ( 1 , v i ) ∈ R 3 1 ≤ i ≤ 7 1\leq i\leq7 1 ≤ i ≤ 7
The space of homogeneous quadratics vanishing at all p i p_i p i
The map on symmetric cubic tensors
T ⟼ ( T ( p i , p i , ⋅ ) ) i = 1 7 T\longmapsto\bigl(T(p_i,p_i,\cdot)\bigr)_{i=1}^{7} T ⟼ ( T ( p i , p i , ⋅ ) ) i = 1 7 is injective.
If a nonzero quadratic q q q 3 , 4 , 5 3,4,5 3 , 4 , 5
q ( R 3 ) ∗ , R q 2 , { 0 } , q(\mathbb{R}^3)^*,\qquad\mathbb{R}q^2,\qquad\{0\}, q ( R 3 ) ∗ , R q 2 , { 0 } , respectively.
These assertions have polynomial conditioning. More precisely, if
∣ p i ∣ ≤ A , ∣ det ( p i , p j , p k ) ∣ ≥ A − 1 , A ≥ 2 , |p_i|\leq A,\qquad|\det(p_i,p_j,p_k)|\geq A^{-1},\qquad A\geq2, ∣ p i ∣ ≤ A , ∣ det ( p i , p j , p k ) ∣ ≥ A − 1 , A ≥ 2 , the inverse of the injective map in part 2 costs at most C A C CA^C C A C q q q A A A C A C CA^C C A C
Proof. All divisibility arguments can be made over C \mathbb{C} C
There is exactly one conic through any five, up to scalar. Indeed, use the first three points as a projective coordinate basis. Quadratics vanishing there have form a x 0 x 1 + b x 0 x 2 + c x 1 x 2 ax_0x_1+bx_0x_2+cx_1x_2 a x 0 x 1 + b x 0 x 2 + c x 1 x 2 ( x 0 x 1 , x 0 x 2 , x 1 x 2 ) (x_0x_1,x_0x_2,x_1x_2) ( x 0 x 1 , x 0 x 2 , x 1 x 2 ) ( a , b , c ) (a,b,c) ( a , b , c )
If T ( p i , p i , ⋅ ) = 0 T(p_i,p_i,\cdot)=0 T ( p i , p i , ⋅ ) = 0 F F F F = 0 F=0 F = 0 ∂ j F = c j q \partial_jF=c_jq ∂ j F = c j q F = q L F=qL F = q L L L L
0 = ∇ F ( p i ) = L ( p i ) ∇ q ( p i ) . 0=\nabla F(p_i)=L(p_i)\nabla q(p_i). 0 = ∇ F ( p i ) = L ( p i ) ∇ q ( p i ) . Nonsingularity gives L ( p i ) = 0 L(p_i)=0 L ( p i ) = 0 L = 0 L=0 L = 0
A cubic vanishing at seven distinct points of a nonsingular conic is divisible by its equation. Here is an elementary justification of this intersection fact. Over C \mathbb{C} C x 0 x 2 − x 1 2 = 0 x_0x_2-x_1^2=0 x 0 x 2 − x 1 2 = 0 [ s : t ] ↦ [ s 2 : s t : t 2 ] [s:t]\mapsto[s^2:st:t^2] [ s : t ] ↦ [ s 2 : s t : t 2 ] x 0 x 2 − x 1 2 x_0x_2-x_1^2 x 0 x 2 − x 1 2 x 0 , x 2 x_0,x_2 x 0 , x 2 x 1 r x_1^r x 1 r q q q
For a quartic F F F q q q F = q H F=qH F = q H H H H
∂ j F = ( ∂ j q ) H + q ∂ j H . \partial_jF=(\partial_jq)H+q\partial_jH. ∂ j F = ( ∂ j q ) H + q ∂ j H . Choose a nonzero linear partial derivative of q q q q q q ∂ j F \partial_jF ∂ j F q q q q ∣ H q\mid H q ∣ H F F F q 2 q^2 q 2 q 2 q^2 q 2
For a quintic in the rank-five kernel, every second derivative is a cubic divisible by q q q q 2 q^2 q 2 F = q 2 L F=q^2L F = q 2 L L L L q 2 q^2 q 2 q ∣ ( ∂ j q ) L q\mid(\partial_jq)L q ∣ ( ∂ j q ) L q ∣ L q\mid L q ∣ L L = 0 L=0 L = 0
For fixed A A A 3 , 1 , 0 3,1,0 3 , 1 , 0 A A A
A positive semialgebraic function of one real parameter has a reciprocal polynomial lower bound at infinity. To see the rate relevant here, decompose its graph into finitely many algebraic branches for large A A A P ( A , s ) = 0 P(A,s)=0 P ( A , s ) = 0 s s s P P P s s s A A A A A A A A A A A A 0 < s ≤ 1 0<s\leq1 0 < s ≤ 1 s ≥ c A − C s\geq cA^{-C} s ≥ c A − C
Lemma 6.7 (Restoration of an approximate real conic). Suppose the points satisfy the conditioning bounds of Lemma 6.6, and a real quadratic q ( x ) = x T Q x q(x)=x^{\mathsf T}\mathsf{Q}x q ( x ) = x T Q x A − 1 ≤ ∥ q ∥ ≤ A A^{-1}\leq\lVert q\rVert\leq A A − 1 ≤ ∥ q ∥ ≤ A ∣ q ( p i ) ∣ ≤ ϵ |q(p_i)|\leq\epsilon ∣ q ( p i ) ∣ ≤ ϵ ϵ \epsilon ϵ A A A
∣ det Q ∣ ≥ c A − C , |\det\mathsf{Q}|\geq cA^{-C}, ∣ det Q ∣ ≥ c A − C , and there are exact real zeros p i ′ = ( 1 , v i ′ ) p_i'=(1,v_i') p i ′ = ( 1 , v i ′ ) q q q
∣ p i ′ − p i ∣ ≤ C A C ϵ |p_i'-p_i|\le CA^{C}\epsilon ∣ p i ′ − p i ∣ ≤ C A C ϵ
For sufficiently small ϵ \epsilon ϵ ( 2 A ) − 1 (2A)^{-1} ( 2 A ) − 1
Proof. For coefficient-normalized quadratics the function
∣ det Q ∣ + ∑ i = 1 7 ∣ q ( p i ) ∣ |\det\mathsf{Q}|+\sum_{i=1}^{7}|q(p_i)| ∣ det Q ∣ + i = 1 ∑ 7 ∣ q ( p i ) ∣ has a positive minimum on the compact point family: a zero would be a nonzero singular quadratic through all seven points, contrary to Lemma 6.6. The semialgebraic argument in that lemma gives a lower bound c A − C cA^{-C} c A − C q q q
It bounds the smallest singular value of Q \mathsf{Q} Q c A − C cA^{-C} c A − C p i p_i p i g i = ∇ v q ( 1 , v i ) g_i=\nabla_vq(1,v_i) g i = ∇ v q ( 1 , v i )
∂ 0 q ( 1 , v i ) + v i ⋅ g i = 2 q ( 1 , v i ) . \partial_0q(1,v_i)+v_i\cdot g_i=2q(1,v_i). ∂ 0 q ( 1 , v i ) + v i ⋅ g i = 2 q ( 1 , v i ) . Since ∣ v i ∣ ≤ A |v_i|\le A ∣ v i ∣ ≤ A q ( 1 , v i ) q(1,v_i) q ( 1 , v i ) ∣ g i ∣ ≥ c A − C |g_i|\ge cA^{-C} ∣ g i ∣ ≥ c A − C e i = ( 0 , g i / ∣ g i ∣ ) e_i=(0,g_i/|g_i|) e i = ( 0 , g i /∣ g i ∣ )
q ( p i + t e i ) = q ( p i ) + ∣ g i ∣ t + b i t 2 , ∣ b i ∣ ≤ C A . q(p_i+te_i)=q(p_i)+|g_i|t+b_it^2,\qquad|b_i|\le CA. q ( p i + t e i ) = q ( p i ) + ∣ g i ∣ t + b i t 2 , ∣ b i ∣ ≤ C A . If q ( p i ) ≠ 0 q(p_i)\ne0 q ( p i ) = 0 t = − 2 q ( p i ) / ∣ g i ∣ t=-2q(p_i)/|g_i| t = − 2 q ( p i ) /∣ g i ∣ ϵ \epsilon ϵ ∣ q ( p i ) ∣ / 2 |q(p_i)|/2 ∣ q ( p i ) ∣/2 q ( p i ) q(p_i) q ( p i ) 2 ∣ q ( p i ) ∣ / ∣ g i ∣ 2|q(p_i)|/|g_i| 2∣ q ( p i ) ∣/∣ g i ∣ q ( p i ) = 0 q(p_i)=0 q ( p i ) = 0 C A 2 max i ∣ p i ′ − p i ∣ CA^2\max_i|p_i'-p_i| C A 2 max i ∣ p i ′ − p i ∣
Restoring quadratic tests
Proposition 6.8 (Degree restoration). The potential fits and heavy intercept groups constructed above yield a candidate with all the properties in Theorem 6.1, without reducing the positive exponent e 2 e_2 e 2 a 2 / a = h e 2 a_2/a=h^{e_2} a 2 / a = h e 2
Proof. We treat the two versions separately.
Version 1. Choose a heavy intercept group. In normalized block time, (6.29) compares P P P X − C X-C X − C
∣ D v 3 P ∣ ≤ K a 2 . |D_{\mathbf v}^3P|\le Ka_2. ∣ D v 3 P ∣ ≤ K a 2 . there. Conditional product sampling and palette averaging fix four separated very fine velocity bins with witnesses on common states carrying an inverse-subpower fraction of the group. Use their centers as directions. The locations in a state differ by at most K b / h Kb/h K b / h u u u P P P K h Kh K h b / h ≪ a 2 / h b/h\ll a_2/h b / h ≪ a 2 / h (6.27) and (6.28) , these common tests hold on inverse-subpower base volume. Polynomial Remez extends them to the enclosing box. A binary cubic is determined by its values in four separated directions ( 1 , V i ) (1,V_i) ( 1 , V i ) K K K ∥ ∇ 3 P ∥ ≤ K a 2 \|\nabla^3P\|\le Ka_2 ∥ ∇ 3 P ∥ ≤ K a 2 P + C P+C P + C K a 2 Ka_2 K a 2 w w w
Branches in version 2. Here w w w w w w
P ∗ ( z , w ) = P ( u ) + C , z = z c + h u . (6.30) P_{\ast}(z,w)=P(u)+C,\qquad z=z_c+hu.\tag{6.30} P ∗ ( z , w ) = P ( u ) + C , z = z c + h u . ( 6.30 ) At path observations the residual is O ( K a 2 ) O(Ka_2) O ( K a 2 ) K − 1 K^{-1} K − 1 K K K K K K
Δ ( u ) = ( P ∗ , w ) 2 − 2 P ∗ , w w ( P ∗ ( z , w ) − P ( u ) − C ) \Delta(u)=(P_{\ast,w})^2-2P_{\ast,ww}\bigl(P_{\ast}(z,w)-P(u)-C\bigr) Δ ( u ) = ( P ∗ , w ) 2 − 2 P ∗ , w w ( P ∗ ( z , w ) − P ( u ) − C ) is independent of w w w w w w K − 1 K^{-1} K − 1 K K K (6.27) gives inverse-subpower base volume. Polynomial Remez bounds the coefficients and derivatives of Δ \Delta Δ K K K Δ \Delta Δ
Each observed w l w_l w l K a 2 Ka_2 K a 2 (6.30) . For a quadratic, choose the root with the same derivative sign as at w l w_l w l K − 1 K^{-1} K − 1 K K K Δ \Delta Δ
Choose a lattice of real comparison cores, each inside a fixed several-fold complex enlargement, with radius common up to K K K ( D , C ) (\mathcal{D},C) ( D , C ) K b / h Kb/h K b / h h / K 1 h/K_1 h / K 1 K K K K h ν Q / K 1 Kh\nu_Q/K_1 K h ν Q / K 1
There are at most K ( a / a 2 ) d K(a/a_2)^d K ( a / a 2 ) d
K − 1 h ν Q ( a 2 / a ) d . (6.31) K^{-1}h\nu_Q(a_2/a)^d.\tag{6.31} K − 1 h ν Q ( a 2 / a ) d . ( 6.31 ) Let U U U f f f h / K h/K h / K U U U f f f
Analytic estimates before tensor inversion. Set
R ( u ) = P s ( z c + h u , f ( u ) ) / a . (6.32) R(u)=P_s(z_c+hu,f(u))/a.\tag{6.32} R ( u ) = P s ( z c + h u , f ( u )) / a . ( 6.32 ) We will apply the two-copy cubic test to ∇ 3 f \nabla^3 f ∇ 3 f R R R R R R
∣ P ( z , w l ) ∣ ≤ K a , ∣ P s , w ( z , w l ) ∣ ≤ K , ∣ P s , w w ∣ ≤ K / a , ∣ w l − f ( u ) ∣ ≤ K a 2 . |P(z,w_l)|\le Ka,\qquad|P_{s,w}(z,w_l)|\le K,\qquad|P_{s,ww}|\le K/a,\qquad|w_l-f(u)|\le Ka_2. ∣ P ( z , w l ) ∣ ≤ K a , ∣ P s , w ( z , w l ) ∣ ≤ K , ∣ P s , w w ∣ ≤ K / a , ∣ w l − f ( u ) ∣ ≤ K a 2 . Its exact quadratic Taylor formula in w w w
∣ R ( u ) ∣ ≤ K ( 1 + a 2 / a + ( a 2 / a ) 2 ) ≤ K . |R(u)|\le K\left(1+a_2/a+(a_2/a)^2\right)\le K. ∣ R ( u ) ∣ ≤ K ( 1 + a 2 / a + ( a 2 / a ) 2 ) ≤ K . The base density of this choice is bounded by the original group ceiling K h ν Q ( a 2 / a ) d K h\nu_Q(a_2/a)^d K h ν Q ( a 2 / a ) d (6.31) , this gives real projection volume at least K − 1 K^{-1} K − 1 R R R K K K
Along a contributing line, compare f f f w l w_l w l R R R P s ( z , w l ) / a P_s(z,w_l)/a P s ( z , w l ) / a K − 1 K^{-1} K − 1
∣ D v 2 f ∣ ≤ K a 2 , ∣ D v 3 R ∣ ≤ K a 2 / a . (6.33) \lvert D_{\mathbf v}^2f\rvert\le K a_2,\qquad\lvert D_{\mathbf v}^3R\rvert\le K a_2/a.\tag{6.33} ∣ D v 2 f ∣ ≤ K a 2 , ∣ D v 3 R ∣ ≤ K a 2 / a . ( 6.33 ) Let
S 0 = inf L a f f i n e ∥ f − L ∥ C 0 ( U ) . S_0=\inf_{L\ \mathrm{affine}}\lVert f-L\rVert_{C^0(U)}. S 0 = L affine inf ∥ f − L ∥ C 0 ( U ) . For S 0 > 0 S_0>0 S 0 > 0 f − L f-L f − L ∥ f − L ∥ C 0 ( U ) ≤ 2 S 0 \lVert f-L\rVert_{C^0(U)}\le2S_0 ∥ f − L ∥ C 0 ( U ) ≤ 2 S 0 K S 0 K S_0 K S 0 S 0 = 0 S_0=0 S 0 = 0 f f f
Sample seven velocities in each retained state, sequentially avoiding all preceding points and joining lines. The prepared breadth gives a positive probability that all triple determinants of v i = ( 1 , v i ) \mathbf{v}_i=(1,v_i) v i = ( 1 , v i ) K − 1 K^{-1} K − 1 K − 1 K^{-1} K − 1 R R R a 2 / a a_2/a a 2 / a f f f K S 0 K S_0 K S 0 h 2 / K h^2/K h 2 / K S 0 S_0 S 0 K − 1 K^{-1} K − 1 U U U
∣ D v i 2 f ∣ ≤ K ( a 2 + S 0 h ) , ∣ D v i 3 R ∣ ≤ K a 2 / a . (6.34) \lvert D_{\mathbf v_i}^2f\rvert\le K(a_2+S_0h),\qquad\lvert D_{\mathbf v_i}^3R\rvert\le K a_2/a.\tag{6.34} ∣ D v i 2 f ∣ ≤ K ( a 2 + S 0 h ) , ∣ D v i 3 R ∣ ≤ K a 2 / a . ( 6.34 ) The weaker S 0 h S_0h S 0 h S 0 h 2 S_0h^2 S 0 h 2 K K K
Differentiate the first tests once and apply the injective cubic two-copy test in Lemma 6.6. Its polynomial conditioning is subpower, so
∥ ∇ 3 f ∥ ≤ K ( a 2 + S 0 h ) . (6.35) \lVert\nabla^3 f\rVert\le K(a_2+S_0h).\tag{6.35} ∥ ∇ 3 f ∥ ≤ K ( a 2 + S 0 h ) . ( 6.35 ) If S 0 ≤ a 2 / h S_0\le a_2/h S 0 ≤ a 2 / h f 2 f_2 f 2 f f f U U U K a 2 K a_2 K a 2 w − f 2 ( ( z − z c ) / h ) w-f_2((z-z_c)/h) w − f 2 (( z − z c ) / h ) S 0 = 0 S_0=0 S 0 = 0
The conic case. Suppose S 0 > a 2 / h S_0>a_2/h S 0 > a 2 / h u c u_c u c L ′ L' L ′ f f f
q ( x ) = 1 2 S 0 ∇ 2 f ( u c ) [ x , x ] . q(x)=\frac{1}{2S_0}\nabla^2 f(u_c)[x,x]. q ( x ) = 2 S 0 1 ∇ 2 f ( u c ) [ x , x ] . Taylor’s theorem and Equation (6.35) give
f ( u ) − L ′ ( u ) S 0 = q ( u − u c ) + O ( K h ) . (6.36) \frac{f(u)-L'(u)}{S_0}=q(u-u_c)+O(Kh).\tag{6.36} S 0 f ( u ) − L ′ ( u ) = q ( u − u c ) + O ( K h ) . ( 6.36 ) The relative derivative bound gives ∥ q ∥ ≤ K \lVert q\rVert\le K ∥ q ∥ ≤ K S 0 S_0 S 0
1 ≤ ∥ q ∥ C 0 ( U − u c ) + K h ≤ K ∥ q ∥ + K h , 1 \le\lVert q\rVert_{C^0(U-u_c)} + Kh \le K\lVert q\rVert+ Kh, 1 ≤ ∥ q ∥ C 0 ( U − u c ) + K h ≤ K ∥ q ∥ + K h , so ∥ q ∥ ≥ K − 1 \lVert q\rVert\ge K^{-1} ∥ q ∥ ≥ K − 1 ∣ q ( v i ) ∣ ≤ K h \lvert q(\mathbf v_i)\rvert\le Kh ∣ q ( v i )∣ ≤ K h A = N o ( 1 ) A=N^{o(1)} A = N o ( 1 ) h h h K − 1 K^{-1} K − 1 q q q v i ′ = ( 1 , v i ′ ) \mathbf v_i'=(1,v_i') v i ′ = ( 1 , v i ′ )
∣ v i ′ − v i ∣ ≤ K h , ∣ det ( v i ′ , v j ′ , v k ′ ) ∣ ≥ K − 1 . \lvert\mathbf v_i'-\mathbf v_i\rvert\le Kh,\qquad\lvert\det(\mathbf v_i',\mathbf v_j',\mathbf v_k')\rvert\ge K^{-1}. ∣ v i ′ − v i ∣ ≤ K h , ∣ det ( v i ′ , v j ′ , v k ′ )∣ ≥ K − 1 . Thus even a nearly singular or nearly definite proposed quadratic is quantitatively controlled before inversion.
The independent derivative bounds for R R R (6.32) show that these shifts alter its rank-three, rank-four and rank-five contractions by at most K h Kh K h a 2 / a = h e 2 a_2/a=h^{e_2} a 2 / a = h e 2 e 2 < 1 e_2<1 e 2 < 1 (6.34) therefore remain K a 2 / a Ka_2/a K a 2 / a R R R K a 2 / a Ka_2/a K a 2 / a u c u_c u c K a 2 / a Ka_2/a K a 2 / a q q q q 2 q^2 q 2
R ( u ) = R 2 ( u ) + q ( u − u c ) L 1 ( u ) + λ 0 q ( u − u c ) 2 + O ( K a 2 / a ) , (6.37) R(u)=R_2(u)+q(u-u_c)L_1(u)+\lambda_0q(u-u_c)^2+O(Ka_2/a),\tag{6.37} R ( u ) = R 2 ( u ) + q ( u − u c ) L 1 ( u ) + λ 0 q ( u − u c ) 2 + O ( K a 2 / a ) , ( 6.37 ) where R 2 R_2 R 2 L 1 L_1 L 1 K K K ∥ q ∥ ≥ K − 1 \lVert q\rVert\ge K^{-1} ∥ q ∥ ≥ K − 1 q q q q 2 q^2 q 2 K K K L 1 L_1 L 1 R 2 R_2 R 2
On the earlier long witnesses, put
H ′ ( u , w ) = w − L ′ ( u ) S 0 . H'(u,w)=\frac{w-L'(u)}{S_0}. H ′ ( u , w ) = S 0 w − L ′ ( u ) . Because a 2 / S 0 < h a_2/S_0<h a 2 / S 0 < h (6.36) and ∣ w − f ( u ) ∣ ≤ K a 2 \lvert w-f(u)\rvert\le Ka_2 ∣ w − f ( u )∣ ≤ K a 2 H ′ = q ( u − u c ) + O ( K h ) H'=q(u-u_c)+O(Kh) H ′ = q ( u − u c ) + O ( K h ) ∣ H ′ ∣ ≤ K \lvert H'\rvert\le K ∣ H ′ ∣ ≤ K
P n e w ( z , w ) = P s ( z , w ) − a ( R 2 ( u ) + H ′ ( u , w ) L 1 ( u ) + λ 0 H ′ ( u , w ) 2 ) , u = ( z − z c ) / h . (6.38) P_{\mathrm{new}}(z,w)=P_s(z,w)-a\left(R_2(u)+H'(u,w)L_1(u)+\lambda_0H'(u,w)^2\right),\qquad u=(z-z_c)/h.\tag{6.38} P new ( z , w ) = P s ( z , w ) − a ( R 2 ( u ) + H ′ ( u , w ) L 1 ( u ) + λ 0 H ′ ( u , w ) 2 ) , u = ( z − z c ) / h . ( 6.38 ) Every term has total degree at most two in ( z , w ) (z,w) ( z , w ) u u u H ′ H' H ′ L ′ L' L ′ L 1 L_1 L 1 H ′ H' H ′ q q q K a h Kah K ah a 2 a_2 a 2 h 1 − e 2 h^{1-e_2} h 1 − e 2 f f f w w w P s P_s P s
K ∣ w − f ∣ + ( K / a ) ∣ w − f ∣ 2 ≤ K a 2 + K a 2 2 / a ≤ K a 2 . K\lvert w-f\rvert+(K/a)\lvert w-f\rvert^2\le Ka_2+Ka_2^2/a\le Ka_2. K ∣ w − f ∣ + ( K / a ) ∣ w − f ∣ 2 ≤ K a 2 + K a 2 2 / a ≤ K a 2 . Together with Equation (6.37) , this proves the K a 2 Ka_2 K a 2
The exact derivative perturbations are
∂ w ( P s − P n e w ) = a S 0 ( L 1 + 2 λ 0 H ′ ) , ∂ w w ( P s − P n e w ) = 2 a λ 0 S 0 2 . \partial_w(P_s-P_{\mathrm{new}})=\frac{a}{S_0}(L_1+2\lambda_0H'),\qquad
\partial_{ww}(P_s-P_{\mathrm{new}})=\frac{2a\lambda_0}{S_0^2}. ∂ w ( P s − P new ) = S 0 a ( L 1 + 2 λ 0 H ′ ) , ∂ w w ( P s − P new ) = S 0 2 2 a λ 0 . Consequently
∣ ∂ w ( P s − P n e w ) ∣ ≤ K a / S 0 < K h 1 − e 2 ≪ K − 1 , (6.39) \lvert\partial_w(P_s-P_{\mathrm{new}})\rvert\le Ka/S_0<Kh^{1-e_2}\ll K^{-1},\tag{6.39} ∣ ∂ w ( P s − P new )∣ ≤ K a / S 0 < K h 1 − e 2 ≪ K − 1 , ( 6.39 ) ∣ ∂ w w P n e w ∣ ≤ K / a + K a / S 0 2 ≤ K / a 2 + ( K / a 2 ) h 2 − e 2 ≤ K / a 2 . (6.40) \lvert\partial_{ww}P_{\mathrm{new}}\rvert\le K/a+Ka/S_0^2\le K/a_2+(K/a_2)h^{2-e_2}\le K/a_2.\tag{6.40} ∣ ∂ w w P new ∣ ≤ K / a + K a / S 0 2 ≤ K / a 2 + ( K / a 2 ) h 2 − e 2 ≤ K / a 2 . ( 6.40 ) The old derivative lower bound at the counted observations survives Equation (6.39) ; the upper bound holds on the earlier witnesses. No magnitude bound for w w w L ′ L' L ′ S 0 S_0 S 0 H ′ H' H ′ S 0 S_0 S 0
The same time block. In both cases the test is an allowed quadratic. On each contributing version 2 trajectory, z ( t ) z(t) z ( t ) w ( t ) w(t) w ( t ) h / K h/K h / K h h h Q s Q_s Q s K K K w w w h / K h/K h / K parent time interval.
The retained choice has mass h ν Q ( a 2 / a ) d / K h\nu_Q(a_2/a)^d/K h ν Q ( a 2 / a ) d / K (6.31) or the heavy intercept floor in version 1. Degree restoration makes no further discard. Dividing by its time length h h h ν Q ( a 2 / a ) d / K \nu_Q(a_2/a)^d/K ν Q ( a 2 / a ) d / K
Conclusion of Theorem 6.1. Start with any substantial residual of the reference path. The physical estimates and Proposition 6.2 retain a substantial reference family. Proposition 6.3 is an assertion of relative o ( 1 ) o(1) o ( 1 ) (6.21) therefore leaves substantial good-state mass. Cartesian interpolation selects a packet and states of mass at least h ν Q / K h\nu_Q/K h ν Q / K (6.20) , its gain has positive limit e 2 k s e_2ks e 2 k s
Corollary 6.9 (Zero horizon in the scalar model). For every fixed admissible choice of angular parameters and every d < 1 d<1 d < 1 C ( d ) C(d) C ( d ) H ( E ) = 0 H(E)=0 H ( E ) = 0
Proof. If a bad version 1 problem existed, the critical-path construction of Proposition 4.3 would give a tangent problem with positive k k k
This scalar conclusion is established before the finite-narrowness analysis. It supplies the version 1 input to Theorem 8.2, whose projection labels are then used in Sections 8 to 10. The unbounded-narrowness case retains its separate limiting construction.
We prove that a row whose evaluation is nearly constant along many lines agrees, on many incidences, with a constant row plus a multiple of J z Jz J z J ( t , Z ) = ( − Z , t ) J(t,Z)=(-Z,t) J ( t , Z ) = ( − Z , t )
The main step is planar rigidity for graphs with small restricted sumsets: many original edges have their endpoints in two parallel strips, of width equal to the mesh up to a subpower factor and with subpower-bounded common slope. Keeping original edges allows this geometric conclusion to be lifted back to incidence mass. We first prove the planar statement, then apply it to the line projections that arise from the row.
Throughout this section h = N − a h = N^{-a} h = N − a a > 0 a > 0 a > 0 K = h − o ( 1 ) K = h^{-o(1)} K = h − o ( 1 ) U U U N h ( U ) N_h(U) N h ( U ) h h h h − 1 h^{-1} h − 1 h 1 − o ( 1 ) h^{1-o(1)} h 1 − o ( 1 )
Graph refinement and common endpoint sets
We begin with the graph form of Balog–Szemerédi–Gowers that records the edges surviving the sumset reduction. The three-path argument follows Sudakov et al. [28 ] (Lemma 4.2 and the proof of Theorem 4.1); compare also Tao and Vu [29 ] (Theorem 2.29 and Exercise 6.4.10) for the graph form and retention of original edges. We give the proof with the polynomial dependence needed after mesh rounding.
Lemma 7.1 (Three paths retain graph edges). Let A , B A, B A , B a 0 = ∣ A ∣ a_0 = |A| a 0 = ∣ A ∣ b 0 = ∣ B ∣ b_0 = |B| b 0 = ∣ B ∣ G ⊂ A × B G \subset A \times B G ⊂ A × B θ a 0 b 0 \theta a_0b_0 θ a 0 b 0 0 < θ ≤ 1 0 < \theta\le1 0 < θ ≤ 1 C = { x + y : ( x , y ) ∈ G } C = \{x + y : (x, y) \in G\} C = { x + y : ( x , y ) ∈ G } A ′ ⊂ A A' \subset A A ′ ⊂ A B ′ ⊂ B B' \subset B B ′ ⊂ B
∣ A ′ ∣ ≥ c θ a 0 , ∣ B ′ ∣ ≥ c θ b 0 , |A'| \ge c\theta a_0,\qquad|B'| \ge c\theta b_0, ∣ A ′ ∣ ≥ c θ a 0 , ∣ B ′ ∣ ≥ c θ b 0 , ∣ G ∩ ( A ′ × B ′ ) ∣ ≥ c θ 2 a 0 b 0 , ∣ A ′ + B ′ ∣ ≤ C 0 θ − 5 ∣ C ∣ 3 a 0 b 0 , |G \cap(A' \times B')| \ge c\theta^2 a_0b_0,\qquad|A' + B'| \le C_0\theta^{-5}\frac{|C|^3}{a_0b_0}, ∣ G ∩ ( A ′ × B ′ ) ∣ ≥ c θ 2 a 0 b 0 , ∣ A ′ + B ′ ∣ ≤ C 0 θ − 5 a 0 b 0 ∣ C ∣ 3 , where c , C 0 > 0 c, C_0 > 0 c , C 0 > 0
Proof. We first choose endpoint sets with many three-edge walks between every pair. The sums along those walks will bound the full sumset, while degree counting will retain many edges of the original graph. Remove vertices of A A A θ b 0 / 2 \theta b_0/2 θ b 0 /2 A 0 A_0 A 0 θ a 0 b 0 / 2 \theta a_0b_0/2 θ a 0 b 0 /2 A 0 2 A_0^2 A 0 2 τ b 0 \tau b_0 τ b 0 τ = θ 3 / 512 \tau= \theta^3/512 τ = θ 3 /512 v 0 ∈ B v_0 \in B v 0 ∈ B U = N ( v 0 ) ∩ A 0 U = N(v_0) \cap A_0 U = N ( v 0 ) ∩ A 0
E ∣ U ∣ ≥ θ a 0 / 2 , E ∣ U ∣ 2 ≥ θ 2 a 0 2 / 4 , E Bad ( U ) ≤ τ a 0 2 , \mathbb{E}|U| \ge\theta a_0/2,\qquad\mathbb{E}|U|^2 \ge\theta^2a_0^2/4,\qquad\mathbb{E}\operatorname{Bad}(U) \le\tau a_0^2, E ∣ U ∣ ≥ θ a 0 /2 , E ∣ U ∣ 2 ≥ θ 2 a 0 2 /4 , E Bad ( U ) ≤ τ a 0 2 , the last bound because a bad pair has probability below τ \tau τ U 2 U^2 U 2 U U U
∣ U ∣ 2 − ( 32 / θ ) Bad ( U ) ≥ 3 θ 2 a 0 2 / 16. |U|^2 - (32/\theta)\operatorname{Bad}(U) \ge3\theta^2a_0^2/16. ∣ U ∣ 2 − ( 32/ θ ) Bad ( U ) ≥ 3 θ 2 a 0 2 /16. It follows that ∣ U ∣ ≥ θ a 0 / ( 2 2 ) |U| \ge\theta a_0/(2\sqrt{2}) ∣ U ∣ ≥ θ a 0 / ( 2 2 ) Bad ( U ) ≤ θ ∣ U ∣ 2 / 32 \operatorname{Bad}(U) \le\theta|U|^2/32 Bad ( U ) ≤ θ ∣ U ∣ 2 /32 θ ∣ U ∣ / 16 \theta|U|/16 θ ∣ U ∣/16 U U U A ′ A' A ′
B ′ = { v ∈ B : ∣ N ( v ) ∩ A ′ ∣ ≥ θ ∣ A ′ ∣ / 4 } . B' = \{v \in B : |N(v) \cap A'| \ge\theta|A'|/4\}. B ′ = { v ∈ B : ∣ N ( v ) ∩ A ′ ∣ ≥ θ ∣ A ′ ∣/4 } . Every member of A ′ A' A ′ θ b 0 / 2 \theta b_0/2 θ b 0 /2 B ∖ B ′ B \setminus B' B ∖ B ′ θ ∣ A ′ ∣ b 0 / 4 \theta|A'|b_0/4 θ ∣ A ′ ∣ b 0 /4
∣ G ∩ ( A ′ × B ′ ) ∣ ≥ θ ∣ A ′ ∣ b 0 / 4 , ∣ B ′ ∣ ≥ θ b 0 / 4. |G \cap(A' \times B')| \ge\theta|A'|b_0/4,\qquad|B'| \ge\theta b_0/4. ∣ G ∩ ( A ′ × B ′ ) ∣ ≥ θ ∣ A ′ ∣ b 0 /4 , ∣ B ′ ∣ ≥ θ b 0 /4. Fix u ∈ A ′ u \in A' u ∈ A ′ v ∈ B ′ v \in B' v ∈ B ′ θ ∣ A ′ ∣ / 4 ≥ θ ∣ U ∣ / 8 \theta|A'|/4 \ge\theta|U|/8 θ ∣ A ′ ∣/4 ≥ θ ∣ U ∣/8 N ( v ) ∩ A ′ N(v) \cap A' N ( v ) ∩ A ′ θ ∣ U ∣ / 16 \theta|U|/16 θ ∣ U ∣/16 u u u u 1 u_1 u 1 τ b 0 \tau b_0 τ b 0 v 1 v_1 v 1 u u u u , v u,v u , v c θ 5 a 0 b 0 c\theta^5a_0b_0 c θ 5 a 0 b 0 u , v 1 , u 1 , v u,v_1,u_1,v u , v 1 , u 1 , v
( u + v 1 , u 1 + v 1 , u 1 + v ) ∈ C 3 , (u + v_1,\ u_1 + v_1,\ u_1 + v) \in C^3, ( u + v 1 , u 1 + v 1 , u 1 + v ) ∈ C 3 , and, for fixed u , v u,v u , v u + v u+v u + v A ′ + B ′ A' + B' A ′ + B ′ c θ 5 a 0 b 0 ∣ A ′ + B ′ ∣ ≤ ∣ C ∣ 3 c\theta^5a_0b_0|A' + B'| \le|C|^3 c θ 5 a 0 b 0 ∣ A ′ + B ′ ∣ ≤ ∣ C ∣ 3
Here is the common finite-mesh input for the next three lemmas. The parameter law may be a law on indices: distinct direction bins with the same rounded parameter are not silently identified. The graph bounds refer to the corresponding original edge sets.
Assumption 7.2 (Planar graph data). Let A , B ⊂ h Z 2 ∩ [ − K , K ] 2 A,B \subset h\mathbb{Z}^{2}\cap[-K,K]^{2} A , B ⊂ h Z 2 ∩ [ − K , K ] 2 h − 1 / K ≤ ∣ A ∣ , ∣ B ∣ ≤ K h − 1 h^{-1}/K \le|A|,|B| \le Kh^{-1} h − 1 / K ≤ ∣ A ∣ , ∣ B ∣ ≤ K h − 1 ν \nu ν κ \kappa κ S > 0 S>0 S > 0
K − 1 ≤ ∣ κ ∣ ≤ K , ν ( I ) ≤ K ∣ I ∣ S ( h ≤ ∣ I ∣ ≤ 1 ) . (7.1) K^{-1} \le|\kappa| \le K,\qquad\nu(I) \le K|I|^{S}\quad(h \le|I| \le1).\tag{7.1} K − 1 ≤ ∣ κ ∣ ≤ K , ν ( I ) ≤ K ∣ I ∣ S ( h ≤ ∣ I ∣ ≤ 1 ) . ( 7.1 ) For each index there is G κ ⊂ A × B G_{\kappa}\subset A\times B G κ ⊂ A × B
∣ G κ ∣ ≥ h − 2 / K , N h { x + κ y : ( x , y ) ∈ G κ } ≤ K h − 1 . (7.2) |G_{\kappa}| \ge h^{-2}/K,\qquad N_{h}\{x+\kappa y:(x,y)\in G_{\kappa}\}\le Kh^{-1}.\tag{7.2} ∣ G κ ∣ ≥ h − 2 / K , N h { x + κ y : ( x , y ) ∈ G κ } ≤ K h − 1 . ( 7.2 ) The uniform law on G κ G_{\kappa} G κ I , I ′ I,I' I , I ′ h h h
P ( x 1 ∈ I , y 1 ∈ I ′ ) ≤ K h 2 . (7.3) \mathbb{P}(x_{1}\in I,y_{1}\in I')\le Kh^{2}.\tag{7.3} P ( x 1 ∈ I , y 1 ∈ I ′ ) ≤ K h 2 . ( 7.3 ) Set L = h − 1 L=h^{-1} L = h − 1 κ \kappa κ A A A κ B \kappa B κ B h Z 2 h\mathbb{Z}^{2} h Z 2 ∣ κ ∣ |\kappa| ∣ κ ∣ K K K K K K K 2 K^{2} K 2 K L KL K L A κ ⊂ A A_{\kappa}\subset A A κ ⊂ A B κ ⊂ B B_{\kappa}\subset B B κ ⊂ B
∣ A κ ∣ , ∣ B κ ∣ ≥ L / K , N h ( A κ + κ B κ ) ≤ K L , ∣ G κ ∩ ( A κ × B κ ) ∣ ≥ L 2 / K . (7.4) |A_{\kappa}|,|B_{\kappa}|\ge L/K,\qquad N_{h}(A_{\kappa}+\kappa B_{\kappa})\le KL,\qquad|G_{\kappa}\cap(A_{\kappa}\times B_{\kappa})|\ge L^{2}/K.\tag{7.4} ∣ A κ ∣ , ∣ B κ ∣ ≥ L / K , N h ( A κ + κ B κ ) ≤ K L , ∣ G κ ∩ ( A κ × B κ ) ∣ ≥ L 2 / K . ( 7.4 ) We use the covering-number forms of the Ruzsa triangle and fixed-iterate Plünnecke–Ruzsa inequalities [29 ] :
N h ( U − W ) N h ( V ) ≤ C N h ( U − V ) N h ( V − W ) , (7.5) N_{h}(U-W)N_{h}(V)\le CN_{h}(U-V)N_{h}(V-W),\tag{7.5} N h ( U − W ) N h ( V ) ≤ C N h ( U − V ) N h ( V − W ) , ( 7.5 ) N h ( m U − n U ) ≤ C m , n K C m , n N h ( U ) if N h ( U − U ) ≤ K N h ( U ) . (7.6) N_{h}(mU-nU)\le C_{m,n}K^{C_{m,n}}N_{h}(U)\quad\text{if }N_{h}(U-U)\le KN_{h}(U).\tag{7.6} N h ( m U − n U ) ≤ C m , n K C m , n N h ( U ) if N h ( U − U ) ≤ K N h ( U ) . ( 7.6 ) These apply to h h h u − w u-w u − w v v v V V V ( u − v , v − w ) (u-v,v-w) ( u − v , v − w ) v v v
N h ( U + V ) ≤ K N h ( V ) ⟹ U ⊂ ⋃ i = 1 C K ( u i + V − V + B ( 0 , C h ) ) . (7.7) N_{h}(U+V)\le KN_{h}(V)\quad\Longrightarrow\quad U\subset\bigcup_{i=1}^{CK}(u_{i}+V-V+B(0,Ch)).\tag{7.7} N h ( U + V ) ≤ K N h ( V ) ⟹ U ⊂ i = 1 ⋃ C K ( u i + V − V + B ( 0 , C h )) . ( 7.7 ) To see this, take a maximal disjoint collection of translates of V + B ( 0 , h ) V+B(0,h) V + B ( 0 , h ) U + V + B ( 0 , h ) U+V+B(0,h) U + V + B ( 0 , h )
The sets in (7.4) depend on the parameter. We next choose one pair of endpoint sets that has large overlap with the sets for many parameters. The Ruzsa inequalities then control sums of fixed numbers of dilates of this one pair, including the scalar expressions needed later in the collision argument.
Lemma 7.3 (A common pair and scalar closure). After restricting ν \nu ν κ 0 \kappa_{0} κ 0 A ∗ ⊂ A A_{*}\subset A A ∗ ⊂ A B ∗ ⊂ B B_{*}\subset B B ∗ ⊂ B R = { κ / κ 0 } \mathcal{R}=\{\kappa/\kappa_{0}\} R = { κ / κ 0 } L 2 / K L^{2}/K L 2 / K G κ 0 G_{\kappa_{0}} G κ 0 A ∗ × B ∗ A_{*}\times B_{*} A ∗ × B ∗ E = A ∗ E=A_{*} E = A ∗ F = − κ 0 B ∗ F=-\kappa_{0}B_{*} F = − κ 0 B ∗ m , H m,H m , H
c 1 E + ⋯ + c m E + d 1 F + ⋯ + d m F (7.8) c_{1}E+\cdots+c_{m}E+d_{1}F+\cdots+d_{m}F\tag{7.8} c 1 E + ⋯ + c m E + d 1 F + ⋯ + d m F ( 7.8 ) has at most K m H , L K^{m_{H,L}} K m H , L h h h H H H H H H R \mathcal{R} R R \mathcal{R} R S S S
Proof. The rectangles A κ × B κ A_\kappa\times B_\kappa A κ × B κ A × B A \times B A × B K 2 L 2 K^2L^2 K 2 L 2
E κ , κ ′ ∣ ( A κ × B κ ) ∩ ( A κ ′ × B κ ′ ) ∣ ≥ ( E κ ∣ A κ × B κ ∣ ) 2 ∣ A × B ∣ ≥ L 2 K 6 . \mathbb{E}_{\kappa,\kappa'}\lvert(A_\kappa\times B_\kappa) \cap(A_{\kappa'} \times B_{\kappa'})\rvert\ge\frac{\left(\mathbb{E}_\kappa\lvert A_\kappa\times B_\kappa\rvert\right)^2}{\lvert A \times B\rvert} \ge\frac{L^2}{K^6}. E κ , κ ′ ∣( A κ × B κ ) ∩ ( A κ ′ × B κ ′ )∣ ≥ ∣ A × B ∣ ( E κ ∣ A κ × B κ ∣ ) 2 ≥ K 6 L 2 . Choose κ 0 \kappa_0 κ 0 L 2 / ( 2 K 6 ) L^2/(2K^6) L 2 / ( 2 K 6 ) 1 / ( 2 K 8 ) 1/(2K^8) 1/ ( 2 K 8 ) A ∗ = A κ 0 A_* = A_{\kappa_0} A ∗ = A κ 0 B ∗ = B κ 0 B_* = B_{\kappa_0} B ∗ = B κ 0 K L KL K L
∣ A κ ∩ A ∗ ∣ , ∣ B κ ∩ B ∗ ∣ ≥ L 2 K 7 . \lvert A_\kappa\cap A_*\rvert,\ \lvert B_\kappa\cap B_*\rvert\ge\frac{L}{2K^7}. ∣ A κ ∩ A ∗ ∣ , ∣ B κ ∩ B ∗ ∣ ≥ 2 K 7 L . The edge assertion follows from (7.4) at κ 0 \kappa_0 κ 0 κ 0 \kappa_0 κ 0
For clarity about the uniform set calculus, put
d h ( U , V ) = log N h ( U − V ) N h ( U ) N h ( V ) . d_h(U,V)=\log\frac{N_h(U-V)}{\sqrt{N_h(U)N_h(V)}}. d h ( U , V ) = log N h ( U ) N h ( V ) N h ( U − V ) . The triangle inequality (7.5) is the triangle inequality for d h d_h d h O ( 1 ) O(1) O ( 1 ) h h h h / K C h/K^C h / K C K C K^C K C O C ( log K ) O_C(\log K) O C ( log K ) O ( K 2 C ) O(K^{2C}) O ( K 2 C ) d h ( A κ , − κ B κ ) = O ( log K ) d_h(A_\kappa,-\kappa B_\kappa)=O(\log K) d h ( A κ , − κ B κ ) = O ( log K ) A κ A_\kappa A κ B κ B_\kappa B κ H κ = A κ ∩ A ∗ H_\kappa=A_\kappa\cap A_* H κ = A κ ∩ A ∗ A κ − H κ A_\kappa-H_\kappa A κ − H κ H κ − A ∗ H_\kappa-A_* H κ − A ∗ ∣ H κ ∣ ≥ L / ( 2 K 7 ) \lvert H_\kappa\rvert\ge L/(2K^7) ∣ H κ ∣ ≥ L / ( 2 K 7 ) H κ H_\kappa H κ d h ( A κ , A ∗ ) = O ( log K ) d_h(A_\kappa,A_*)=O(\log K) d h ( A κ , A ∗ ) = O ( log K ) B B B d h ( B κ , B ∗ ) = O ( log K ) d_h(B_\kappa,B_*)=O(\log K) d h ( B κ , B ∗ ) = O ( log K )
d h ( E , F ) = O ( log K ) , d h ( E , t F ) = O ( log K ) ( t ∈ R ) . (7.9) d_h(E,F)=O(\log K),\qquad d_h(E,tF)=O(\log K)\quad(t\in\mathcal{R}).\tag{7.9} d h ( E , F ) = O ( log K ) , d h ( E , tF ) = O ( log K ) ( t ∈ R ) . ( 7.9 ) Here the second relation uses t F = − κ B ∗ tF=-\kappa B_* tF = − κ B ∗ F F F E E E t F tF tF
d h ( E , t E ) ≤ d h ( E , t F ) + d h ( t F , t E ) + O ( 1 ) = d h ( E , t F ) + d h ( F , E ) + O ( log K ) = O ( log K ) . d_h(E,tE)\le d_h(E,tF)+d_h(tF,tE)+O(1)
= d_h(E,tF)+d_h(F,E)+O(\log K)=O(\log K). d h ( E , tE ) ≤ d h ( E , tF ) + d h ( tF , tE ) + O ( 1 ) = d h ( E , tF ) + d h ( F , E ) + O ( log K ) = O ( log K ) . The dilation error is uniform because both ∣ t ∣ \lvert t\rvert ∣ t ∣ ∣ t ∣ − 1 \lvert t\rvert^{-1} ∣ t ∣ − 1 p j = t 1 ⋯ t j p_j=t_1\cdots t_j p j = t 1 ⋯ t j t i ∈ R t_i\in\mathcal{R} t i ∈ R p 0 = 1 p_0=1 p 0 = 1 j j j
d h ( E , p j E ) ≤ d h ( E , p j − 1 E ) + d h ( p j − 1 E , p j − 1 t j E ) + O ( 1 ) ≤ d h ( E , p j − 1 E ) + d h ( E , t j E ) + O j ( log K ) = O j ( log K ) , d h ( E , p j F ) ≤ d h ( E , p j E ) + d h ( p j E , p j F ) + O ( 1 ) ≤ d h ( E , p j E ) + d h ( E , F ) + O j ( log K ) = O j ( log K ) . \begin{aligned}
d_h(E,p_jE)&\le d_h(E,p_{j-1}E)+d_h(p_{j-1}E,p_{j-1}t_jE)+O(1)\\
&\le d_h(E,p_{j-1}E)+d_h(E,t_jE)+O_j(\log K)=O_j(\log K),\\
d_h(E,p_jF)&\le d_h(E,p_jE)+d_h(p_jE,p_jF)+O(1)\\
&\le d_h(E,p_jE)+d_h(E,F)+O_j(\log K)=O_j(\log K).
\end{aligned} d h ( E , p j E ) d h ( E , p j F ) ≤ d h ( E , p j − 1 E ) + d h ( p j − 1 E , p j − 1 t j E ) + O ( 1 ) ≤ d h ( E , p j − 1 E ) + d h ( E , t j E ) + O j ( log K ) = O j ( log K ) , ≤ d h ( E , p j E ) + d h ( p j E , p j F ) + O ( 1 ) ≤ d h ( E , p j E ) + d h ( E , F ) + O j ( log K ) = O j ( log K ) . Here the first bound is inductive, starting with the small self-difference of E E E p j p_j p j j j j
To include signed products, (7.6) with m = 2 m=2 m = 2 n = 0 n=0 n = 0 N h ( E + E ) ≤ K L N_h(E+E) \le KL N h ( E + E ) ≤ K L d h ( E , − E ) = O ( log K ) d_h(E,-E)=O(\log K) d h ( E , − E ) = O ( log K ) X = E X=E X = E F F F
d h ( E , − p j X ) ≤ d h ( E , − E ) + d h ( − E , − p j X ) + O ( 1 ) = d h ( E , − E ) + d h ( E , p j X ) + O ( 1 ) = O j ( log K ) . d_h(E,-p_jX) \le d_h(E,-E)+d_h(-E,-p_jX)+O(1)
= d_h(E,-E)+d_h(E,p_jX)+O(1)=O_j(\log K). d h ( E , − p j X ) ≤ d h ( E , − E ) + d h ( − E , − p j X ) + O ( 1 ) = d h ( E , − E ) + d h ( E , p j X ) + O ( 1 ) = O j ( log K ) . More concretely, for every signed product dilate U U U E E E F F F N h ( U − E ) ≤ K H L N_h(U-E) \le K_HL N h ( U − E ) ≤ K H L N h ( E ) ≥ L / K N_h(E) \ge L/K N h ( E ) ≥ L / K − E -E − E U U U K H K_H K H E − E + B ( 0 , K H h ) E-E+B(0,K_Hh) E − E + B ( 0 , K H h ) K m , H K_{m,H} K m , H E − E E-E E − E K m , H L K_{m,H}L K m , H L c = ∑ ℓ = 1 q ϵ ℓ α ℓ c=\sum_{\ell=1}^{q}\epsilon_\ell\alpha_\ell c = ∑ ℓ = 1 q ϵ ℓ α ℓ q ≤ H q\le H q ≤ H ϵ ℓ ∈ { − 1 , 1 } \epsilon_\ell\in\{-1,1\} ϵ ℓ ∈ { − 1 , 1 } α ℓ \alpha_\ell α ℓ H H H R \mathcal{R} R
c X ⊆ ϵ 1 α 1 X + ⋯ + ϵ q α q X , X = E or F . cX\subseteq\epsilon_1\alpha_1X+\cdots+\epsilon_q\alpha_qX,\qquad X=E\text{ or }F. c X ⊆ ϵ 1 α 1 X + ⋯ + ϵ q α q X , X = E or F . Indeed the left side uses the same point of X X X 2 m 2m 2 m H H H c = 0 c=0 c = 0
Flat scalar coefficients
We now seek random coefficients with two properties: their values belong to the scalar class just constructed, and their probabilities on intervals of length h h h h h h h − 1 − o ( 1 ) h^{-1-o(1)} h − 1 − o ( 1 ) h 1 + o ( 1 ) h^{1+o(1)} h 1 + o ( 1 )
The analytic input is a finite-scale Fourier-decay theorem. For a probability measure λ \lambda λ
I s ( λ ) = ∬ ∣ x − y ∣ − s d λ ( x ) d λ ( y ) , I s r ( λ ) = I s ( λ ∗ P r ) , I_s(\lambda)=\iint|x-y|^{-s}\,d\lambda(x)\,d\lambda(y),\qquad I_s^r(\lambda)=I_s(\lambda*P_r), I s ( λ ) = ∬ ∣ x − y ∣ − s d λ ( x ) d λ ( y ) , I s r ( λ ) = I s ( λ ∗ P r ) , where P r P_r P r r r r [25 ] says: for fixed n ≥ 2 n\ge2 n ≥ 2 s j ∈ ( 0 , 1 ] s_j\in(0,1] s j ∈ ( 0 , 1 ] ∑ j s j > 1 \sum_j s_j>1 ∑ j s j > 1 r 0 , ϵ 0 , τ > 0 r_0,\epsilon_0,\tau>0 r 0 , ϵ 0 , τ > 0 λ j \lambda_j λ j [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] I s j r ( λ j ) ≤ r − ϵ 0 I_{s_j}^r(\lambda_j)\le r^{-\epsilon_0} I s j r ( λ j ) ≤ r − ϵ 0 0 < r < r 0 0<r<r_0 0 < r < r 0
∣ ( λ 1 ⊠ ⋯ ⊠ λ n ) ∧ ( ξ ) ∣ ≤ ∣ ξ ∣ − τ ( r − 1 ≤ ∣ ξ ∣ ≤ 2 r − 1 ) . (7.10) \left|(\lambda_1\boxtimes\cdots\boxtimes\lambda_n)^{\wedge}(\xi)\right|\le|\xi|^{-\tau}\qquad(r^{-1}\le|\xi|\le2r^{-1}).\tag{7.10} ∣ ( λ 1 ⊠ ⋯ ⊠ λ n ) ∧ ( ξ ) ∣ ≤ ∣ ξ ∣ − τ ( r − 1 ≤ ∣ ξ ∣ ≤ 2 r − 1 ) . ( 7.10 ) where ⊠ \boxtimes ⊠ 1 1 1
Lemma 7.4 (Flat generated coefficients). Let ν R \nu_{\mathcal R} ν R e > 0 e>0 e > 0 n , k n,k n , k R ≥ 1 R\ge1 R ≥ 1 T i j ∼ ν R T_{ij}\sim\nu_{\mathcal R} T ij ∼ ν R
C = ∑ i = 1 k ∏ j = 1 n ( T i j / R ) (7.11) C=\sum_{i=1}^{k}\prod_{j=1}^{n}(T_{ij}/R)\tag{7.11} C = i = 1 ∑ k j = 1 ∏ n ( T ij / R ) ( 7.11 ) satisfies
sup x P ( C ∈ [ x − h , x + h ] ) ≤ C e h 1 − e . (7.12) \sup_x\mathbb{P}(C\in[x-h,x+h])\le C_eh^{1-e}.\tag{7.12} x sup P ( C ∈ [ x − h , x + h ]) ≤ C e h 1 − e . ( 7.12 ) Every value in its support is an expression allowed in (7.8) , with the common subpower dilation R − n R^{-n} R − n C C C
Proof. Take R ≥ sup ∣ T ∣ R \ge\sup|T| R ≥ sup ∣ T ∣ R R R λ \lambda λ T / R T/R T / R [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] (7.1) , also for intervals of length above 1 / R 1/R 1/ R λ ( B ( x , u ) ) ≤ K ′ u S \lambda(B(x,u)) \le K'u^{S} λ ( B ( x , u )) ≤ K ′ u S h ≤ u ≤ 1 h \le u \le1 h ≤ u ≤ 1 K ′ = h − o ( 1 ) K'=h^{-o(1)} K ′ = h − o ( 1 ) 0 < s < min ( S , 1 ) 0 < s < \min(S,1) 0 < s < min ( S , 1 ) n ≥ 2 n \ge2 n ≥ 2 n s > 1 ns > 1 n s > 1 h ≤ r ≤ 1 h \le r \le1 h ≤ r ≤ 1 r r r C s max ( r , ∣ x − y ∣ ) − s C_s\max(r,|x-y|)^{-s} C s max ( r , ∣ x − y ∣ ) − s r r r
I s r ( λ ) ≤ C s K ′ ( r S − s + ∑ j : 2 j r ≤ 1 ( 2 j r ) S − s + 1 ) ≤ C s ′ K ′ . (7.13) I_s^r(\lambda) \le C_sK'\left(r^{S-s}+\sum_{j:2^jr\le1}(2^jr)^{S-s}+1\right)\le C_s'K'.\tag{7.13} I s r ( λ ) ≤ C s K ′ r S − s + j : 2 j r ≤ 1 ∑ ( 2 j r ) S − s + 1 ≤ C s ′ K ′ . ( 7.13 ) Fix 0 < e 0 < min ( e / 2 , 1 ) 0 < e_0 < \min(e/2,1) 0 < e 0 < min ( e /2 , 1 ) r 0 , ϵ 0 , τ r_0,\epsilon_0,\tau r 0 , ϵ 0 , τ (7.10) are now fixed. For small enough h h h r ∈ [ h , h e 0 ] r \in[h,h^{e_0}] r ∈ [ h , h e 0 ] r 0 r_0 r 0 K ′ K' K ′ (7.13) at most r − ϵ 0 r^{-\epsilon_0} r − ϵ 0 (7.10) with r = ∣ ξ ∣ − 1 r=|\xi|^{-1} r = ∣ ξ ∣ − 1 ω = λ ⊠ n \omega=\lambda^{\boxtimes n} ω = λ ⊠ n
∣ ω ^ ( ξ ) ∣ ≤ ∣ ξ ∣ − τ ( h − e 0 ≤ ∣ ξ ∣ ≤ h − 1 ) . |\widehat{\omega}(\xi)|\le|\xi|^{-\tau}\qquad(h^{-e_0}\le|\xi|\le h^{-1}). ∣ ω ( ξ ) ∣ ≤ ∣ ξ ∣ − τ ( h − e 0 ≤ ∣ ξ ∣ ≤ h − 1 ) . up to an inessential fixed adjustment at the lower endpoint. Choose fixed k k k k τ > 2 k\tau>2 k τ > 2 η = ω ∗ k \eta=\omega^{*k} η = ω ∗ k (7.11) . Choose a nonnegative integrable ψ ≥ 1 \psi\ge1 ψ ≥ 1 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] x x x
η ( [ x − h , x + h ] ) ≤ ∫ ψ ( ( u − x ) / h ) d η ( u ) ≤ C h ∫ ∣ ξ ∣ ≤ h − 1 ∣ ω ^ ( ξ ) ∣ k d ξ ≤ C h ( h − e 0 + ∫ h − e 0 h − 1 t − k τ d t ) ≤ C h 1 − e 0 ≤ C e h 1 − e . \begin{aligned}
\eta([x-h,x+h])&\le\int\psi((u-x)/h)\,d\eta(u) \\
&\le Ch\int_{|\xi|\le h^{-1}}|\widehat{\omega}(\xi)|^k\,d\xi\\
&\le Ch\left(h^{-e_0}+\int_{h^{-e_0}}^{h^{-1}}t^{-k\tau}\,dt\right)\le Ch^{1-e_0}\le C_eh^{1-e}.
\end{aligned} η ([ x − h , x + h ]) ≤ ∫ ψ (( u − x ) / h ) d η ( u ) ≤ C h ∫ ∣ ξ ∣ ≤ h − 1 ∣ ω ( ξ ) ∣ k d ξ ≤ C h ( h − e 0 + ∫ h − e 0 h − 1 t − k τ d t ) ≤ C h 1 − e 0 ≤ C e h 1 − e . The integers n , k n,k n , k e e e h → 0 h\to0 h → 0 R − n R^{-n} R − n
Parallel strips from collision probabilities
Lemma 7.5 (Weighted parallel strips). Let E , F E,F E , F μ \mu μ h − 2 / K h^{-2}/K h − 2 / K G κ 0 ∩ ( A ∗ × B ∗ ) G_{\kappa_0}\cap(A_*\times B_*) G κ 0 ∩ ( A ∗ × B ∗ ) y ↦ − κ 0 y y\mapsto-\kappa_0y y ↦ − κ 0 y (7.3) holds for μ \mu μ K K K h 1 − o ( 1 ) h^{1-o(1)} h 1 − o ( 1 ) h o ( 1 ) h^{o(1)} h o ( 1 ) μ \mu μ h − o ( 1 ) h^{-o(1)} h − o ( 1 )
Proof. We prove that a fixed-power failure of strip concentration would force most sampled endpoint differences to include a transverse pair. The generated coefficients then give incompatible lower and upper bounds for the probability that two endpoint sums occupy the same mesh cell. Let D ≤ K D\le K D ≤ K
p ( w ) = sup { μ ( S E × S F ) : S E , S F parallel strips of width w } . p(w)=\sup\{\mu(S_E\times S_F):S_E,S_F\text{ parallel strips of width }w\}. p ( w ) = sup { μ ( S E × S F ) : S E , S F parallel strips of width w } . Fix 0 < ε < 1 0 < \varepsilon< 1 0 < ε < 1 c > 0 c > 0 c > 0 p ( w ) ≤ h c p(w) \le h^c p ( w ) ≤ h c w = h 1 − ε w = h^{1-\varepsilon} w = h 1 − ε m m m ( a i , b i ) , ( a i ′ , b i ′ ) ∼ μ (a_i,b_i),(a_i',b_i') \sim\mu ( a i , b i ) , ( a i ′ , b i ′ ) ∼ μ
v i = a i − a i ′ , u i = b i − b i ′ , q = h ε / 2 , Δ = q w / 10. v_i = a_i-a_i', \qquad u_i = b_i-b_i', \qquad q = h^{\varepsilon/2}, \qquad\Delta= qw/10. v i = a i − a i ′ , u i = b i − b i ′ , q = h ε /2 , Δ = q w /10. Summing the rectangle cap over the O ( D / h ) O(D/h) O ( D / h ) P ( ( a i ) 1 ∈ I ) ≤ C K D h \mathbb{P}((a_i)_1 \in I) \le CKDh P (( a i ) 1 ∈ I ) ≤ C K D h h h h I I I P ( ∣ v i ∣ < q ) ≤ C K D ( q + h ) \mathbb{P}(|v_i| < q) \le CKD(q+h) P ( ∣ v i ∣ < q ) ≤ C K D ( q + h )
P ( max r , s ∈ { v i , u i : 1 ≤ i ≤ m } ∣ r ∧ s ∣ ≤ Δ ) ≤ [ C K D ( q + h ) ] m + m p ( w ) m − 1 . (7.14) \mathbb{P}\left(\max_{r,s\in\{v_i,u_i:1\le i\le m\}} |r\wedge s| \le\Delta\right) \le[CKD(q+h)]^m+mp(w)^{m-1}.\tag{7.14} P ( r , s ∈ { v i , u i : 1 ≤ i ≤ m } max ∣ r ∧ s ∣ ≤ Δ ) ≤ [ C K D ( q + h ) ] m + m p ( w ) m − 1 . ( 7.14 ) The first term covers the event that every ∣ v i ∣ < q |v_i|<q ∣ v i ∣ < q i i i ∣ v i ∣ ≥ q |v_i|\ge q ∣ v i ∣ ≥ q v j , u j v_j,u_j v j , u j w / 2 w/2 w /2 R v i \mathbb{R}v_i R v i j ≠ i j\ne i j = i w w w p ( w ) p(w) p ( w ) m m m m ε / 2 > 3 m\varepsilon/2>3 m ε /2 > 3 c ( m − 1 ) > 3 c(m-1)>3 c ( m − 1 ) > 3 K , D K,D K , D h 3 h^3 h 3 h h h
Fix e < ε / 8 e<\varepsilon/8 e < ε /8 C i , D i C_i,D_i C i , D i 1 ≤ i ≤ m 1\le i\le m 1 ≤ i ≤ m
Z = ∑ i = 1 m ( C i a i + D i b i ) , Z ′ = ∑ i = 1 m ( C i a i ′ + D i b i ′ ) . Z=\sum_{i=1}^{m}(C_i a_i+D_i b_i), \qquad Z'=\sum_{i=1}^{m}(C_i a_i'+D_i b_i'). Z = i = 1 ∑ m ( C i a i + D i b i ) , Z ′ = i = 1 ∑ m ( C i a i ′ + D i b i ′ ) . For every fixed coefficient tuple, Lemma 7.3 bounds the support of each sum by K m h − 1 K_mh^{-1} K m h − 1 Z , Z ′ Z,Z' Z , Z ′
P ( ∣ Z − Z ′ ∣ ∞ ≤ h ∣ ( C i , D i ) i = 1 m ) > h / K m . (7.15) \mathbb{P}(|Z-Z'|_\infty\le h\mid(C_i,D_i)_{i=1}^{m})>h/K_m.\tag{7.15} P ( ∣ Z − Z ′ ∣ ∞ ≤ h ∣ ( C i , D i ) i = 1 m ) > h / K m . ( 7.15 ) Averaging over the coefficient tuple gives the same unconditional lower bound. For the upper bound, first fix the sampled endpoints. Outside the event in (7.14), choose two difference vectors r , s r,s r , s { v i , u i } \{v_i,u_i\} { v i , u i } ∣ r ∧ s ∣ > Δ |r\wedge s|>\Delta ∣ r ∧ s ∣ > Δ r , s r,s r , s i i i ∣ Z − Z ′ ∣ ∞ ≤ h |Z-Z'|_\infty\le h ∣ Z − Z ′ ∣ ∞ ≤ h O ( h 2 / Δ ) O(h^2/\Delta) O ( h 2 /Δ ) O ( D h / Δ ) O(Dh/\Delta) O ( D h /Δ ) C ( ( 1 + D ) / Δ + 1 ) C((1+D)/\Delta+1) C (( 1 + D ) /Δ + 1 ) h h h C e 2 h 2 − 2 e C_e^2h^{2-2e} C e 2 h 2 − 2 e
P ( ∣ Z − Z ′ ∣ ∞ ≤ h ) ≤ h 3 + K m h 2 − 2 e Δ = h 3 + K m h 1 + ε / 2 − 2 e . (7.16) \mathbb{P}(|Z-Z'|_\infty\le h)\le h^3+K_m\frac{h^{2-2e}}{\Delta}=h^3+K_mh^{1+\varepsilon/2-2e}.\tag{7.16} P ( ∣ Z − Z ′ ∣ ∞ ≤ h ) ≤ h 3 + K m Δ h 2 − 2 e = h 3 + K m h 1 + ε /2 − 2 e . ( 7.16 ) This contradicts (7.15), since ε / 2 − 2 e > 0 \varepsilon/2-2e>0 ε /2 − 2 e > 0 K m K_m K m
Thus for every fixed ε , c > 0 \varepsilon,c>0 ε , c > 0 p ( h 1 − ε ) > h c p(h^{1-\varepsilon})>h^c p ( h 1 − ε ) > h c ε , c ↓ 0 \varepsilon,c\downarrow0 ε , c ↓ 0 w = h 1 − o ( 1 ) w=h^{1-o(1)} w = h 1 − o ( 1 ) m 0 = h o ( 1 ) m_0=h^{o(1)} m 0 = h o ( 1 ) q 0 q_0 q 0 [ − D , D ] 2 [-D,D]^2 [ − D , D ] 2 C ( D ∣ ( q 0 ) 1 ∣ + w ) C(D|(q_0)_1|+w) C ( D ∣ ( q 0 ) 1 ∣ + w ) h h h
m 0 ≤ C K ( D ∣ ( q 0 ) 1 ∣ + w + h ) 2 . m_0\le CK\bigl(D|(q_0)_1|+w+h\bigr)^2. m 0 ≤ C K ( D ∣ ( q 0 ) 1 ∣ + w + h ) 2 . Since m 0 m_0 m 0 K , D K,D K , D w → 0 w\to0 w → 0 ∣ ( q 0 ) 1 ∣ ≥ h o ( 1 ) |(q_0)_1|\ge h^{o(1)} ∣ ( q 0 ) 1 ∣ ≥ h o ( 1 ) ( q 0 ) 2 / ( q 0 ) 1 (q_0)_2/(q_0)_1 ( q 0 ) 2 / ( q 0 ) 1 □ \square □
Proposition 7.6 (Retained-edge planar rigidity). Under Assumption 7.2, after increasing to a new subpower K ∗ K_\ast K ∗ κ 0 \kappa_0 κ 0 h − 2 / K ∗ h^{-2}/K_\ast h − 2 / K ∗ G κ 0 G_{\kappa_0} G κ 0 ∣ λ ∣ + ∣ c A ∣ + ∣ c B ∣ ≤ K ∗ |\lambda|+|c_A|+|c_B|\le K_\ast ∣ λ ∣ + ∣ c A ∣ + ∣ c B ∣ ≤ K ∗
∣ x 2 − λ x 1 − c A ∣ + ∣ y 2 − λ y 1 − c B ∣ ≤ K ∗ h . |x_2-\lambda x_1-c_A|+|y_2-\lambda y_1-c_B|\le K_\ast h. ∣ x 2 − λ x 1 − c A ∣ + ∣ y 2 − λ y 1 − c B ∣ ≤ K ∗ h . No fixed-power dependence of K ∗ K_\ast K ∗ K K K
Proof. The rounding and Lemma 7.1 give (7.4). Lemma 7.3 fixes κ 0 \kappa_0 κ 0 A ∗ A_\ast A ∗ B ∗ B_\ast B ∗ h − 2 / K h^{-2}/K h − 2 / K G κ 0 G_{\kappa_0} G κ 0 h − 2 / K h^{-2}/K h − 2 / K ∣ G κ 0 ∣ ≤ K 2 h − 2 |G_{\kappa_0}|\le K^2h^{-2} ∣ G κ 0 ∣ ≤ K 2 h − 2 y ↦ − κ 0 y y\mapsto-\kappa_0y y ↦ − κ 0 y h h h K K K h h h K ∗ K_\ast K ∗ □ \square □
An affine–skew row from line incidences
We apply the planar rigidity theorem to a row evaluated along moving lines. In a fixed direction, the line intercept and the row’s evaluation are nearly constant. Long line segments therefore give small planar projection sets. Two directions provide the endpoints of a graph, and a third direction provides its small restricted sumset.
Lemma 7.7 (Affine-skew row fit). Let z ( t ) = ( t , Z ( t ) ) z(t)=(t,Z(t)) z ( t ) = ( t , Z ( t )) 0 ≤ t ≤ 1 0\le t\le1 0 ≤ t ≤ 1 ( 1 , v ) (1,v) ( 1 , v ) K K K μ \mu μ 1 / K 1/K 1/ K K K K ξ \xi ξ π \pi π π ( I ) ≤ K ∣ I ∣ S \pi(I)\le K|I|^S π ( I ) ≤ K ∣ I ∣ S S > 0 S>0 S > 0 ξ ≤ ∣ I ∣ ≤ 1 \xi\le|I|\le1 ξ ≤ ∣ I ∣ ≤ 1
μ ( z ξ , v ξ ) ≤ K ξ 2 π ( v ξ ) . \mu(z_\xi,v_\xi)\le K\xi^2\pi(v_\xi). μ ( z ξ , v ξ ) ≤ K ξ 2 π ( v ξ ) . Let T ∈ [ − K , K ] 2 T\in[-K,K]^2 T ∈ [ − K , K ] 2 K K K E ≥ ξ E\ge\xi E ≥ ξ z ξ z_\xi z ξ T ( 1 , v ) T(1,v) T ( 1 , v ) K E KE K E 1 / K 1/K 1/ K
T = B + λ J z + O ( K E ) , J ( t , Z ) = ( − Z , t ) , ∣ B ∣ + ∣ λ ∣ ≤ K . T=B+\lambda Jz+O(KE),\qquad J(t,Z)=(-Z,t),\qquad|B|+|\lambda|\le K. T = B + λ J z + O ( K E ) , J ( t , Z ) = ( − Z , t ) , ∣ B ∣ + ∣ λ ∣ ≤ K . The subpower factor K K K
Proof. We may assume ξ ≤ E ≤ 1 \xi\le E\le1 ξ ≤ E ≤ 1 E > 1 E>1 E > 1 V 0 = ( ξ / E ) T V_0=(\xi/E)T V 0 = ( ξ / E ) T ( z , V 0 ) ξ (z,V_0)_\xi ( z , V 0 ) ξ K K K K ξ − 2 K\xi^{-2} K ξ − 2 d d d ξ \xi ξ v d v_d v d u d = ( 1 , v d ) u_d=(1,v_d) u d = ( 1 , v d )
P u d ( z , V 0 ) = ( z ⋅ J u d , V 0 ⋅ u d ) . \mathcal{P}_{u_d}(z,V_0)=(z\cdot Ju_d,V_0\cdot u_d). P u d ( z , V 0 ) = ( z ⋅ J u d , V 0 ⋅ u d ) . For the true direction, its first coordinate is constant on a line; the second is constant to error K ξ K\xi K ξ
Projection covers and point-mass witnesses. Freeze the present incidence law as μ c o v \mu_{\mathrm{cov}} μ cov μ c o v \mu_{\mathrm{cov}} μ cov 1 / K 1/K 1/ K 1 / ( K ξ ) 1/(K\xi) 1/ ( K ξ ) K ξ K\xi K ξ K ξ K\xi K ξ
For each direction bin, projected values separated by a sufficiently large multiple of K ξ K\xi K ξ K ξ − 2 K\xi^{-2} K ξ − 2 K ξ − 1 K\xi^{-1} K ξ − 1 μ c o v \mu_{\mathrm{cov}} μ cov
Discard slope bins of palette weight below ξ 2 \xi^2 ξ 2 K / ξ K/\xi K / ξ M M M 1 / K 1/K 1/ K K ρ S ′ K\rho^{S'} K ρ S ′ S ′ > 0 S'>0 S ′ > 0 ξ ≤ ρ ≤ 1 \xi\leq\rho\leq1 ξ ≤ ρ ≤ 1
Normalize this current incidence law to probability and freeze it as μ p t \mu_{\mathrm{pt}} μ pt K ξ 2 / M K\xi^2/M K ξ 2 / M ξ 2 / K 1 \xi^2/K_1 ξ 2 / K 1 ξ 2 / ( K 1 M ) \xi^2/(K_1M) ξ 2 / ( K 1 M ) K 1 K_1 K 1 ξ − 2 M / K \xi^{-2}M/K ξ − 2 M / K ξ − 2 / K \xi^{-2}/K ξ − 2 / K M / K M/K M / K
Every surviving point still has its floor in the frozen μ p t \mu_{\mathrm{pt}} μ pt ξ − 2 / K \xi^{-2}/K ξ − 2 / K μ p t \mu_{\mathrm{pt}} μ pt
Two projections and the third-direction sumset. At each surviving point there are at least M 3 / K M^3/K M 3 / K 1 / K 1/K 1/ K v 1 , v 2 v_1,v_2 v 1 , v 2 ξ − 2 M / K \xi^{-2}M/K ξ − 2 M / K ξ − 2 / K \xi^{-2}/K ξ − 2 / K M / K M/K M / K
Let A \mathcal A A B \mathcal B B P u 1 \mathcal P_{u_1} P u 1 P u 2 \mathcal P_{u_2} P u 2 ξ \xi ξ ∣ A ∣ , ∣ B ∣ ≤ K ξ − 1 |\mathcal A|,|\mathcal B|\leq K\xi^{-1} ∣ A ∣ , ∣ B ∣ ≤ K ξ − 1 K K K 1 / K 1/K 1/ K K K K G κ ⊂ A × B G_\kappa\subset\mathcal A\times\mathcal B G κ ⊂ A × B ξ − 2 / K \xi^{-2}/K ξ − 2 / K
For that direction write
u d = c 1 u 1 + c 2 u 2 , c 1 = v 2 − v d v 2 − v 1 , c 2 = v d − v 1 v 2 − v 1 , κ = c 2 c 1 = v d − v 1 v 2 − v d . \mathbf u_d=c_1\mathbf u_1+c_2\mathbf u_2,\qquad c_1=\frac{v_2-v_d}{v_2-v_1},\qquad c_2=\frac{v_d-v_1}{v_2-v_1},\qquad\kappa=\frac{c_2}{c_1}=\frac{v_d-v_1}{v_2-v_d}. u d = c 1 u 1 + c 2 u 2 , c 1 = v 2 − v 1 v 2 − v d , c 2 = v 2 − v 1 v d − v 1 , κ = c 1 c 2 = v 2 − v d v d − v 1 . The map P u \mathcal P_{\mathbf u} P u u \mathbf u u a , b a,b a , b
a + κ b = c 1 − 1 P u d ( z , V 0 ) . a+\kappa b=c_1^{-1}\mathcal P_{\mathbf u_d}(z,V_0). a + κb = c 1 − 1 P u d ( z , V 0 ) . Both c i c_i c i K K K
N ξ { a + κ b : ( a , b ) ∈ G κ } ≤ K ξ − 1 . N_\xi\{a+\kappa b:(a,b)\in G_\kappa\}\leq K\xi^{-1}. N ξ { a + κb : ( a , b ) ∈ G κ } ≤ K ξ − 1 . We verify the remaining planar hypotheses. The graph sizes imply ∣ A ∣ , ∣ B ∣ ≥ ξ − 1 / K |\mathcal{A}|,|\mathcal{B}| \ge\xi^{-1}/K ∣ A ∣ , ∣ B ∣ ≥ ξ − 1 / K K K K K − 1 ≤ ∣ κ ∣ ≤ K K^{-1} \le|\kappa| \le K K − 1 ≤ ∣ κ ∣ ≤ K K K K
∣ v d − v d ′ ∣ ≤ K ∣ κ d − κ d ′ ∣ |v_d-v_{d'}| \le K|\kappa_d-\kappa_{d'}| ∣ v d − v d ′ ∣ ≤ K ∣ κ d − κ d ′ ∣ on the retained choices. Finally, prescribing length-ξ \xi ξ ( a 1 , b 1 ) (a_1,b_1) ( a 1 , b 1 ) z z z K K K K K K K K K ξ − 2 / K \xi^{-2}/K ξ − 2 / K ξ − 2 / K \xi^{-2}/K ξ − 2 / K
Recovering the row and restoring incidence mass. Apply Proposition 7.6 with h = ξ h=\xi h = ξ κ 0 \kappa_0 κ 0 ξ − 2 / K ∗ \xi^{-2}/K_* ξ − 2 / K ∗ G κ 0 G_{\kappa_0} G κ 0 K ∗ ξ K_*\xi K ∗ ξ K ∗ K_* K ∗ K K K K ∗ K_* K ∗
On the corresponding point cells,
V 0 ⋅ u i = b i + λ 0 z ⋅ J u i + O ( K ξ ) , i = 1 , 2. V_0\cdot\mathbf{u}_i=b_i+\lambda_0 z\cdot J\mathbf{u}_i+O(K\xi), \qquad i=1,2. V 0 ⋅ u i = b i + λ 0 z ⋅ J u i + O ( K ξ ) , i = 1 , 2. The estimate depends only on ( z , V 0 ) (z,V_0) ( z , V 0 ) V 0 = ( ξ / E ) T V_0=(\xi/E)T V 0 = ( ξ / E ) T T = B + λ J z + O ( K E ) T=B+\lambda Jz+O(KE) T = B + λ J z + O ( K E )
Bounded endpoint multiplicity leaves at least ξ − 2 / K \xi^{-2}/K ξ − 2 / K μ p t \mu_{\mathrm{pt}} μ pt ξ 2 / K 1 \xi^2/K_1 ξ 2 / K 1 μ p t \mu_{\mathrm{pt}} μ pt
It remains to bound the coefficients after undoing the scaling. If ∣ λ ∣ |\lambda| ∣ λ ∣ ∣ T ∣ ≤ K |T|\le K ∣ T ∣ ≤ K z z z K / ∣ λ ∣ K/|\lambda| K /∣ λ ∣
K ( K / ∣ λ ∣ + ξ ) 2 , K(K/|\lambda|+\xi)^2, K ( K /∣ λ ∣ + ξ ) 2 , a contradiction. Hence ∣ λ ∣ ≤ K |\lambda|\le K ∣ λ ∣ ≤ K ∣ B ∣ |B| ∣ B ∣ K K K □ \square □
Finite narrowness and scalar projections
We treat a tangent parent with positive finite narrowness. Scalar projection first supplies full-time graphs. Wide projected intervals produce a time improvement; the other case forces a horizontal affine model, whose critical rates are treated in Section 9. Throughout this section, K = N o ( 1 ) K=N^{o(1)} K = N o ( 1 ) K − 1 K^{-1} K − 1
Remark 8.1 (Reference weights and current masses). We use the reference-weight pruning of Lemma 4.8 in the following scaled form. Suppose reference weights R e R_e R e ∑ e R e ≤ K ( 0 ) M \sum_e R_e\le K^{(0)}M ∑ e R e ≤ K ( 0 ) M M / K ( 2 ) M/K^{(2)} M / K ( 2 ) K ( 3 ) K^{(3)} K ( 3 ) e e e R e / K ( 1 ) R_e/K^{(1)} R e / K ( 1 )
K ( 3 ) K ( 1 ) ∑ e R e ≤ K ( 0 ) K ( 3 ) K ( 1 ) M . \frac{K^{(3)}}{K^{(1)}}\sum_e R_e\le\frac{K^{(0)}K^{(3)}}{K^{(1)}}M. K ( 1 ) K ( 3 ) e ∑ R e ≤ K ( 1 ) K ( 0 ) K ( 3 ) M . Choose the subpower K ( 1 ) K^{(1)} K ( 1 ) K ( 0 ) , K ( 2 ) , K ( 3 ) K^{(0)}, K^{(2)}, K^{(3)} K ( 0 ) , K ( 2 ) , K ( 3 ) b b b K π ( b ) R e K\pi(b)R_e K π ( b ) R e K K ( 1 ) π ( b ) KK^{(1)}\pi(b) K K ( 1 ) π ( b )
A scalar projection theorem at finite meshes
Theorem 8.2 (Scalar mesh projection). Let ν \nu ν ( t , y ( t ) , x ( t ) ) (t,y(t),x(t)) ( t , y ( t ) , x ( t )) y y y x x x K = N o ( 1 ) K=N^{o(1)} K = N o ( 1 ) μ a c t ≤ K ν ⊗ d t \mu_{\mathrm{act}}\le K\nu\otimes dt μ act ≤ K ν ⊗ d t 1 / K 1/K 1/ K a = N − s a=N^{-s} a = N − s s > 0 s>0 s > 0 σ ∼ log , N N − M \sigma\sim_{\log,N}N^{-M} σ ∼ l o g , N N − M ( t , y ) (t,y) ( t , y ) M > s M>s M > s c > 0 c>0 c > 0 d < 1 d<1 d < 1
each ( t , y ) (t,y) ( t , y ) x x x p = N − r p=N^{-r} p = N − r 0 ≤ r ≤ s 0\le r\le s 0 ≤ r ≤ s K σ 2 c p d K\sigma^2cp^d K σ 2 c p d
the number of occupied ( σ , σ , a ) (\sigma,\sigma,a) ( σ , σ , a ) σ 2 \sigma^2 σ 2 K c − 1 a − d Kc^{-1}a^{-d} K c − 1 a − d
The depth hypotheses use fixed powers, with the rounding and interpolation conventions of Section 2. They imply d ≥ 0 d\ge0 d ≥ 0
After a dense restriction and passage to a subsequence, there are labels with disjoint trajectory assignments and a common scale σ / K ≤ β ≤ K \sigma/K\le\beta\le K σ / K ≤ β ≤ K
a moving affine interval of width K β K\beta K β y y y
a quadratic F ( t , y ) F(t,y) F ( t , y ) ∣ x − F ( t , y ) ∣ ≤ K a \lvert x-F(t,y)\rvert\le Ka ∣ x − F ( t , y )∣ ≤ K a
assigned active mass at least c a d β / K ca^d\beta/K c a d β / K
The quadratic norm is at most K K K β \beta β
If, at r = s r=s r = s y ′ y' y ′ N − u N^{-u} N − u K N − S 1 u KN^{-S_1u} K N − S 1 u 0 < u ≤ u 1 0<u\le u_1 0 < u ≤ u 1 S 1 , u 1 > 0 S_1,u_1>0 S 1 , u 1 > 0 β ∼ log , N 1 \beta\sim_{\log,N}1 β ∼ l o g , N 1
Proof.
Smoothing the mesh law. Let ν \nu ν μ a c t ≤ K ν ⊗ d t \mu_{\mathrm{act}}\le K\nu\otimes dt μ act ≤ K ν ⊗ d t y , x y,x y , x σ ′ ≪ σ a 2 \sigma'\ll\sigma a^2 σ ′ ≪ σ a 2 j j j ω j \omega_j ω j ν \nu ν j j j σ \sigma σ I I I
μ a c t ( j × I ) ≥ ω j σ K 1 . \mu_{\mathrm{act}}(j\times I)\ge\frac{\omega_j\sigma}{K_1}. μ act ( j × I ) ≥ K 1 ω j σ . Write M j M_j M j K 1 K_1 K 1
Let ρ σ \rho_\sigma ρ σ σ \sigma σ ( j , u ) (j,\mathbf u) ( j , u ) j j j u \mathbf u u
ν j i t = ∑ j ω j δ j ⊗ ρ σ , d μ m a r k ( j , u , t ) = 1 M j ( t ) d ν j i t ( j , u ) d t . \nu_{\mathrm{jit}}=\sum_j\omega_j\delta_j\otimes\rho_\sigma,\qquad
d\mu_{\mathrm{mark}}(j,\mathbf u,t)=\mathbf1_{M_j}(t)\,d\nu_{\mathrm{jit}}(j,\mathbf u)\,dt. ν jit = j ∑ ω j δ j ⊗ ρ σ , d μ mark ( j , u , t ) = 1 M j ( t ) d ν jit ( j , u ) d t . Thus the noise distribution is the same for every tag; the independence is between its five coordinates.
The domination
μ a c t { ( j , t ) : t ∈ M j } ≤ K ∑ j ω j ∣ M j ∣ = K μ m a r k ( a l l ) \mu_{\mathrm{act}}\{(j,t):t\in M_j\}\le K\sum_j\omega_j|M_j|=K\mu_{\mathrm{mark}}(\mathrm{all}) μ act {( j , t ) : t ∈ M j } ≤ K j ∑ ω j ∣ M j ∣ = K μ mark ( all ) shows that μ m a r k \mu_{\mathrm{mark}} μ mark N N N w = x w=x w = x
For the marked noisy law, the exact-base density in an x p x_p x p K c p d Kcp^d K c p d ( t , y ) (t,y) ( t , y ) K σ c p d K\sigma cp^d K σ c p d K σ K\sigma K σ K σ K\sigma K σ y y y K / σ K/\sigma K / σ
The independent slope noise gives a further factor K min ( 1 , ρ / σ ) K\min(1,\rho/\sigma) K min ( 1 , ρ / σ ) y y y ρ \rho ρ x x x
K c a d N − S 1 u Kc a^d N^{-S_1u} K c a d N − S 1 u for the joint exact-base density with a velocity bin of width N − u N^{-u} N − u u < M u<M u < M
The noisy terminal support, measured by base Lebesgue measure and counting x a x_a x a K c − 1 a − d Kc^{-1}a^{-d} K c − 1 a − d K K K C 0 = lim log N c C_0=\lim\log_N c C 0 = lim log N c
n ( s ) ∼ log , N c a d , f ( r ) ≥ d r − C 0 , f ( s ) − f ( r ) ≤ d ( s − r ) . n(s)\sim_{\log,N} ca^d,\qquad f(r)\ge dr-C_0,\qquad f(s)-f(r)\le d(s-r). n ( s ) ∼ l o g , N c a d , f ( r ) ≥ d r − C 0 , f ( s ) − f ( r ) ≤ d ( s − r ) . If the angular estimate was assumed, the resulting problem belongs to C ( d ) C(d) C ( d ) M M M
Obtaining an angular gain. When no angular gain is assumed, also homogenize the joint profiles f ( r , u ) f(r,u) f ( r , u ) U = 4 ( M + s + 1 ) U=4(M+s+1) U = 4 ( M + s + 1 ) R = ( 1 − d ) / 2 > 0 R=(1-d)/2>0 R = ( 1 − d ) /2 > 0 ϵ > 0 \epsilon>0 ϵ > 0
( 1 − R ) r − f ( r , u ) + ϵ u , ( r , u ) ∈ [ 0 , s ] × [ 0 , U ] . (1-R)r-f(r,u)+\epsilon u,\qquad(r,u)\in[0,s]\times[0,U]. ( 1 − R ) r − f ( r , u ) + ϵ u , ( r , u ) ∈ [ 0 , s ] × [ 0 , U ] . At ( s , 0 ) (s,0) ( s , 0 ) C 0 + R s C_0+Rs C 0 + R s C 0 + R r + ϵ u − ( u − M ) + C_0+Rr+\epsilon u-(u-M)_+ C 0 + R r + ϵ u − ( u − M ) + ( r 0 , u 0 ) (r_0,u_0) ( r 0 , u 0 )
s − r 0 ≤ ϵ U / R , u 0 ≤ M / ( 1 − ϵ ) , f ( r 0 , u 0 ) ≤ f ( s , 0 ) + ϵ U . s-r_0\le\epsilon U/R,\qquad u_0\le M/(1-\epsilon),\qquad f(r_0,u_0)\le f(s,0)+\epsilon U. s − r 0 ≤ ϵ U / R , u 0 ≤ M / ( 1 − ϵ ) , f ( r 0 , u 0 ) ≤ f ( s , 0 ) + ϵ U . In particular r 0 > 0 r_0>0 r 0 > 0 u 0 + r 0 ≤ U u_0+r_0\le U u 0 + r 0 ≤ U u = u 0 u=u_0 u = u 0 1 − R 1-R 1 − R r 0 r_0 r 0 r = r 0 r=r_0 r = r 0
f ( r 0 , u 0 + b ) − f ( r 0 , u 0 ) ≥ ϵ b , 0 ≤ b ≤ r 0 . f(r_0,u_0+b)-f(r_0,u_0)\ge\epsilon b,\qquad0\le b\le r_0. f ( r 0 , u 0 + b ) − f ( r 0 , u 0 ) ≥ ϵ b , 0 ≤ b ≤ r 0 . Crop the noisy trajectories by the intercept and velocity of y y y G G G N − u 0 N^{-u_0} N − u 0 C \mathcal{C} C p C p_{\mathcal{C}} p C y y y G G G p C = N − α + o ( 1 ) p_\mathcal{C} = N^{-\alpha+o(1)} p C = N − α + o ( 1 )
The new pure density exponent is
f ′ ( r ) = f ( r , u 0 ) + u 0 − α . f'(r) = f(r,u_0) + u_0 - \alpha. f ′ ( r ) = f ( r , u 0 ) + u 0 − α . Indeed, given exact ( t , y ) (t,y) ( t , y ) y ( 0 ) = y ( t ) − t y ′ y(0)=y(t)-ty' y ( 0 ) = y ( t ) − t y ′ G / p C G/p_\mathcal{C} G / p C r 0 r_0 r 0 b b b f ( r , u 0 ) f(r,u_0) f ( r , u 0 ) f ( r 0 , u 0 + b ) f(r_0,u_0+b) f ( r 0 , u 0 + b ) C ( 1 − R ) C(1-R) C ( 1 − R ) ϵ \epsilon ϵ
Lifting such packets to original noisy coordinates, write β \beta β G G G
K − 1 p C N − f ′ ( r 0 ) ( β / G ) ≥ β c a d N − ϵ U − o ( 1 ) . K^{-1}p_{\mathcal C}N^{-f'(r_0)}(\beta/G) \ge\beta c a^d N^{-\epsilon U-o(1)}. K − 1 p C N − f ′ ( r 0 ) ( β / G ) ≥ β c a d N − ϵ U − o ( 1 ) . by the floor in Definition 2.8 and the profile formulas. They have an explicit quadratic fit over that time block at thickness N − r 0 + o ( 1 ) N^{-r_0+o(1)} N − r 0 + o ( 1 ) t , y t,y t , y
Full-time fits and return to the original law. In the cropped construction, let ϵ \epsilon ϵ q ≥ 1 / K q \ge1/K q ≥ 1/ K β ≤ K \beta\le K β ≤ K q c a d β / K qca^d\beta/K q c a d β / K K a Ka K a
The interval scale satisfies β ≥ σ / K \beta\ge\sigma/K β ≥ σ / K y ′ y' y ′ K β / q ≤ K β K\beta/q \le K\beta K β / q ≤ K β K K K x a x_a x a K q β Kq\beta K q β u u u β \beta β
The quadratic predictor also has norm at most K K K ( y − y c ( t ) ) / β (y-y_c(t))/\beta ( y − y c ( t )) / β ∣ x ∣ ≤ K |x| \le K ∣ x ∣ ≤ K ∣ F ∣ ≤ K |F| \le K ∣ F ∣ ≤ K 1 / K 1/K 1/ K K K K q ≥ 1 / K q \ge1/K q ≥ 1/ K K K K K K K
Keep one block’s assignments carrying dense selected μ m a r k \mu_{\mathrm{mark}} μ mark K K K q ≥ 1 / K q \ge1/K q ≥ 1/ K ∑ l a b e l s c a d β ≤ K \sum_{\mathrm{labels}} ca^d\beta\le K ∑ labels c a d β ≤ K u \mathbf{u} u one common coefficient jitter vector for which the assigned marked-time mass, measured by ∑ j ω j δ j ⊗ d t \sum_j \omega_j\delta_j \otimes dt ∑ j ω j δ j ⊗ d t
Undo this common jitter in every label. If its affine and quadratic components are Δ y ( t ) , Δ x ( t ) \Delta y(t),\Delta x(t) Δ y ( t ) , Δ x ( t ) y c − Δ y y_c-\Delta y y c − Δ y F ( t , y + Δ y ( t ) ) − Δ x ( t ) F(t,y+\Delta y(t))-\Delta x(t) F ( t , y + Δ y ( t )) − Δ x ( t ) O ( σ ′ ) O(\sigma') O ( σ ′ ) a a a β ≥ σ / K \beta\ge\sigma/K β ≥ σ / K
It remains to return from marked-time mass to μ a c t \mu_{\mathrm{act}} μ act ω j σ / K 1 \omega_j\sigma/K_1 ω j σ / K 1 K 1 − 1 K_1^{-1} K 1 − 1 c a d β ca^d\beta c a d β β \beta β
Subpower conclusions over an ensemble of eligible models with common exponent bounds can be taken uniformly: a fixed-power failure along a sequence of individually chosen models would contradict existence with subpower losses on a further subsequence. Equivalently use the worst infimum log-loss over models at each stage then diagonalize. Theorem 8.2 can thus be used anew on dense restrictions meeting its hypotheses.
Projection of a finite-narrowness parent
Assume now 0 < ℓ < ∞ 0 < \ell< \infty 0 < ℓ < ∞ y = ( Z , Y ) y=(Z,Y) y = ( Z , Y )
v = Z ′ , V 1 = ( 1 , v ) , a s = N − s , h s = N − k s , g s = N − ℓ s , b s = h s g s . v=Z',\qquad V_1=(1,v),\qquad a_s=N^{-s},\qquad h_s=N^{-ks},\qquad g_s=N^{-\ell s},\qquad b_s=h_sg_s. v = Z ′ , V 1 = ( 1 , v ) , a s = N − s , h s = N − k s , g s = N − ℓ s , b s = h s g s . The true label Q s Q_s Q s q s ∼ log , N h s q_s\sim_{\log,N}h_s q s ∼ l o g , N h s
∣ Y − Y Q ( t , Z ) ∣ ≤ K b s , ∣ Y ′ − A Q V 1 ∣ ≤ K g s . |Y-Y_{Q}(t,Z)|\le Kb_s,\qquad|Y'-A_QV_1|\le Kg_s. ∣ Y − Y Q ( t , Z ) ∣ ≤ K b s , ∣ Y ′ − A Q V 1 ∣ ≤ K g s . where Y Q Y_Q Y Q A Q A_Q A Q K K K K K K Z Z Z q s q_s q s U = ( Y − Y Q ( t , Z ) ) / b s U=(Y-Y_Q(t,Z))/b_s U = ( Y − Y Q ( t , Z )) / b s K K K
Start with any substantial residual and choose a typical parent with dense success and the prepared upper bounds. A candidate in such a parent suffices. Use its restricted incidence measure, dominated by K ν ⊗ d t K\nu\otimes dt K ν ⊗ d t π \pi π ( t , Z , Y , X p , [ v ] χ ) (t,Z,Y,X_p,[v]_\chi) ( t , Z , Y , X p , [ v ] χ ) K p d π ( [ v ] χ ) Kp^d\pi([v]_\chi) K p d π ([ v ] χ )
We will integrate these caps over moving triangular-coordinate boxes whose inverse widths and distortions have fixed-power bounds. Do so on much finer base meshes, holding the velocity bin fixed. A scalar-bin offset may vary with the base, provided its variation in one such cell costs at most K K K K π K\pi K π
The projected groups. Choose a sufficiently fine fixed-power χ \chi χ
α = ( [ v ] χ , [ Z ( 0 ) ] X ) , Z = Z α ( t ) + O ( K χ ) , w α = π ( [ v ] χ ) χ . \alpha=([v]_\chi,[Z(0)]_X),\qquad Z=Z_\alpha(t)+O(K\chi),\qquad w_\alpha=\pi([v]_\chi)\chi. α = ([ v ] χ , [ Z ( 0 ) ] X ) , Z = Z α ( t ) + O ( K χ ) , w α = π ([ v ] χ ) χ . Here Z α Z_\alpha Z α w α w_\alpha w α s s s ( Y ( 0 ) , Y ′ ) (Y(0),Y') ( Y ( 0 ) , Y ′ ) g s g_s g s γ \gamma γ p α γ p_{\alpha\gamma} p α γ p α γ p_{\alpha\gamma} p α γ Y Y Y g s g_s g s
c = w α g s p α γ , a = a s . c = \frac{w_\alpha g_s}{p_{\alpha\gamma}}, \qquad a = a_s. c = p α γ w α g s , a = a s . Use a time and normalized-Y Y Y σ ∼ N − M \sigma\sim N^{-M} σ ∼ N − M M > s M>s M > s s , k , ℓ s,k,\ell s , k , ℓ χ \chi χ Z Z Z X p X_p X p
K p d σ 2 g s χ π ( [ v ] χ ) = K σ 2 w α g s p d . Kp^d\sigma^2g_s\chi\,\pi([v]_\chi)=K\sigma^2w_\alpha g_sp^d. K p d σ 2 g s χ π ([ v ] χ ) = K σ 2 w α g s p d . This is the required projected density bound.
The projected support. Fix a true Q s Q_s Q s K h s / χ Kh_s/\chi K h s / χ Z ( 0 ) Z(0) Z ( 0 ) K K K γ \gamma γ A Q A_{Q} A Q K g s Kg_s K g s Y Q ( t , Z α ( t ) ) Y_Q(t,Z_\alpha(t)) Y Q ( t , Z α ( t )) K b s Kb_s K b s ( t , Y ) (t,Y) ( t , Y ) K q s b s / g s Kq_sb_s/g_s K q s b s / g s K K K
The stable graph lists of Q s Q_s Q s K K K X a s X_{a_s} X a s K ( σ + χ ) / b s K(\sigma+\chi)/b_s K ( σ + χ ) / b s a s a_s a s K K K w α g s w_\alpha g_s w α g s K q s h s b s Kq_sh_sb_s K q s h s b s
ν Q ∼ log , N a s d h s b s , ∑ Q q s ν Q ≤ 1 , \nu_Q \sim_{\log,N} a_s^d h_sb_s,\qquad\sum_Q q_s\nu_Q\le1, ν Q ∼ l o g , N a s d h s b s , Q ∑ q s ν Q ≤ 1 , the total cost is at most K a s − d Ka_s^{-d} K a s − d
Discard groups whose weighted support exceeds p α γ a s − d p_{\alpha\gamma}a_s^{-d} p α γ a s − d c c c c − 1 c^{-1} c − 1
We obtain labels Φ s \Phi_s Φ s ( α , γ ) (\alpha,\gamma) ( α , γ ) w α a s d β s / K w_\alpha a_s^d\beta_s/K w α a s d β s / K β s \beta_s β s Y Y Y
g s σ / K ≤ β s ≤ K g s , ∣ X − F Φ s ( t , Y ) ∣ ≤ K a s . g_s\sigma/K\le\beta_s\le Kg_s,\qquad|X-F_{\Phi_s}(t,Y)|\le Ka_s. g s σ / K ≤ β s ≤ K g s , ∣ X − F Φ s ( t , Y ) ∣ ≤ K a s . The positions and slopes lie in the corresponding affine packets, and F Φ s F_{\Phi_s} F Φ s K K K ∑ Φ s w α a s d β s ≤ K \sum_{\Phi_s}w_\alpha a_s^d\beta_s\le K ∑ Φ s w α a s d β s ≤ K s s s M M M χ \chi χ
To compare the two kinds of label, subdivide the true time blocks into dyadic intervals of length
q ∼ log , N min ( h s , β s / g s ) , m ∼ log , N q g s . q\sim_{\log,N}\min(h_s,\beta_s/g_s),\qquad m\sim_{\log,N}qg_s. q ∼ l o g , N min ( h s , β s / g s ) , m ∼ l o g , N q g s . A domain D D D Y ′ Y' Y ′ g s g_s g s Y Y Y m m m Y ′ Y' Y ′
Lemma 8.3 (Comparison on common stable domains). For the true labels Q s Q_s Q s Φ s \Phi_s Φ s D D D K a s K a_s K a s ( α , D , Q s ) (\alpha,D,Q_s) ( α , D , Q s ) K K K Φ s \Phi_s Φ s K K K ( α , D , Φ s ) (\alpha,D,\Phi_s) ( α , D , Φ s )
Proof. We put both lists on common domains before pairing their graphs. This allows each graph to keep its own marginal witnesses for the later list count, even when its mass in the current pair law is small.
Common domains and marginal witnesses. Fix α \alpha α
T D = q w α a s d m . T_D = q w_\alpha a_s^d m. T D = q w α a s d m . These weights have a summable budget. For a true label Q s Q_s Q s α \alpha α K K K α \alpha α K b s / m K b_s/m K b s / m K q s ν Q K q_s \nu_Q K q s ν Q Φ s \Phi_s Φ s K β s / m K \beta_s/m K β s / m K K K K w α a s d β s K w_\alpha a_s^d \beta_s K w α a s d β s ∑ T D ≤ K \sum T_D \le K ∑ T D ≤ K T D / K T_D/K T D / K
Evaluate the local graphs for Q s Q_s Q s Z α ( t ) Z_\alpha(t) Z α ( t ) X X X K a s K a_s K a s D D D K K K
∣ Y D ′ − A Q ( 1 , v α ) ∣ ≤ K g s , q g s , m ≤ K b s . \left|Y_D^\prime-A_Q(1,v_\alpha)\right|\le K g_s,\qquad q g_s,m\le K b_s. ∣ Y D ′ − A Q ( 1 , v α ) ∣ ≤ K g s , q g s , m ≤ K b s . Replacing Z Z Z Z α ( t ) Z_\alpha(t) Z α ( t ) K χ / b s K\chi/b_s K χ / b s a s a_s a s
Choose, before pairing any graphs, a grid in normalized D D D K K K T D / K 1 T_D/K_1 T D / K 1 K 1 K_1 K 1
On these domains both kinds of graph are algebraic of bounded degree and have norm at most K K K Φ s \Phi_s Φ s K β s K\beta_s K β s α \alpha α Z Z Z K a s K a_s K a s K T D K T_D K T D ( t , Y ) (t,Y) ( t , Y ) K q m Kqm K q m K a s K a_s K a s
Matching with separate reference weights. Take norms and scaled center jets on the common comparison balls, with their fixed holomorphic margins. Suppose that graph discrepancies exceed a s a_s a s M 0 M_0 M 0 ( α , D ) (\alpha,D) ( α , D ) O ( M 0 ) O(M_0) O ( M 0 ) O ( log N ) O(\log N) O ( log N ) K K K K K K
Each color has at most K ( M 0 / a s ) d K(M_0/a_s)^d K ( M 0 / a s ) d K M 0 KM_0 K M 0 X X X T D / K T_D/K T D / K M 0 M_0 M 0
Let B B B W c = ∑ i ∈ I c T i W_c=\sum_{i\in\mathcal{I}_c}T_i W c = ∑ i ∈ I c T i c = 1 , 2 c=1,2 c = 1 , 2
B ≥ ( W 1 + W 2 ) / K . B \ge(W_1+W_2)/K. B ≥ ( W 1 + W 2 ) / K . Restriction preserves the index-base ceiling f i ( z ) ≤ K T i f_i(z)\le KT_i f i ( z ) ≤ K T i p i = T i / W c p_i=T_i/W_c p i = T i / W c
f i ( z ) B ≤ K W c B T i W c ≤ K p i . \frac{f_i(z)}{B}\le K\frac{W_c}{B}\frac{T_i}{W_c}\le Kp_i. B f i ( z ) ≤ K B W c W c T i ≤ K p i . Integrating the same ceiling also gives B ≤ K W c B\le KW_c B ≤ K W c R = M 0 / a s R=M_0/a_s R = M 0 / a s K a s Ka_s K a s
For fixed ( α , D , Q s ) (\alpha,D,Q_s) ( α , D , Q s ) K a s Ka_s K a s K K K Q s Q_s Q s K a s Ka_s K a s K K K Φ s \Phi_s Φ s K K K
Three wide moving intervals
Proposition 8.4 (Wide-interval improvement). Use the finite-narrowness parent data of Section 8.2: the joint caps, conditional palette domination at fine point and path states, and the true Q s Q_s Q s X X X s s s s + u s+u s + u s + c 2 s+c_2 s + c 2
a = a s , a 1 = a N − u , a 2 = a N − c 2 . a=a_s,\qquad a_1=aN^{-u},\qquad a_2=aN^{-c_2}. a = a s , a 1 = a N − u , a 2 = a N − c 2 . Suppose that simultaneous projected labels Φ , Φ 1 , Φ 2 \Phi,\Phi_1,\Phi_2 Φ , Φ 1 , Φ 2 α \alpha α β min , β max \beta_{\min},\beta_{\max} β m i n , β m a x 0 < c 2 < ( 1 − d ) u / 4 0<c_2<(1-d)u/4 0 < c 2 < ( 1 − d ) u /4
β max ≤ K q s , b s / β min + h s β max / β min ≤ N − ( C d + 3 ) u , C d = 10 / ( 1 − d ) , \beta_{\max}\le Kq_s,\qquad b_s/\beta_{\min}+h_s\beta_{\max}/\beta_{\min}\le N^{-(C_d+3)u},\qquad C_d=10/(1-d), β m a x ≤ K q s , b s / β m i n + h s β m a x / β m i n ≤ N − ( C d + 3 ) u , C d = 10/ ( 1 − d ) , up to subpower factors, with common scale exponents. Take all comparison meshes, including the intercept mesh χ \chi χ h s , b s , a 1 h_s,b_s,a_1 h s , b s , a 1 a 1 a_1 a 1
Then some true Q = Q s Q=Q_s Q = Q s ( t , Z , Y ) (t,Z,Y) ( t , Z , Y ) X X X K a 2 Ka_2 K a 2
q s ν Q ( a 2 / a ) d / K . q_s\nu_Q(a_2/a)^d/K. q s ν Q ( a 2 / a ) d / K . Here ν Q \nu_Q ν Q
Proof. We first show that the finer projected fits differ from a coarse fit by nearly constant offsets on a true block. We then find common nonconstant transverse coefficients, subtract them, and apply scalar projection in ( t , Z ) (t,Z) ( t , Z )
Use first the short domains in Lemma 8.3, now at the Q s Q_s Q s b s b_s b s g s g_s g s Q = Q s Q=Q_s Q = Q s α \alpha α K K K K K K Φ \Phi Φ α \alpha α D D D Φ \Phi Φ K K K Φ i \Phi_i Φ i K − 1 q s w α a i d b s K^{-1}q_sw_\alpha a_i^d b_s K − 1 q s w α a i d b s ≤ K g s \le K g_s ≤ K g s
Comparing the projected fits at two domain sizes. Compare Φ i \Phi_i Φ i Φ \Phi Φ ρ ∼ log , N β min / β max \rho\sim_{\log,N}\beta_{\min}/\beta_{\max} ρ ∼ l o g , N β m i n / β m a x β max \beta_{\max} β m a x β min \beta_{\min} β m i n A A A i i i
T i = A w α a i d . T_i=Aw_\alpha a_i^d. T i = A w α a i d . As in Lemma 8.3, slicing gives a summable budget of these weights and retained marginal floors T i / K T_i/K T i / K K T i KT_i K T i
The polynomial discrepancies on these long domains are at most K a N C d u KaN^{C_d u} K a N C d u M 0 M_0 M 0 a a a i i i K ( M 0 / a i ) d K(M_0/a_i)^d K ( M 0 / a i ) d a 1 = a N − u a_1=aN^{-u} a 1 = a N − u M 0 / a ≥ N C d u M_0/a\ge N^{C_d u} M 0 / a ≥ N C d u
( M 0 / a i ) d ≤ ( M 0 / a ) d N d u ≤ ( M 0 / a ) d + d / C d ≤ ( M 0 / a ) ( 1 + d ) / 2 . (M_0/a_i)^d\le(M_0/a)^dN^{du}\le(M_0/a)^{d+d/C_d}\le(M_0/a)^{(1+d)/2}. ( M 0 / a i ) d ≤ ( M 0 / a ) d N d u ≤ ( M 0 / a ) d + d / C d ≤ ( M 0 / a ) ( 1 + d ) /2 . The strict inequality d + d / C d < ( 1 + d ) / 2 < 1 d+d/C_d<(1+d)/2<1 d + d / C d < ( 1 + d ) /2 < 1 T i / W i T_i/W_i T i / W i B B B W i W_i W i a N C d u aN^{C_d u} a N C d u a i d a_i^d a i d
On the whole short domain of a retained pairing this difference F Φ i − F Φ F_{\Phi_i}-F_\Phi F Φ i − F Φ ≪ a i \ll a_i ≪ a i K ( h s / ρ + b s / β min ) K(h_s/\rho+b_s/\beta_{\min}) K ( h s / ρ + b s / β m i n ) K g s Kg_s K g s Z = Z α ( t ) + O ( K χ ) Z=Z_\alpha(t)+O(K\chi) Z = Z α ( t ) + O ( K χ ) Y = Y Q ( t , Z ) + O ( K b s ) Y=Y_Q(t,Z)+O(Kb_s) Y = Y Q ( t , Z ) + O ( K b s )
X = F Φ ( t , Y ) + D i + O ( K a i ) X=F_\Phi(t,Y)+D_i+O(Ka_i) X = F Φ ( t , Y ) + D i + O ( K a i ) where D i D_i D i Q Q Q α \alpha α Φ \Phi Φ Φ i \Phi_i Φ i ∣ D i ∣ ≤ K a |D_i|\le Ka ∣ D i ∣ ≤ K a
There are at most K ( a / a i ) d K(a/a_i)^d K ( a / a i ) d Φ i \Phi_i Φ i Q Q Q α \alpha α Φ \Phi Φ Φ i \Phi_i Φ i K a Ka K a F Φ F_\Phi F Φ α \alpha α Φ i \Phi_i Φ i K K K Φ \Phi Φ
The reverse bound and the true-label comparison list give only K K K Φ 1 \Phi_1 Φ 1 ( α , D , Φ 1 , Φ , Q ) (\alpha,D,\Phi_1,\Phi,Q) ( α , D , Φ 1 , Φ , Q ) K − 1 q s w α a 1 d b s K^{-1}q_sw_\alpha a_1^d b_s K − 1 q s w α a 1 d b s ( t , Y ) (t,Y) ( t , Y ) ≤ K \le K ≤ K
Use horizontal coordinates z = ( t − t ∗ , Z − Z ∗ ) / q s z=(t-t_*,Z-Z_*)/q_s z = ( t − t ∗ , Z − Z ∗ ) / q s Q Q Q U = ( Y − Y Q ( t , Z ) ) / b s U=(Y-Y_Q(t,Z))/b_s U = ( Y − Y Q ( t , Z )) / b s
F Φ ( t , Y ) = f 0 Φ ( z ) + f 1 Φ ( z ) U + f 2 Φ U 2 . F_\Phi(t,Y)=f_0^\Phi(z)+f_1^\Phi(z)U+f_2^\Phi U^2. F Φ ( t , Y ) = f 0 Φ ( z ) + f 1 Φ ( z ) U + f 2 Φ U 2 . Here f 1 Φ f_1^\Phi f 1 Φ f 2 Φ f_2^\Phi f 2 Φ
P n c ( z , U ) = C 1 ( z ) U + C 2 U 2 , e = a N − ( 1 − d ) u / 2 , P_{\mathrm{nc}}(z,U)=C_1(z)U+C_2U^2,\qquad e=aN^{-(1-d)u/2}, P nc ( z , U ) = C 1 ( z ) U + C 2 U 2 , e = a N − ( 1 − d ) u /2 , with C 1 C_1 C 1 C 2 C_2 C 2 ( f 1 Φ ( z ) , f 2 Φ ) = ( C 1 ( z ) , C 2 ) + O ( K e ) (f_1^\Phi(z),f_2^\Phi)=(C_1(z),C_2)+O(Ke) ( f 1 Φ ( z ) , f 2 Φ ) = ( C 1 ( z ) , C 2 ) + O ( K e ) K e ≪ a 2 Ke\ll a_2 K e ≪ a 2 z z z D 2 D_2 D 2 U U U ( t , Z ) (t,Z) ( t , Z )
The coefficients f 1 Φ , f 2 Φ f_1^\Phi,f_2^\Phi f 1 Φ , f 2 Φ K K K K b s Kb_s K b s Q Q Q K β min − O ( 1 ) K\beta_{\min}^{-O(1)} K β m i n − O ( 1 ) K K K χ \chi χ Z Z Z Z α ( t ) Z_\alpha(t) Z α ( t )
Coefficient agreement at a fine state. Condition on a fine state consisting of Q Q Q ( z , U , X ) (z,U,X) ( z , U , X ) v v v K π K\pi K π K K K
Normalize only the whole dense current law to sample a source event, then sample a target independently conditional on the same state. Source entries ( α , D , Φ 1 , Φ , Q ) (\alpha,D,\Phi_1,\Phi,Q) ( α , D , Φ 1 , Φ , Q ) unnormalized Φ 1 \Phi_1 Φ 1 q s w α a 1 d b s q_sw_\alpha a_1^db_s q s w α a 1 d b s K K K K π ( d v ) K\pi(dv) K π ( d v ) K e Ke K e
Fix a source entry and a target velocity bin d v d_v d v χ \chi χ K π ( d v ) K\pi(d_v) K π ( d v )
First restrict source time to a sufficiently short fixed-power bin. The fine state ties the source and target base positions much more closely than χ \chi χ K χ K\chi K χ Z α ( t ) − t v d v Z_\alpha(t)-t v_{d_v} Z α ( t ) − t v d v K K K α \alpha α U , X U,X U , X Q Q Q K K K K K K Φ \Phi Φ K N d u KN^{du} K N d u D 1 D_1 D 1
Use source variables t , U α , U α = ( Y − Y Q ( t , Z α ( t ) ) ) / b s t,U_\alpha,U_\alpha=(Y-Y_Q(t,Z_\alpha(t)))/b_s t , U α , U α = ( Y − Y Q ( t , Z α ( t ))) / b s ∣ U α ∣ ≤ K |U_\alpha|\le K ∣ U α ∣ ≤ K both candidate polynomials for these variables at source Z α ( t ) Z_\alpha(t) Z α ( t ) i = 1 i=1 i = 1 K a 1 Ka_1 K a 1 X , t , Z , Y X,t,Z,Y X , t , Z , Y ( t , Y ) (t,Y) ( t , Y ) z z z K a 1 Ka_1 K a 1 χ \chi χ
Put τ = ( 1 − d ) u / 8 \tau=(1-d)u/8 τ = ( 1 − d ) u /8 e e e e N − τ eN^{-\tau} e N − τ U α U_\alpha U α K N − τ KN^{-\tau} K N − τ e / K e/K e / K K K K
On the complement, monotonicity on a bounded number of intervals for each polynomial and offset bounds the total length where values can match to K a 1 Ka_1 K a 1 K N d u a 1 N τ / e KN^{du}a_1N^\tau/e K N d u a 1 N τ / e
K N − τ , K N d u a 1 N τ e = K N − 3 ( 1 − d ) u / 8 . KN^{-\tau},\qquad\frac{KN^{du}a_1N^\tau}{e}=KN^{-3(1-d)u/8}. K N − τ , e K N d u a 1 N τ = K N − 3 ( 1 − d ) u /8 . Both save a fixed power. The errors in freezing the coefficients are already within K a 1 Ka_1 K a 1 K e K e K e K N d u KN^{du} K N d u
Integrate the fine time bins with their lengths. No factor equal to their number is incurred. State deletion only decreases the source entry’s unnormalized ceiling, and normalization of the whole dense sampling law costs K K K
At such fine source meshes in normalized time and U α U_\alpha U α ≤ K q s w α a 1 d b s \le Kq_sw_\alpha a_1^db_s ≤ K q s w α a 1 d b s Φ 1 \Phi_1 Φ 1
A common coefficient field. Conditional averaging now assigns a deterministic coefficient pair to each fine state, agreeing to K e K e K e Q Q Q q s ν Q q_s\nu_Q q s ν Q ∑ Q q s ν Q ≤ 1 \sum_Q q_s\nu_Q \le1 ∑ Q q s ν Q ≤ 1 z z z K K K ( C 1 ( z ) , C 2 ) (C_1(z), C_2) ( C 1 ( z ) , C 2 ) C 1 C_1 C 1 C 2 C_2 C 2 K K K
Choose first a fixed-power grid step ω ≪ χ \omega\ll\chi ω ≪ χ ≪ e \ll e ≪ e K K K ω \omega ω χ \chi χ α \alpha α
Now sample from π \pi π 1 / K 1/K 1/ K 1 / K 1/K 1/ K ( 1 , v 1 ) , ( 1 , v 2 ) (1,v^1),(1,v^2) ( 1 , v 1 ) , ( 1 , v 2 ) ω \omega ω z z z
Along a grid fiber parallel to either axis, its witnesses have only K K K α \alpha α ( t , Z ) (t,Z) ( t , Z ) K q s ω Kq_s\omega K q s ω ω \omega ω K K K χ \chi χ K K K Φ \Phi Φ Q Q Q K e K e K e
There are now at least K − 1 ω − 2 K^{-1}\omega^{-2} K − 1 ω − 2 ω 2 / K \omega^2/K ω 2 / K z z z ≤ K / ω \le K/\omega ≤ K / ω K K K ≥ 1 / ( K ω ) \ge1/(K\omega) ≥ 1/ ( K ω ) ω \omega ω ≥ ω − 2 / K \ge\omega^{-2}/K ≥ ω − 2 / K
Bilinear interpolation from the four anchors fits the first entry at the target vertices to K e K e K e K e K e K e
The fiber constants compare through their crossings with one anchor row. Occupied anchor bounds and the gaps bound both coefficient sets by K K K
It remains to remove the mixed bilinear term. Let μ Q \mu_Q μ Q Q Q Q Q Q Q μ Q ( a l l ) ≤ K m Q \mu_Q(\mathrm{all}) \le K m_Q μ Q ( all ) ≤ K m Q m Q = q s ν Q m_Q = q_s \nu_Q m Q = q s ν Q v j v^j v j
μ Q { ∣ v − v 1 ∣ < r or ∣ v − v 2 ∣ < r } ≤ K m Q ∑ j = 1 2 π ( B ( v j , 2 r ) ) ≤ K m Q r s ′ \mu_Q\{ |v-v^1|<r\ \text{or}\ |v-v^2|<r \}\le Km_Q\sum_{j=1}^{2}\pi(B(v^j,2r))\le Km_Qr^{s'} μ Q { ∣ v − v 1 ∣ < r or ∣ v − v 2 ∣ < r } ≤ K m Q j = 1 ∑ 2 π ( B ( v j , 2 r )) ≤ K m Q r s ′ This upper bound survives every subsequent restriction. If the final fitted law has mass m Q / K r e t m_Q/K_{\mathrm{ret}} m Q / K ret r r r K K r e t r s ′ = o ( 1 ) K K_{\mathrm{ret}}r^{s'}=o(1) K K ret r s ′ = o ( 1 ) K K K Q Q Q ν Q \nu_Q ν Q q s / K q_s/K q s / K α \alpha α Q Q Q K K K Φ \Phi Φ q s / K q_s/K q s / K K e Ke K e f 1 Φ f_1^\Phi f 1 Φ ( 1 , v ) (1,v) ( 1 , v )
v 2 − v v 2 − v 1 , v − v 1 v 2 − v 1 . \frac{v^2-v}{v^2-v^1},\qquad\frac{v-v^1}{v^2-v^1}. v 2 − v 1 v 2 − v , v 2 − v 1 v − v 1 . Both are bounded below in absolute value by a reciprocal subpower. The quadratic coefficient of the bilinear interpolant on this line is its mixed coefficient times their product. Remez on the retained time set therefore bounds the mixed coefficient by K e Ke K e C 1 C_1 C 1 C 2 C_2 C 2
Scalar support after subtraction. The common field just proved supplies the proposed P n c = C 1 ( z ) U + C 2 U 2 P_{\mathrm{nc}}=C_1(z)U+C_2U^2 P nc = C 1 ( z ) U + C 2 U 2 X X X Q Q Q z z z U U U K ( a / a 2 ) d K(a/a_2)^d K ( a / a 2 ) d a 2 a_2 a 2 [ v ] χ [v]_\chi [ v ] χ f 0 Φ ( z ) + D 2 + O ( K a 2 ) f_0^\Phi(z)+D_2+O(Ka_2) f 0 Φ ( z ) + D 2 + O ( K a 2 ) K K K α , Φ \alpha,\Phi α , Φ K ( a / a 2 ) d K(a/a_2)^d K ( a / a 2 ) d D 2 D_2 D 2 K e Ke K e c 2 < ( 1 − d ) u / 4 c_2<(1-d)u/4 c 2 < ( 1 − d ) u /4
To sum this list count over velocities, use the output state consisting of the z z z Q Q Q K K K P n c P_{\mathrm{nc}} P nc Q Q Q K q s ν Q Kq_s\nu_Q K q s ν Q 1 / K 1/K 1/ K π \pi π z z z χ \chi χ K K K f 0 Φ ( z ) f_0^\Phi(z) f 0 Φ ( z ) K a 2 Ka_2 K a 2
Normalize the current mass by q s ν Q q_s\nu_Q q s ν Q Q Q Q Z Z Z a 2 ≤ p ≤ 1 a_2\le p\le1 a 2 ≤ p ≤ 1 K ( p / a ) d K(p/a)^d K ( p / a ) d π \pi π z z z σ ∗ \sigma_* σ ∗ ∣ U ∣ ≤ K |U|\le K ∣ U ∣ ≤ K K q s 2 σ ∗ 2 Kq_s^2\sigma_*^2 K q s 2 σ ∗ 2 X X X P n c P_{\mathrm{nc}} P nc ν Q ∼ log , N a d h s \nu_Q\sim_{\log,N}a^d h_s ν Q ∼ l o g , N a d h s v v v
The prior for the next projection is the true assignment of Q Q Q Q Q Q K K K ( X − P n c , Z ) (X-P_{\mathrm{nc}},Z) ( X − P nc , Z ) c = a − d c=a^{-d} c = a − d a 2 a_2 a 2 c a 2 d = ( a 2 / a ) d ca_2^d=(a_2/a)^d c a 2 d = ( a 2 / a ) d K K K ( t , Z ) (t,Z) ( t , Z ) ( a 2 / a ) d / K (a_2/a)^d/K ( a 2 / a ) d / K P n c P_{\mathrm{nc}} P nc
These last comparison meshes may have arbitrarily fine fixed-power widths. The equations with errors O ( a i ) O(a_i) O ( a i )
For the original parent observable X X X s = 1 s=1 s = 1 u > 0 u>0 u > 0 s , k , ℓ , d s,k,\ell,d s , k , ℓ , d 0 < c 2 < ( 1 − d ) u / 4 0<c_2<(1-d)u/4 0 < c 2 < ( 1 − d ) u /4 β s ′ ∼ log , N g s ′ \beta_s'\sim_{\log,N} g_s' β s ′ ∼ l o g , N g s ′ s s s
β s = N − B 0 + o ( 1 ) , ℓ s < B 0 ≤ ℓ s + M . \beta_s=N^{-B_0+o(1)},\qquad\ell_s<B_0\leq\ell_s+M. β s = N − B 0 + o ( 1 ) , ℓ s < B 0 ≤ ℓ s + M . We now deal with this remaining case. In choosing χ \chi χ
The horizontal model
Proposition 8.5 (Reduction to the horizontal model). Suppose β s = N − B 0 + o ( 1 ) \beta_s=N^{-B_0+o(1)} β s = N − B 0 + o ( 1 ) ℓ s < B 0 ≤ ℓ s + M \ell_s<B_0\leq\ell_s+M ℓ s < B 0 ≤ ℓ s + M On every substantial residual either there is a strip-improvement candidate, or a true-parent normalization gives k = ℓ k=\ell k = ℓ (8.1) , with fixed e 0 > 0 e_0>0 e 0 > 0 H H H H 2 H^2 H 2 (8.2) . Mass is normalized by the chosen parent’s full assignment times its time length.
Proof.
From projected intervals to a row list. Choose r > s r>s r > s ( k + ℓ ) r > B 0 > ℓ r (k+\ell)r>B_0>\ell r ( k + ℓ ) r > B 0 > ℓ r Q r Q_r Q r Q s Q_s Q s K K K β s \beta_s β s Y ′ Y' Y ′ α \alpha α Q s Q_s Q s Φ s \Phi_s Φ s K K K Q r Q_r Q r Y ′ Y' Y ′ K b r K b_r K b r Q s Q_s Q s g s g_s g s K g s q K g_s q K g s q b r ≲ log , N min ( b s , β s ) b_r\lesssim_{\log,N}\min(b_s,\beta_s) b r ≲ l o g , N min ( b s , β s ) K K K K K K Y ′ Y' Y ′ K β s K\beta_s K β s χ \chi χ
Normalize the true parent Q = Q r Q=Q_r Q = Q r Z Z Z q r q_r q r V 1 = ( 1 , v ) V_1=(1,v) V 1 = ( 1 , v ) Y Q Y_Q Y Q q r g r q_r g_r q r g r
Y ′ − A Q V 1 g r , \frac{Y'-A_QV_1}{g_r}, g r Y ′ − A Q V 1 , bounded by K K K E = N − ( B 0 − ℓ r ) E=N^{-(B_0-\ell r)} E = N − ( B 0 − ℓ r )
T = ( A d e e p − A Q ) / g r T=(A_{\mathrm{deep}}-A_Q)/g_r T = ( A deep − A Q ) / g r whose evaluation T V 1 T V_1 T V 1 ≪ E \ll E ≪ E ξ ≪ E \xi\ll E ξ ≪ E
We verify the remaining row-fit hypotheses. At a state consisting of the deep row label and fine horizontal location, use sufficiently deep histories, append this state, and delete light states to obtain conditional velocity domination by K π K\pi K π 1 / K 1/K 1/ K 1 / K 1/K 1/ K T T T K K K
For any fixed such pair at a horizontal cell, the two lists for the normalized transverse velocity constrain compatible E E E K K K 1 / K 1/K 1/ K K K K
The horizontal joint cell cap, before dividing mass by q r ν Q q_r \nu_Q q r ν Q
K ξ 2 q r ν Q π ( v ξ ) . K\xi^2q_r\nu_Q\pi(v_\xi). K ξ 2 q r ν Q π ( v ξ ) . It follows by summing the joint state bounds with their earlier weights on which the Q Q Q ξ \xi ξ Q r Q_r Q r q r ν Q q_r\nu_Q q r ν Q
The horizontal equation and its scales. The row fit gives normalized transverse velocity ( B + λ J z ) ⋅ V 1 + O ( K E ) (B+\lambda Jz)\cdot V_1+O(KE) ( B + λ J z ) ⋅ V 1 + O ( K E ) ∣ λ ∣ |\lambda| ∣ λ ∣ g r g_r g r A Q V 1 A_QV_1 A Q V 1 ∣ λ ∣ ∼ log , N 1 |\lambda|\sim_{\log,N}1 ∣ λ ∣ ∼ l o g , N 1 B ⋅ z B\cdot z B ⋅ z λ \lambda λ ( t , Z , Y ) (t,Z,Y) ( t , Z , Y )
∣ Y ′ − J z ⋅ V 1 ∣ ≤ N − e 0 (8.1) |Y'-Jz\cdot V_1|\le N^{-e_0}\tag{8.1} ∣ Y ′ − J z ⋅ V 1 ∣ ≤ N − e 0 ( 8.1 ) for some fixed e 0 > 0 e_0>0 e 0 > 0
At a small relative depth s ′ > 0 s'>0 s ′ > 0 Q r + s ′ Q_{r+s'} Q r + s ′ H = q r + s ′ / q r ∼ log , N h s ′ H=q_{r+s'}/q_r\sim_{\log,N}h^{s'} H = q r + s ′ / q r ∼ l o g , N h s ′ K H KH K H K N − ℓ s ′ KN^{-\ell s'} K N − ℓ s ′ J z Jz J z K ( N − ℓ s ′ + N − e 0 ) K(N^{-\ell s'}+N^{-e_0}) K ( N − ℓ s ′ + N − e 0 ) Q r Q_r Q r K q r ν Q Kq_r\nu_Q K q r ν Q
Choose s ′ s' s ′ N − e 0 N^{-e_0} N − e 0 H ν Q r + s ′ / ( K ν Q ) H\nu_{Q_{r+s'}}/(K\nu_Q) H ν Q r + s ′ / ( K ν Q ) H / K H/K H / K J z Jz J z t t t K N − ℓ s ′ KN^{-\ell s'} K N − ℓ s ′
N − k s ′ ≤ K N − ℓ s ′ , N^{-ks'}\le KN^{-\ell s'}, N − k s ′ ≤ K N − ℓ s ′ , so k ≥ ℓ k\ge\ell k ≥ ℓ k > ℓ k>\ell k > ℓ J z Jz J z (8.1) by its value at an active child center. The error from the horizontal diameter is smaller by a power than the nominal velocity width, giving a strip candidate at Q r + s ′ Q_{r+s'} Q r + s ′ ∣ λ ∣ g r |\lambda|g_r ∣ λ ∣ g r H H H ν Q \nu_Q ν Q k = ℓ k=\ell k = ℓ
Whole horizontal boxes. Choose an occupied center ( z c , Y c ) (z_c,Y_c) ( z c , Y c )
z ^ = ( z − z c ) / H , Y ^ = ( Y − Y c − J z c ⋅ ( z − z c ) ) / H 2 . (8.2) \widehat{z}=(z-z_c)/H,\qquad\widehat{Y}=(Y-Y_c-Jz_c\cdot(z-z_c))/H^2.\tag{8.2} z = ( z − z c ) / H , Y = ( Y − Y c − J z c ⋅ ( z − z c )) / H 2 . ( 8.2 ) The true strip row is within K H KH K H J z c Jz_c J z c K H KH K H k = ℓ k=\ell k = ℓ K H 2 KH^2 K H 2 K K K K H KH K H K H KH K H K H 2 KH^2 K H 2 K K K
The two-velocity comparison can use common-location meshes much finer than the box widths: the row is fixed, and the variation of J z Jz J z Q r Q_r Q r
Affine approximation and residual normalization
The horizontal equation first forces the local hidden sheets to be affine to their fitted accuracy. We then subtract one parent affine fit. This gives a bounded residual with caps and state lists at arbitrarily fine prescribed meshes; those preparations will not require the horizontal equation at the same accuracy. Finally, comparison of nested affine fits gives the slope bounds used in the next section.
Lemma 8.6 (Affine approximation of the hidden sheets). In Proposition 8.5, for finitely many sufficiently small relative depths s ′ s' s ′ K K K K a r a s ′ K a_r a_{s'} K a r a s ′ s ′ = 0 s'=0 s ′ = 0
Proof. Fix one of the required depths, including depth zero. Until we return to the parent at the end of the proof, ( t , Z , Y ) (t,Z,Y) ( t , Z , Y ) (8.2) , and z = ( t , Z ) z=(t,Z) z = ( t , Z ) w w w Δ = a r a s ′ \Delta=a_r a_{s'} Δ = a r a s ′ K K K
W = ( 1 , v , J z ⋅ V 1 + O ( K N − e 0 / h s ′ ) ) . \mathbf W=(1,v,Jz\cdot V_1+O(KN^{-e_0}/h_{s'})). W = ( 1 , v , J z ⋅ V 1 + O ( K N − e 0 / h s ′ )) . Indeed subtracting the central plane and rescaling changes the horizontal row to J ( z o l d − z c ) / H J(z_{\mathrm{old}}-z_c)/H J ( z old − z c ) / H Q r Q_r Q r (8.1) suffice. Choose the required depths small enough that the displayed velocity error is at most N − e N^{-e} N − e e > 0 e>0 e > 0
Use the stable-ball preparation of Proposition 4.7 for the hidden test sheet f f f w w w K Δ K\Delta K Δ K K K K K K
In each ball/branch entry, remove trajectories whose associated time set has relative length below a sufficiently small reciprocal subpower. Charge this deletion to the full true trajectory weights using prior-times-time domination. Also discard entries too light relative to block length times their true assignment weight. These thresholds sum within Q r Q_r Q r
Let M f M_f M f K M f K M_f K M f W \mathbf{W} W K Δ K\Delta K Δ 1 / K 1/K 1/ K K Δ K\Delta K Δ K K K w w w M f M_f M f
Use fine states consisting of the path label, fine base position, and ball/branch choice. They have K K K w w w ( 1 , v , J z ⋅ V 1 ) (1,v,Jz \cdot V_1) ( 1 , v , J z ⋅ V 1 ) K M f N − e KM_fN^{-e} K M f N − e
Put
X = ∂ t − Z ∂ Y , Z = ∂ Z + t ∂ Y . \mathcal{X} = \partial_t - Z\partial_Y,\qquad\mathcal{Z} = \partial_Z + t\partial_Y. X = ∂ t − Z ∂ Y , Z = ∂ Z + t ∂ Y . For fixed v v v ( X + v Z ) 2 f (\mathcal{X}+v\mathcal{Z})^2f ( X + v Z ) 2 f ∂ Y \partial_Y ∂ Y − Z + t v -Z+tv − Z + t v
( X + v Z ) ( − Z + t v ) = − v + v = 0. (\mathcal{X}+v\mathcal{Z})(-Z+tv)=-v+v=0. ( X + v Z ) ( − Z + t v ) = − v + v = 0. In particular an affine function contributes zero before the actual direction is perturbed. The error above depends on M f M_f M f
Interpolate the quadratic in v v v
X 2 f , Z 2 f , ( X Z + Z X ) f = O ( K ( Δ + M f N − e ) ) . \mathcal{X}^2f,\quad\mathcal{Z}^2f,\quad(\mathcal{X}\mathcal{Z}+\mathcal{Z}\mathcal{X})f=O\bigl(K(\Delta+M_fN^{-e})\bigr). X 2 f , Z 2 f , ( XZ + ZX ) f = O ( K ( Δ + M f N − e ) ) . These expressions are algebraic analytic functions of bounded relation degree on the stable enlargement. The Remez and derivative bounds in Lemma 3.1 extend the estimates to the real comparison ball and to every fixed further derivative needed below.
The central commutator T = [ X , Z ] = 2 ∂ Y \mathcal{T}=[\mathcal{X},\mathcal{Z}]=2\partial_Y T = [ X , Z ] = 2 ∂ Y T \mathcal{T} T
X 2 Z = Z X 2 + 2 T X , X Z X = Z X 2 + T X . \mathcal{X}^2\mathcal{Z}=\mathcal{Z}\mathcal{X}^2+2\mathcal{T}\mathcal{X},\qquad\mathcal{X}\mathcal{Z}\mathcal{X}=\mathcal{Z}\mathcal{X}^2+\mathcal{T}\mathcal{X}. X 2 Z = Z X 2 + 2 T X , XZX = Z X 2 + T X . Consequently
X ( X Z + Z X ) − 2 Z X 2 = 3 T X , \mathcal{X}(\mathcal{X}\mathcal{Z}+\mathcal{Z}\mathcal{X})-2\mathcal{Z}\mathcal{X}^2=3\mathcal{T}\mathcal{X}, X ( XZ + ZX ) − 2 Z X 2 = 3 T X , Z ( X Z + Z X ) − 2 X Z 2 = − 3 T Z . \mathcal{Z}(\mathcal{X}\mathcal{Z}+\mathcal{Z}\mathcal{X})-2\mathcal{X}\mathcal{Z}^2=-3\mathcal{T}\mathcal{Z}. Z ( XZ + ZX ) − 2 X Z 2 = − 3 T Z . The preceding differentiated bounds therefore control T X f \mathcal{T}\mathcal{X}f T X f T Z f \mathcal{T}\mathcal{Z}f T Z f
T 2 f = X T Z f − Z T X f . \mathcal{T}^2f=\mathcal{X}\mathcal{T}\mathcal{Z}f-\mathcal{Z}\mathcal{T}\mathcal{X}f. T 2 f = X T Z f − Z T X f . With E f = K ( Δ + M f N − e ) E_f = K(\Delta+ M_fN^{-e}) E f = K ( Δ + M f N − e ) E f E_f E f K K K
T 2 f = 4 f Y Y , T X f = 2 ( f t Y − Z f Y Y ) , T Z f = 2 ( f Z Y + t f Y Y ) , X 2 f = f t t − 2 Z f t Y + Z 2 f Y Y , Z 2 f = f Z Z + 2 t f Z Y + t 2 f Y Y , 1 2 ( X Z + Z X ) f = f t Z + t f t Y − Z f Z Y − t Z f Y Y . \begin{aligned}
\mathcal{T}^{2}f &= 4f_{YY},\\
\mathcal{T}\mathcal{X}f &= 2(f_{tY}-Zf_{YY}), & \mathcal{T}\mathcal{Z}f &= 2(f_{ZY}+tf_{YY}),\\
\mathcal{X}^{2}f &= f_{tt}-2Zf_{tY}+Z^{2}f_{YY}, & \mathcal{Z}^{2}f &= f_{ZZ}+2tf_{ZY}+t^{2}f_{YY},\\
\frac{1}{2}(\mathcal{X}\mathcal{Z}+\mathcal{Z}\mathcal{X})f &= f_{tZ}+tf_{tY}-Zf_{ZY}-tZf_{YY}.
\end{aligned} T 2 f T X f X 2 f 2 1 ( XZ + ZX ) f = 4 f Y Y , = 2 ( f t Y − Z f Y Y ) , = f tt − 2 Z f t Y + Z 2 f Y Y , = f tZ + t f t Y − Z f Z Y − tZ f Y Y . T Z f Z 2 f = 2 ( f Z Y + t f Y Y ) , = f Z Z + 2 t f Z Y + t 2 f Y Y , Since the comparison coordinates are bounded by K K K K ( Δ + M f N − e ) K(\Delta+ M_fN^{-e}) K ( Δ + M f N − e )
M f ≤ K Δ + K M f N − e . M_f \le K\Delta+ KM_fN^{-e}. M f ≤ K Δ + K M f N − e . Absorb the second term for sufficiently large N N N M f M_f M f
The resulting affine fit agrees with w w w K Δ K\Delta K Δ K K K K K K
Lemma 8.7 (Residual normalization and fine-state bounds). Choose one successful parent affine fit L L L x = ( w − L ) / a r x=(w-L)/a_r x = ( w − L ) / a r x x x ∣ x ∣ + ∣ x ′ ∣ ≤ K |x|+|x'|\le K ∣ x ∣ + ∣ x ′ ∣ ≤ K
Let μ \mu μ Q r Q_r Q r q r ν Q r q_r\nu_{Q_r} q r ν Q r
μ { ( t , Z , Y ) ∈ C , x ∈ J p } ≤ K ∣ C ∣ p d \mu\{(t,Z,Y)\in C,\ x\in J_p\}\le K|C|p^d μ {( t , Z , Y ) ∈ C , x ∈ J p } ≤ K ∣ C ∣ p d μ { ( t , Z , Y ) ∈ C , x ∈ J p , v ∈ b χ } ≤ K ∣ C ∣ p d π ( b χ ) . (8.3) \mu\{(t,Z,Y)\in C,\ x\in J_p,\ v\in b_\chi\}\le K|C|p^d\pi(b_\chi).\tag{8.3} μ {( t , Z , Y ) ∈ C , x ∈ J p , v ∈ b χ } ≤ K ∣ C ∣ p d π ( b χ ) . ( 8.3 ) Here C C C ∣ C ∣ |C| ∣ C ∣ J p J_p J p 0 < p ≤ 1 0<p\le1 0 < p ≤ 1 b χ b_\chi b χ χ \chi χ π \pi π K K K
Sufficiently deep true labels determine these base and residual bins with K K K
Proof. Return to the Q r Q_r Q r (8.1) . The full-block parent fit makes x = ( w − L ( z , Y ) ) / a r x=(w-L(z,Y))/a_r x = ( w − L ( z , Y )) / a r ∣ x ′ ∣ ≤ K |x'|\le K ∣ x ′ ∣ ≤ K
Freeze the dense law after choosing L L L x x x R R R Q r Q_r Q r K q r ν Q r Kq_r\nu_{Q_r} K q r ν Q r x x x
g R = ( f R − L ) / a r g_R=(f_R-L)/a_r g R = ( f R − L ) / a r is bounded by K K K K K K L L L R R R K K K
Now append the recovered state to each earlier end history e e e R e R_e R e ∑ e R e ≤ K q r ν Q r \sum_e R_e \le K q_r \nu_{Q_r} ∑ e R e ≤ K q r ν Q r K π ( b ) R e K\pi(b)R_e K π ( b ) R e K K K R e / K 1 R_e/K_1 R e / K 1 K 1 K_1 K 1 Q r Q_r Q r L L L
The pure residual cap is K p d Kp^d K p d w w w p a r p a_r p a r L L L Q r Q_r Q r K p d π ( b χ ) Kp^d\pi(b_\chi) K p d π ( b χ ) (8.3) . No slow variation of L L L
If a probability prior on bounded participating trajectories is needed, conditioning away the other trajectories costs at most K K K Q r Q_r Q r a s ′ d h s ′ 3 a_{s'}^dh_{s'}^3 a s ′ d h s ′ 3 K K K
For subsequent counts let ν ♭ \nu^\flat ν ♭ Q r Q_r Q r L L L μ \mu μ K ν ♭ ⊗ d t K\nu^\flat \otimes dt K ν ♭ ⊗ d t Q s Q_s Q s s s s Q r Q_r Q r H s = q r + s / q r ∼ log , N h s H_s=q_{r+s}/q_r\sim_{\log,N}h_s H s = q r + s / q r ∼ l o g , N h s H / H 2 H/H^2 H / H 2 k = ℓ k=\ell k = ℓ
ν ♭ ( Q s ) ∼ log , N a s d h s 3 , m Q s = H s ν ♭ ( Q s ) , ∑ Q s m Q s ≤ 1. (8.4) \nu^\flat(Q_s)\sim_{\log,N}a_s^d h_s^3,\qquad m_{Q_s}=H_s\nu^\flat(Q_s),\qquad\sum_{Q_s}m_{Q_s}\le1.\tag{8.4} ν ♭ ( Q s ) ∼ l o g , N a s d h s 3 , m Q s = H s ν ♭ ( Q s ) , Q s ∑ m Q s ≤ 1. ( 8.4 ) These full assignment weights remain the reference budgets after restrictions of the current incidence law.
Lemma 8.8 (Slopes of nested affine fits). Use the residual x x x s s s Q r Q_r Q r L s ( t , Z , Y ) L_s(t,Z,Y) L s ( t , Z , Y ) x x x
∣ ∇ z L s + ( ∂ Y L s ) J z ∣ ≤ K ( 1 + a s / h s ) , ∣ ∂ Y L s ∣ ≤ K ( 1 + a s / h s 2 ) . (8.5) \lvert\nabla_z L_s+(\partial_Y L_s)Jz\rvert\le K(1+a_s/h_s),\qquad\lvert\partial_Y L_s\rvert\le K(1+a_s/h_s^2).\tag{8.5} ∣ ∇ z L s + ( ∂ Y L s ) J z ∣ ≤ K ( 1 + a s / h s ) , ∣ ∂ Y L s ∣ ≤ K ( 1 + a s / h s 2 ) . ( 8.5 ) The predictions have error K a s K a_s K a s K K K ( t , Z , Y ) (t,Z,Y) ( t , Z , Y ) Q r Q_r Q r
Proof. The affine predictions follow by subtracting L L L a r a_r a r s i − 1 , s i s_{i-1},s_i s i − 1 , s i K K K K p a i r K_{\mathrm{pair}} K pair ϵ m c h i l d / K p a i r \epsilon m_{\mathrm{child}}/K_{\mathrm{pair}} ϵ m child / K pair
ϵ ∑ c h i l d m c h i l d ≤ ϵ . \epsilon\sum_{\mathrm{child}}m_{\mathrm{child}}\le\epsilon. ϵ child ∑ m child ≤ ϵ . Choose ϵ \epsilon ϵ
The child base cap gives inverse-subpower filling for every retained pair’s full-domain witnesses. Both affine fits are close to x x x K a s i − 1 K a_{s_{i-1}} K a s i − 1 K a s i − 1 / h s i K a_{s_{i-1}}/h_{s_i} K a s i − 1 / h s i K a s i − 1 / h s i 2 K a_{s_{i-1}}/h_{s_i}^{2} K a s i − 1 / h s i 2 J z Jz J z K h s i K h_{s_i} K h s i
Start at relative depth zero with the zero prediction and the bound ∣ x ∣ ≤ K \lvert x\rvert\le K ∣ x ∣ ≤ K p = 1 , 2 p=1,2 p = 1 , 2 s i ≤ s s_i \le s s i ≤ s
a s i − 1 h s i p ≤ K ( 1 + a s h s p ) . \frac{a_{s_{i-1}}}{h_{s_i}^{p}} \le K\left(1+\frac{a_s}{h_s^{p}}\right). h s i p a s i − 1 ≤ K ( 1 + h s p a s ) . The sum of the slope increments therefore gives (8.5) . All losses remain subpower on a slowly growing grid. The same witness deletion also prepares nonconsecutive ancestor/child pairs when needed.
For later counts, one may replace Y Y Y Y 0 = q 0 − t p Y^{0}=q_0-tp Y 0 = q 0 − tp
Z = p + t v , P = ( p , q 0 ) = ( Z ( 0 ) , Y ( 0 ) ) , Z=p+tv,\qquad P=(p,q_0)=(Z(0),Y(0)), Z = p + t v , P = ( p , q 0 ) = ( Z ( 0 ) , Y ( 0 )) , with error at most N − e 0 N^{-e_0} N − e 0 x x x Y Y Y w = L + a r x w=L+a_rx w = L + a r x
Full-time coefficient counts in the horizontal model
For the critical and subcritical rates, the next lemma supplies one last consequence of the prepared model: when k ≤ 1 / 2 k \le1/2 k ≤ 1/2
Lemma 8.9 (Horizontal coefficient count). Suppose k = ℓ ≤ 1 / 2 k=\ell\le1/2 k = ℓ ≤ 1/2 x x x s > 0 s>0 s > 0 D = a s u D=a_s^{u} D = a s u u ≥ 1 u\ge1 u ≥ 1 d 0 d_0 d 0 D D D P = ( Z ( 0 ) , Y ( 0 ) ) P=(Z(0),Y(0)) P = ( Z ( 0 ) , Y ( 0 )) W G = π ( d 0 ) D 2 W_G=\pi(d_0)D^2 W G = π ( d 0 ) D 2 W G W_G W G K K K ( t , x ) (t,x) ( t , x ) K a s − 1 − d K a_s^{-1-d} K a s − 1 − d a s a_s a s K a s − d K a_s^{-d} K a s − d a s a_s a s W G a s d / K W_G a_s^d/K W G a s d / K η ≪ a s \eta\ll a_s η ≪ a s x a s x_{a_s} x a s K K K
Proof. Work with the full prior ν \nu ν μ \mu μ (8.4) . Restrictions may be taken anew on dense mass in this normalization. The depths are sufficiently small for the horizontal equation, centered boxes, and affine preparations, simultaneously at the finitely many required scales. When k < 1 / 2 k<1/2 k < 1/2 (8.5) and the bounded parent positions bound every ordinary slope of L s L_s L s K K K ( t , Z , Y , x , v ) (t,Z,Y,x,v) ( t , Z , Y , x , v )
Fix small s > 0 s>0 s > 0 D = a s D=a_s D = a s D = a s u D=a_s^u D = a s u u ≥ 1 u\ge1 u ≥ 1 v v v d 0 d_0 d 0 P D P_D P D W G = π ( d 0 ) D 2 W_G=\pi(d_0)D^2 W G = π ( d 0 ) D 2 G G G (8.1) are taken smaller than D D D ( p c , q c ) (p_c,q_c) ( p c , q c ) v d 0 v_{d_0} v d 0 K D KD K D
z c ( t ) = ( t , p c + t v d 0 ) , Y c ( t ) = q c − t p c z_c(t)=(t,p_c+tv_{d_0}),\qquad Y_c(t)=q_c-tp_c z c ( t ) = ( t , p c + t v d 0 ) , Y c ( t ) = q c − t p c on occurrences of the group. Thus their integrated mass over any required time interval I I I x x x ϱ \varrho ϱ a s ≤ ϱ ≤ 1 a_s \leq\varrho\leq1 a s ≤ ϱ ≤ 1
K ∣ I ∣ W G ϱ d . K|I|W_G\varrho^d. K ∣ I ∣ W G ϱ d . Sum the joint caps at fixed d 0 d_0 d 0 ≤ K D 2 \leq KD^2 ≤ K D 2 K ϱ K\varrho K ϱ t t t ≤ K \leq K ≤ K K K K μ ( G ) ≤ K W G \mu(G) \leq KW_G μ ( G ) ≤ K W G x x x K K K W G / K W_G/K W G / K ν ♭ ( G ) \nu^\flat(G) ν ♭ ( G ) ∑ G W G ≤ K \sum_G W_G \leq K ∑ G W G ≤ K P P P K K K K K K W G W_G W G
Counting support by whole true boxes. Before the group restrictions, discard true Q s Q_s Q s m Q s / K 1 m_{Q_s}/K_1 m Q s / K 1 (8.4) makes the loss at most 1 / K 1 1/K_1 1/ K 1 L L L
Fix a time bin of length a s a_s a s K D KD K D H s H_s H s Y c ′ = J z c ⋅ ( 1 , v d 0 ) Y'_c=Jz_c\cdot(1,v_{d_0}) Y c ′ = J z c ⋅ ( 1 , v d 0 ) D ≤ h s 2 D\leq h_s^2 D ≤ h s 2 J z c ( t ∗ ) Jz_c(t_*) J z c ( t ∗ ) K h s Kh_s K h s K h s 2 Kh_s^2 K h s 2 K h s 4 Kh_s^4 K h s 4
The pure base cap on the frozen bounded-residual law is K K K ∣ x ∣ ≤ K |x|\leq K ∣ x ∣ ≤ K h s 4 a s d / K h_s^4a_s^d/K h s 4 a s d / K K a s − d Ka_s^{-d} K a s − d a s a_s a s K K K D ≤ a s D\leq a_s D ≤ a s K K K ( t , x ) (t,x) ( t , x ) K a s − 1 − d Ka_s^{-1-d} K a s − 1 − d a s a_s a s
From support to heavy coefficient bins. Normalize one typical group’s weights by W G W_G W G x x x ϱ ≥ a s \varrho\geq a_s ϱ ≥ a s K ϱ d K\varrho^d K ϱ d ϱ \varrho ϱ K K K K ϱ d K\varrho^d K ϱ d
Apply Lemma 5.1 with p = a s p=a_s p = a s a s d / K a_s^d/K a s d / K K K K
This conclusion holds on every dense residual, so all but negligible mass can be assigned to heavy bins. Explicitly, if for a fixed δ > 0 \delta>0 δ > 0 W G a s d N − δ W_Ga_s^dN^{-\delta} W G a s d N − δ W G W_G W G δ \delta δ W G a s d / K W_Ga_s^d/K W G a s d / K K a s − d K a_s^{-d} K a s − d x x x K a s K a_s K a s
Finally subdivide time into bins of fixed-power length η ≪ a s \eta\ll a_s η ≪ a s η W G a s d / K ′ \eta W_G a_s^d/K' η W G a s d / K ′ K a s − d K a_s^{-d} K a s − d K ′ K' K ′
Fix a group, a time subbin, and an x a s x_{a_s} x a s ℓ 0 \ell_0 ℓ 0 t 0 t_0 t 0 ℓ \ell ℓ t t t
∣ x ℓ ( t ) − x ℓ 0 ( t 0 ) ∣ ≤ C a s + K ∣ t − t 0 ∣ ≤ C a s + K η ≤ K a s . \lvert x_\ell(t)-x_{\ell_0}(t_0)\rvert\le C a_s+K\lvert t-t_0\rvert\le C a_s+K\eta\le K a_s. ∣ x ℓ ( t ) − x ℓ 0 ( t 0 )∣ ≤ C a s + K ∣ t − t 0 ∣ ≤ C a s + K η ≤ K a s . Thus the entry’s entire stored subbin mass lies in the enlarged scalar band. The time/band cap is K η W G a s d K\eta W_G a_s^d K η W G a s d K K K
The critical horizontal rate and the return to wide intervals
We complete the finite-narrowness argument in the horizontal model of Proposition 8.5. All depths in this section are relative to the selected true parent Q r Q_r Q r
a s = N − s , h s = N − k s , 0 < k = ℓ ≤ 1 , m Q s = H s ν ♭ ( Q s ) ≍ log , N h s 4 a s d . a_s=N^{-s},\qquad h_s=N^{-ks},\qquad0<k=\ell\le1,\qquad m_{Q_s}=H_s\nu^\flat(Q_s)\asymp_{\log,N} h_s^4a_s^d. a s = N − s , h s = N − k s , 0 < k = ℓ ≤ 1 , m Q s = H s ν ♭ ( Q s ) ≍ l o g , N h s 4 a s d . where 0 ≤ d < 1 0\le d<1 0 ≤ d < 1 H s H_s H s ν ♭ \nu^\flat ν ♭ K K K q r ν ( Q r ) q_r\nu(Q_r) q r ν ( Q r )
Write z = ( t , Z ) z=(t,Z) z = ( t , Z ) V 1 = ( 1 , v ) V_1=(1,v) V 1 = ( 1 , v ) J ( t , Z ) = ( − Z , t ) J(t,Z)=(-Z,t) J ( t , Z ) = ( − Z , t ) x = ( w − L ) / a r x=(w-L)/a_r x = ( w − L ) / a r ∣ x ∣ + ∣ x ′ ∣ ≤ K \lvert x\rvert+\lvert x'\rvert\le K ∣ x ∣ + ∣ x ′ ∣ ≤ K x ′ x' x ′ Q r Q_r Q r (8.1) and (8.5) and Lemma 8.9. In particular, with P = ( p , q 0 ) = ( Z ( 0 ) , Y ( 0 ) ) P=(p,q_0)=(Z(0),Y(0)) P = ( p , q 0 ) = ( Z ( 0 ) , Y ( 0 ))
Z = p + t v , Y = q 0 − t p + O ( N − e 0 ) (9.1) Z=p+tv,\qquad Y=q_0-tp+O(N^{-e_0})\tag{9.1} Z = p + t v , Y = q 0 − tp + O ( N − e 0 ) ( 9.1 ) holds throughout the participating lines for a fixed e 0 > 0 e_0>0 e 0 > 0 π \pi π v v v
At the critical rate 2 k = 1 2k=1 2 k = 1 x x x 2 k ≠ 1 2k\ne1 2 k = 1 h 2 h_2 h 2
Here is the elementary deletion principle used to make several finite families of labels available at once.
Lemma 9.1 (Simultaneous marginal floors). Let a finite measure of mass M ≥ K − 1 M\ge K^{-1} M ≥ K − 1 T j T_j T j K K K K 1 K_1 K 1 9 M / 10 9M/10 9 M /10 T j / K 1 T_j/K_1 T j / K 1
Proof. Use the fixed-threshold deletion argument of Lemma 2.11, with thresholds T j / K 1 T_j/K_1 T j / K 1 K 1 K_1 K 1 ∑ j T j / K 1 < M / 10 \sum_j T_j/K_1 < M/10 ∑ j T j / K 1 < M /10 M / 10 M/10 M /10 K 1 K_1 K 1
The critical rate
Proposition 9.2 (Critical-rate improvement). If 2 k = 1 2k=1 2 k = 1 s > 0 s>0 s > 0 0 < c < k s / 2 0<c<ks/2 0 < c < k s /2 Q s Q_s Q s K a r + s + c K a_{r+s+c} K a r + s + c
K − 1 q r ν ( Q r ) m Q s N − d c K^{-1}q_r\nu(Q_r)m_{Q_s}N^{-dc} K − 1 q r ν ( Q r ) m Q s N − d c in the initial tangent parent. Its fit holds throughout the selected block on the assigned trajectories.
Proof. An affine primitive for the trajectory derivatives.
We first find an affine base function F ∗ F_* F ∗ x ′ x' x ′ Q s Q_s Q s a ′ = a s N − c a'=a_sN^{-c} a ′ = a s N − c ω \omega ω K ω − 3 ( a s / a ′ ) d K\omega^{-3}(a_s/a')^d K ω − 3 ( a s / a ′ ) d K − 1 ω − 3 K^{-1}\omega^{-3} K − 1 ω − 3 K ( a s / a ′ ) d K(a_s/a')^d K ( a s / a ′ ) d
Fix s > 0 s>0 s > 0 s s s 100 s < e 0 100s<e_0 100 s < e 0 a = a s = h s 2 a=a_s=h_s^2 a = a s = h s 2 D = a D=a D = a d 0 , η d_0,\eta d 0 , η N − 100 s N^{-100s} N − 100 s 0 < c < k s / 2 0<c<ks/2 0 < c < k s /2 s + c , 3 s s+c,3s s + c , 3 s Q s Q_s Q s
At an occurrence in Q s Q_s Q s H = H s H=H_s H = H s z ^ = ( z − z c ) / H \widehat z=(z-z_c)/H z = ( z − z c ) / H Q s , z ^ ξ , v ξ Q_s,\widehat z_\xi,v_\xi Q s , z ξ , v ξ ξ ≪ a \xi\ll a ξ ≪ a d 0 , η d_0,\eta d 0 , η K K K x ′ x' x ′ a a a Q r Q_r Q r p p p Z = p + t v Z=p+tv Z = p + t v Y Y Y K H 2 KH^2 K H 2 Q s , ξ Q_s,\xi Q s , ξ q 0 q_0 q 0 q 0 = Y + t p + O ( N − e 0 ) q_0=Y+tp+O(N^{-e_0}) q 0 = Y + tp + O ( N − e 0 ) K K K P 0 P_0 P 0 D = a = h s 2 D=a=h_s^2 D = a = h s 2 L s L_s L s x a x_a x a K K K d 0 d_0 d 0 x a x_a x a
We next turn the derivative lists into a list of possible rows. On this law,
x ′ = G ∗ ⋅ V 1 + O ( K a ) , G ∗ = ∇ z L 3 s + ( ∂ Y L 3 s ) J z , ∣ G ∗ ∣ ≤ K , x'=G_*\cdot V_1+O(Ka),\qquad G_*=\nabla_zL_{3s}+(\partial_YL_{3s})Jz,\qquad|G_*|\le K, x ′ = G ∗ ⋅ V 1 + O ( K a ) , G ∗ = ∇ z L 3 s + ( ∂ Y L 3 s ) J z , ∣ G ∗ ∣ ≤ K , by Equations (8.1) and (8.5) , and differentiating the affine fits along lines (error ≤ K a 3 s / h 3 s \le Ka_{3s}/h_{3s} ≤ K a 3 s / h 3 s
Use typical states of ( Q s , z ^ ξ , [ G ∗ ] a ) (Q_s,\widehat z_\xi,[G_*]_a) ( Q s , z ξ , [ G ∗ ] a ) K π K\pi K π K K K L 3 s L_{3s} L 3 s K K K (8.5) here controls row variation.
The end-label domination with earlier masses summing inside Q r Q_r Q r ≤ K q r ν Q r \le Kq_r\nu_{Q_r} ≤ K q r ν Q r
Each surviving such state has supporting pairs of velocity bins, separated by at least 1 / K 1/K 1/ K ≥ 1 / K \ge1/K ≥ 1/ K Q s , z ^ ξ Q_s,\widehat{z}_\xi Q s , z ξ x ′ x' x ′ K K K K a Ka K a K K K a a a G ∗ G_* G ∗ Q s , z ^ ξ Q_s,\widehat{z}_\xi Q s , z ξ
The remaining input for Lemma 7.7 is the joint cell cap. Pick one Q s Q_s Q s m Q s m_{Q_s} m Q s v ξ v_\xi v ξ K H 2 KH^2 K H 2 K K K x x x K a Ka K a m Q s m_{Q_s} m Q s K ( H ξ ) 2 H 2 a d π ( v ξ ) K(H\xi)^2H^2a^d\pi(v_\xi) K ( H ξ ) 2 H 2 a d π ( v ξ ) K ξ 2 π ( v ξ ) K\xi^2\pi(v_\xi) K ξ 2 π ( v ξ )
In centered coordinates horizontal velocity remains V 1 V_1 V 1 T = G ∗ , z = z ^ , E = a T=G_*,z=\widehat{z},E=a T = G ∗ , z = z , E = a x ′ x' x ′ m Q s m_{Q_s} m Q s G ∗ = B + λ J z ^ + O ( K a ) G_*=B+\lambda J\widehat{z}+O(Ka) G ∗ = B + λ J z + O ( K a ) ∣ B ∣ + ∣ λ ∣ ≤ K |B|+|\lambda|\le K ∣ B ∣ + ∣ λ ∣ ≤ K
F ∗ ( z , Y ) = ( B − λ J z c / H ) ⋅ z + λ Y / H F_*(z,Y)=(B-\lambda Jz_c/H)\cdot z+\lambda Y/H F ∗ ( z , Y ) = ( B − λ J z c / H ) ⋅ z + λY / H has slopes at most K / H K/H K / H
∣ x ′ − d d t F ∗ ( z , Y ) ∣ ≤ K a . \left|x'-\frac{d}{dt}F_*(z,Y)\right|\le Ka. x ′ − d t d F ∗ ( z , Y ) ≤ K a . Indeed use (8.1) with N − e 0 / H ≪ a N^{-e_0}/H\ll a N − e 0 / H ≪ a
Intercepts and the upper cell count. Take a ′ = a N − c a'=aN^{-c} a ′ = a N − c l l l c l = ( x − F ∗ ) ( t m i d ) c_l=(x-F_*)(t_{\mathrm{mid}}) c l = ( x − F ∗ ) ( t mid ) C = [ c l ] a ′ C=[c_l]_{a'} C = [ c l ] a ′
x ( t ) − F ∗ ( z ( t ) , Y ( t ) ) = c l + O ( K a h s ) , K a h s ≪ a ′ , x(t)-F_*(z(t),Y(t))=c_l+O(Kah_s),\qquad Kah_s\ll a', x ( t ) − F ∗ ( z ( t ) , Y ( t )) = c l + O ( K a h s ) , K a h s ≪ a ′ , throughout the block, since c < k s / 2 c<ks/2 c < k s /2 x x x C C C c l c_l c l O ( a ′ ) O(a') O ( a ′ )
In centered coordinates ( z ^ , Y ^ ) (\widehat{z},\widehat{Y}) ( z , Y ) J z c Jz_c J z c H 2 H^2 H 2 ω ≪ a ′ \omega\ll a' ω ≪ a ′ N − 5 s N^{-5s} N − 5 s
K ω − 3 ( a / a ′ ) d . K\omega^{-3}(a/a')^d. K ω − 3 ( a / a ′ ) d . Indeed the number of true descendants Q s + c Q_{s+c} Q s + c
K ( a a ′ ) d ( h s h s + c ) 4 , K\left(\frac{a}{a'}\right)^d\left(\frac{h_s}{h_{s+c}}\right)^4, K ( a ′ a ) d ( h s + c h s ) 4 , by the time-length ratios and disjoint nested trajectory assignments. Each descendant’s domain has volume at most K ( h s + c / h s ) 4 K(h_{s+c}/h_s)^4 K ( h s + c / h s ) 4 Q s Q_s Q s K K K Q s Q_s Q s ω \omega ω O ( ω − 3 ) O(\omega^{-3}) O ( ω − 3 ) (8.5) and the slopes of F ∗ F_* F ∗ L s + c L_{s+c} L s + c K K K
Three horizontal traversals and the lower cell count. We now prove the complementary estimate: intercept bins carrying almost all the fitted mass each occupy at least K − 1 ω − 3 K^{-1}\omega^{-3} K − 1 ω − 3 K ( a / a ′ ) d K(a/a')^d K ( a / a ′ ) d
The switching state records C C C x x x ω \omega ω K K K K K K C C C F ∗ F_* F ∗ x − F ∗ = c l + O ( K a h s ) x-F_* = c_l+O(Kah_s) x − F ∗ = c l + O ( K a h s ) K a h s ≪ a ′ Kah_s\ll a' K a h s ≪ a ′
Apply the reference-weight pruning of Remark 8.1 to these appended states inside the selected Q s Q_s Q s ∑ e R e ≤ K m Q s \sum_e R_e\le Km_{Q_s} ∑ e R e ≤ K m Q s e e e K π ( b ) R e K\pi(b)R_e K π ( b ) R e μ l a s t \mu_{\mathrm{last}} μ last
m l a s t : = μ l a s t ( a l l ) ≥ m Q s / K m_{\mathrm{last}}:=\mu_{\mathrm{last}}(\mathrm{all})\ge m_{Q_s}/K m last := μ last ( all ) ≥ m Q s / K and conditional velocity-bin weights at most K π ( b ) K\pi(b) K π ( b ) μ ~ = μ l a s t / m l a s t \tilde{\mu}=\mu_{\mathrm{last}}/m_{\mathrm{last}} μ ~ = μ last / m last K K K Q s Q_s Q s
Call a trajectory good when its μ ~ \tilde{\mu} μ ~ 1 / K 1 1/K_1 1/ K 1 K 1 K_1 K 1 K K K K K K
Starting with an event of law μ ~ \tilde{\mu} μ ~ μ ~ \tilde{\mu} μ ~ C C C i i i c i c_i c i τ i \tau_i τ i
Retain for the estimate only paths whose three fresh trajectories are good and for which
∣ c 2 − c 1 ∣ , ∣ c 3 − c 2 ∣ , ∣ τ 2 − τ 1 ∣ ≥ 1 / K ′ . |c_2-c_1|,\ |c_3-c_2|,\ |\tau_2-\tau_1|\ge1/K'. ∣ c 2 − c 1 ∣ , ∣ c 3 − c 2 ∣ , ∣ τ 2 − τ 1 ∣ ≥ 1/ K ′ . Here K ′ K' K ′ o ( 1 ) o(1) o ( 1 ) ( c i , τ i ) (c_i,\tau_i) ( c i , τ i ) K π ⊗ 3 ⊗ d τ 1 d τ 2 d τ 3 K\pi^{\otimes3}\otimes d\tau_1d\tau_2d\tau_3 K π ⊗ 3 ⊗ d τ 1 d τ 2 d τ 3 K π ⊗ 3 ⊗ d τ 1 d τ 2 d τ 3 K\pi^{\otimes3}\otimes d\tau_1d\tau_2d\tau_3 K π ⊗ 3 ⊗ d τ 1 d τ 2 d τ 3
Up to error o ( ω ) o(\omega) o ( ω )
d Z ^ d t ^ = c i , d Y ^ d t ^ = − Z ^ + t ^ c i . \frac{d\widehat{Z}}{d\widehat{t}}=c_i,\qquad
\frac{d\widehat{Y}}{d\widehat{t}}=-\widehat{Z}+\widehat{t}c_i. d t d Z = c i , d t d Y = − Z + t c i . The refresh changes the start of each traversal only within the tiny base mesh. The horizontal equation contributes error O ( K N − e 0 / H ) O(KN^{-e_0}/H) O ( K N − e 0 / H ) s s s o ( ω ) o(\omega) o ( ω )
Fix the starting event ( τ 0 , Z 0 , Y 0 ) (\tau_0,Z_0,Y_0) ( τ 0 , Z 0 , Y 0 ) D j = c j + 1 − c j D_j=c_{j+1}-c_j D j = c j + 1 − c j
Z f = Z 0 + c 1 ( τ 3 − τ 0 ) + ∑ j = 1 2 D j ( τ 3 − τ j ) , Z_f = Z_0 + c_1(\tau_3-\tau_0) + \sum_{j=1}^{2} D_j(\tau_3-\tau_j), Z f = Z 0 + c 1 ( τ 3 − τ 0 ) + j = 1 ∑ 2 D j ( τ 3 − τ j ) , Y f = Y 0 + ( − Z 0 + τ 0 c 1 ) ( τ 3 − τ 0 ) + ∑ j = 1 2 D j τ j ( τ 3 − τ j ) . (9.2) Y_f = Y_0 + (-Z_0+\tau_0c_1)(\tau_3-\tau_0) + \sum_{j=1}^{2} D_j\tau_j(\tau_3-\tau_j).\tag{9.2} Y f = Y 0 + ( − Z 0 + τ 0 c 1 ) ( τ 3 − τ 0 ) + j = 1 ∑ 2 D j τ j ( τ 3 − τ j ) . ( 9.2 ) A switch at τ j \tau_j τ j τ j D j \tau_jD_j τ j D j
det ∂ ( τ 3 , Z f , Y f ) ∂ ( τ 1 , τ 2 , τ 3 ) = det ( − D 1 − D 2 D 1 ( τ 3 − 2 τ 1 ) D 2 ( τ 3 − 2 τ 2 ) ) = 2 D 1 D 2 ( τ 2 − τ 1 ) . (9.3) \det\frac{\partial(\tau_3,Z_f,Y_f)}{\partial(\tau_1,\tau_2,\tau_3)}
=
\det
\begin{pmatrix}
-D_1 & -D_2 \\
D_1(\tau_3-2\tau_1) & D_2(\tau_3-2\tau_2)
\end{pmatrix}
=2D_1D_2(\tau_2-\tau_1).\tag{9.3} det ∂ ( τ 1 , τ 2 , τ 3 ) ∂ ( τ 3 , Z f , Y f ) = det ( − D 1 D 1 ( τ 3 − 2 τ 1 ) − D 2 D 2 ( τ 3 − 2 τ 2 ) ) = 2 D 1 D 2 ( τ 2 − τ 1 ) . ( 9.3 ) This determinant is bounded below by an inverse subpower on the retained paths. At fixed τ 3 \tau_3 τ 3 Z f Z_f Z f Y f Y_f Y f K ω 3 K\omega^3 K ω 3 π ⊗ 3 \pi^{\otimes3} π ⊗ 3
P ( regular endpoint in a specified base cell ∣ start ) ≤ K ω 3 . (9.4) \mathbb{P}(\text{regular endpoint in a specified base cell}\mid\text{start}) \le K\omega^3.\tag{9.4} P ( regular endpoint in a specified base cell ∣ start ) ≤ K ω 3 . ( 9.4 ) The simulation error requires only a bounded enlargement of the cell. Write q C = μ ~ ( C ) q_C=\tilde{\mu}(C) q C = μ ~ ( C ) e C e_C e C C C C ∑ C q C e C = η = o ( 1 ) \sum_C q_Ce_C=\eta=o(1) ∑ C q C e C = η = o ( 1 ) B = { C : e C ≤ 1 / 2 } \mathcal{B}=\{C:e_C\le1/2\} B = { C : e C ≤ 1/2 } 1 − 2 η 1-2\eta 1 − 2 η (9.4) over its conditional starting law. Every endpoint still has intercept bin C C C 1 / 2 1/2 1/2
# { base cells occupied by C } ≥ 1 2 K ω 3 , C ∈ B . \#\{\text{base cells occupied by }C\}\ge\frac{1}{2K\omega^3}, \qquad C\in \mathcal{B}. # { base cells occupied by C } ≥ 2 K ω 3 1 , C ∈ B . This is the lower cell count on bins carrying almost all the fitted mass.
The heavy candidate in the original mass units. The upper count of (base cell, C C C C ∈ B C\in \mathcal{B} C ∈ B # B ≤ K ( a / a ′ ) d \#\mathcal{B}\le K(a/a')^d # B ≤ K ( a / a ′ ) d ∑ C ∈ B q C ≥ 1 − 2 η \sum_{C\in \mathcal{B}}q_C\ge1-2\eta ∑ C ∈ B q C ≥ 1 − 2 η
μ l a s t ( C ) = m l a s t q C ≥ K − 1 m Q s ( a ′ / a ) d = K − 1 m Q s N − d c . \mu_{\mathrm{last}}(C)=m_{\mathrm{last}}q_C\ge K^{-1}m_{Q_s}(a'/a)^d=K^{-1}m_{Q_s}N^{-dc}. μ last ( C ) = m last q C ≥ K − 1 m Q s ( a ′ / a ) d = K − 1 m Q s N − d c . This is mass on the original residual in horizontal-problem units, before probability normalization. Its x − F ∗ x-F_* x − F ∗ K a ′ Ka' K a ′ w w w w − L − a r ( F ∗ + offset ) w-L-a_r(F_*+\text{offset}) w − L − a r ( F ∗ + offset ) K a r a ′ Ka_ra' K a r a ′ K K K q r ν Q r q_r\nu_{Q_r} q r ν Q r K − 1 q r ν Q r m Q s N − d c K^{-1}q_r\nu_{Q_r}m_{Q_s}N^{-dc} K − 1 q r ν Q r m Q s N − d c r + s r+s r + s
Full-time parameter estimates below the critical rate
The coefficient count gives predictors at thickness a s a_s a s a s a_s a s h s 2 = a s 2 k h_s^2=a_s^{2k} h s 2 = a s 2 k
Definition 9.3 (Full-time parameter estimates). Suppose 2 k < 1 2k < 1 2 k < 1 s max > 0 s_{\max} > 0 s m a x > 0 e 0 e_0 e 0 s max s_{\max} s m a x d 0 d_0 d 0 χ ≤ N − e 0 \chi\le N^{-e_0} χ ≤ N − e 0
For 2 k ≤ u ≤ 1 2k \le u \le1 2 k ≤ u ≤ 1 G ( u ) \mathcal{G}(u) G ( u ) 0 < s < s max 0 < s < s_{\max} 0 < s < s m a x ( d 0 , P D ) (d_0,P_D) ( d 0 , P D ) D = a s u D = a_s^u D = a s u
K − 1 π ( d 0 ) D 2 a s d , K^{-1}\pi(d_0)D^2a_s^d, K − 1 π ( d 0 ) D 2 a s d , and each label has a predictor
x = H ( t , P ) + O ( K a s ) x = H(t,P) + O(Ka_s) x = H ( t , P ) + O ( K a s ) throughout unit time on its assigned trajectories. The function H H H P P P K K K
Lemma 9.4 (Comparison with full-time predictors). Suppose 2 k < 1 2k < 1 2 k < 1 2 k ≤ u ≤ 1 2k \le u \le1 2 k ≤ u ≤ 1 G ( u ) \mathcal{G}(u) G ( u ) s 0 s_0 s 0 a 0 = a s 0 a_0 = a_{s_0} a 0 = a s 0 D 0 = a 0 u D_0 = a_0^u D 0 = a 0 u G = ( d 0 , P D 0 ) G = (d_0,P_{D_0}) G = ( d 0 , P D 0 ) s 0 s_0 s 0 G ( u ) \mathcal{G}(u) G ( u ) H 0 H_0 H 0 s 0 s_0 s 0
For each Q s 0 Q_{s_0} Q s 0 I I I
L s 0 ( t , P ) = L s 0 ( t , p + t v d 0 , q 0 − t p ) . \mathcal L_{s_0}(t,P) = L_{s_0}(t,p+tv_{d_0},q_0-tp). L s 0 ( t , P ) = L s 0 ( t , p + t v d 0 , q 0 − tp ) . Every retained pairing of H 0 H_0 H 0 ( Q s 0 , L s 0 ) (Q_{s_0},L_{s_0}) ( Q s 0 , L s 0 )
∥ L s 0 − H 0 ∥ ≤ K a 0 \lVert \mathcal L_{s_0}-H_0\rVert\le Ka_0 ∥ L s 0 − H 0 ∥ ≤ K a 0 on the whole product of I I I H 0 , I H_0,I H 0 , I K K K s s s m Q s / K m_{Q_s}/K m Q s / K ∥ L s − L s 0 ∥ ≤ K a 0 \lVert L_s-L_{s_0}\rVert\le Ka_0 ∥ L s − L s 0 ∥ ≤ K a 0
Proof. For nested true labels we use Lemma 9.1 to retain full-domain witnesses for paired affine entries at s 0 s_0 s 0 s s s (8.5) before the small slices or full-time selections: discard light pairs compared with m Q s = H s ν ♭ ( Q s ) m_{Q_s}=H_s\nu^\flat(Q_s) m Q s = H s ν ♭ ( Q s ) ≤ K \le K ≤ K ≥ m Q s / K \ge m_{Q_s}/K ≥ m Q s / K L s − L s 0 L_s-L_{s_0} L s − L s 0 K a 0 Ka_0 K a 0
Slice by G = ( d 0 , P D 0 ) G=(d_0,P_{D_0}) G = ( d 0 , P D 0 ) Q s 0 Q_{s_0} Q s 0 I I I W G = π ( d 0 ) D 0 2 W_G=\pi(d_0)D_0^2 W G = π ( d 0 ) D 0 2 H 0 H_0 H 0 ( Q s 0 , L s 0 ) (Q_{s_0},L_{s_0}) ( Q s 0 , L s 0 ) T G I / K T_{GI}/K T G I / K T G I = ∣ I ∣ W G a 0 d T_{GI}=|I|W_Ga_0^d T G I = ∣ I ∣ W G a 0 d H 0 H_0 H 0 ∑ I ∣ I ∣ = 1 \sum_I |I|=1 ∑ I ∣ I ∣ = 1
For each Q s 0 , d 0 Q_{s_0},d_0 Q s 0 , d 0 K h s 0 3 / D 0 2 Kh_{s_0}^3/D_0^2 K h s 0 3 / D 0 2 P P P K h s 0 Kh_{s_0} K h s 0 K h s 0 2 Kh_{s_0}^2 K h s 0 2 (8.1) . Thus, up to K ( χ + N − e 0 ) ≪ D 0 K(\chi+N^{-e_0})\ll D_0 K ( χ + N − e 0 ) ≪ D 0 P ↦ ( p + t v d 0 , q 0 − t p ) P \mapsto(p + tv_{d_0}, q_0 - tp) P ↦ ( p + t v d 0 , q 0 − tp ) v d 0 v_{d_0} v d 0 K K K K K K D 0 ≤ K h s 0 2 D_0 \leq Kh_{s_0}^2 D 0 ≤ K h s 0 2
Sum the thresholds using this count, the K K K ∑ π ( d 0 ) = 1 \sum\pi(d_0) = 1 ∑ π ( d 0 ) = 1 ∑ m Q s 0 ≤ 1 \sum m_{Q_{s_0}} \leq1 ∑ m Q s 0 ≤ 1 T G I T_{GI} T G I ≤ K \leq K ≤ K K K K
On this product domain, H 0 H_0 H 0 L s 0 \mathcal{L}_{s_0} L s 0 x x x K a 0 Ka_0 K a 0 (8.1) and (8.5) . We use Corollary 3.4 to upgrade this agreement to the whole domain. The same calligraphic notation will be used for true affine entries at other depths.
For either index on sufficiently fine ( t , P ) (t,P) ( t , P ) K T G I KT_{GI} K T G I d 0 d_0 d 0 x a 0 x_{a_0} x a 0 P ↦ ( p + t v d 0 , q 0 − t p ) P \mapsto(p + tv_{d_0}, q_0 - tp) P ↦ ( p + t v d 0 , q 0 − tp ) K ( χ + N − e 0 ) K(\chi+ N^{-e_0}) K ( χ + N − e 0 ) N − C s max N^{-C s_{\max}} N − C s m a x C C C C s max Cs_{\max} C s m a x e 0 e_0 e 0 a 0 a_0 a 0 L s 0 \mathcal{L}_{s_0} L s 0 K K K H 0 H_0 H 0 ≤ K \leq K ≤ K D 0 D_0 D 0
If a power-excess norm discrepancy occurs on dense mass, pigeonhole a comparable discrepancy M 0 M_0 M 0 a 0 a_0 a 0 K K K M 0 M_0 M 0 K K K ≤ K ( M 0 / a 0 ) d \leq K(M_0/a_0)^d ≤ K ( M 0 / a 0 ) d T G I / K T_{GI}/K T G I / K K M 0 KM_0 K M 0 T G I T_{GI} T G I ≤ K \leq K ≤ K K K K R = M 0 / a 0 R = M_0/a_0 R = M 0 / a 0 K a 0 Ka_0 K a 0 H 0 , I H_0,I H 0 , I K a 0 Ka_0 K a 0 K K K
Proposition 9.5 (Parameter bootstrap below the critical rate). If 2 k < 1 2k < 1 2 k < 1 s max > 0 s_{\max} > 0 s m a x > 0 G ( 2 k ) \mathcal{G}(2k) G ( 2 k ) s max s_{\max} s m a x
Proof. At u = 1 u = 1 u = 1 P P P 2 k < u ≤ 1 2k < u \leq1 2 k < u ≤ 1
Fix 2 k < u ≤ 1 2k < u \leq1 2 k < u ≤ 1 G ( u ) \mathcal{G}(u) G ( u ) s 1 < s max s_1 < s_{\max} s 1 < s m a x s 0 = s 1 / ( 1 + θ ) s_0 = s_1/(1+\theta) s 0 = s 1 / ( 1 + θ ) θ \theta θ
a 0 = a s 0 , D 0 = a 0 u , W G = π ( d 0 ) D 0 2 . a_0 = a_{s_0}, \qquad D_0 = a_0^u, \qquad W_G = \pi(d_0)D_0^2. a 0 = a s 0 , D 0 = a 0 u , W G = π ( d 0 ) D 0 2 . Prepare the nearby true depths, and then take the full-time predictors H 0 H_0 H 0 G ( u ) \mathcal{G}(u) G ( u ) s 0 s_0 s 0 a 0 a_0 a 0 a s 1 a_{s_1} a s 1
The conditional prior and scalar support. Choose one such paired L s 0 , Q s 0 L_{s_0},Q_{s_0} L s 0 , Q s 0 H 0 H_0 H 0 L s 0 \mathcal L_{s_0} L s 0 H 0 H_0 H 0 H 0 H_0 H 0
ν ♭ ( H 0 ) ∼ log , N W G a 0 d . \nu^\flat(H_0)\sim_{\log,N}W_Ga_0^d. ν ♭ ( H 0 ) ∼ l o g , N W G a 0 d . Indeed restore floors or discard now-light H 0 H_0 H 0
ν H = ν ♭ ( ⋅ ∣ H 0 ) \nu_H=\nu^\flat(\cdot\mid H_0) ν H = ν ♭ ( ⋅ ∣ H 0 ) throughout this bootstrap step. Let λ H \lambda_H λ H ν ♭ ( H 0 ) \nu^\flat(H_0) ν ♭ ( H 0 )
d λ H = f H d ν H d t , 0 ≤ f H ≤ K p r e , δ H = λ H ( a l l ) ≥ 1 / K , d\lambda_H=f_H\,d\nu_H\,dt,\qquad0\leq f_H\leq K_{\mathrm{pre}},\qquad\delta_H=\lambda_H(\mathrm{all})\geq1/K, d λ H = f H d ν H d t , 0 ≤ f H ≤ K pre , δ H = λ H ( all ) ≥ 1/ K , where K p r e K_{\mathrm{pre}} K pre
y = ( P − P c ) / D 0 , x ˉ = ( x − H 0 ( t , P ) ) / a 0 . y=(P-P_c)/D_0,\qquad\bar{x}=(x-H_0(t,P))/a_0. y = ( P − P c ) / D 0 , x ˉ = ( x − H 0 ( t , P )) / a 0 . Thus y ∈ R 2 y\in\mathbb{R}^2 y ∈ R 2 x ˉ \bar{x} x ˉ ≤ K \leq K ≤ K F ( t ) = ∇ y ( L s 0 − H 0 ) / a 0 F(t)=\nabla_y(\mathcal L_{s_0}-H_0)/a_0 F ( t ) = ∇ y ( L s 0 − H 0 ) / a 0 K K K S ( t ) = x ˉ ( t ) − F ( t ) y S(t)=\bar{x}(t)-F(t)y S ( t ) = x ˉ ( t ) − F ( t ) y
The immediate target is a short list of residual values and slopes at one time. With w = N − θ s 0 w=N^{-\theta s_0} w = N − θ s 0 θ \theta θ A ( t ) A(t) A ( t ) t ∗ t_* t ∗ K w − d Kw^{-d} K w − d
( S l ( t ∗ ) w , ( x ˉ l ′ − A ( t ∗ ) y l ) w ) \left(S_l(t_*)_w,\;(\bar{x}_l'-A(t_*)y_l)_w\right) ( S l ( t ∗ ) w , ( x ˉ l ′ − A ( t ∗ ) y l ) w ) are needed. Since y l y_l y l x ˉ l \bar{x}_l x ˉ l y y y K w Kw K w H 0 H_0 H 0 K a 0 w = K a s 1 Ka_0w=Ka_{s_1} K a 0 w = K a s 1
D 0 = a 0 u = a s 1 u / ( 1 + θ ) . D_0=a_0^u=a_{s_1}^{u/(1+\theta)}. D 0 = a 0 u = a s 1 u / ( 1 + θ ) . This is how the step improves G ( u ) \mathcal{G}(u) G ( u ) G ( u / ( 1 + θ ) ) \mathcal{G}(u/(1+\theta)) G ( u / ( 1 + θ )) E = ( u − 2 k ) s 0 / 10 E=(u-2k)s_0/10 E = ( u − 2 k ) s 0 /10 ζ = N − E \zeta=N^{-E} ζ = N − E
D 0 / h s 0 + E 2 ≪ ζ . D_0/h_{s_0+E}^2\ll\zeta. D 0 / h s 0 + E 2 ≪ ζ . We first obtain integrated mixed caps for λ H \lambda_H λ H ν H \nu_H ν H r , w ∈ [ ζ , 1 ] r,w\in[\zeta,1] r , w ∈ [ ζ , 1 ] d 0 d_0 d 0 a 0 w a_0w a 0 w H 0 H_0 H 0 P P P K π ( d 0 ) ( D 0 r ) 2 ( a 0 w ) d K\pi(d_0)(D_0r)^2(a_0w)^d K π ( d 0 ) ( D 0 r ) 2 ( a 0 w ) d P P P s max s_{\max} s m a x K / ( D 0 a 0 ) K/(D_0a_0) K / ( D 0 a 0 ) I I I K ∣ I ∣ r 2 w d K|I|r^{2}w^{d} K ∣ I ∣ r 2 w d
Set E l 0 = { t : f H ( l , t ) ≥ δ H / 2 } E_l^0=\{t:f_H(l,t)\ge\delta_H/2\} E l 0 = { t : f H ( l , t ) ≥ δ H /2 } d ρ H 0 = 1 E l 0 ( t ) d ν H ( l ) d t d\rho_H^0=1_{E_l^0}(t)d\nu_H(l)dt d ρ H 0 = 1 E l 0 ( t ) d ν H ( l ) d t λ H \lambda_H λ H δ H / 2 \delta_H/2 δ H /2
( δ H / 2 ) d ρ H 0 ≤ d ( λ H ∣ E 0 ) ≤ K p r e d ρ H 0 . (\delta_H/2)\,d\rho_H^0\le d(\lambda_H|_{E^0})\le K_{\mathrm{pre}}\,d\rho_H^0. ( δ H /2 ) d ρ H 0 ≤ d ( λ H ∣ E 0 ) ≤ K pre d ρ H 0 . In particular ρ H 0 \rho_H^0 ρ H 0 δ H / ( 2 K p r e ) \delta_H/(2K_{\mathrm{pre}}) δ H / ( 2 K pre )
Choose one tiny time partition, fine enough for the current finite width grid that K ∣ I ∣ ≪ ζ K|I|\ll\zeta K ∣ I ∣ ≪ ζ l l l I I I E l 0 ∩ I E_l^0\cap I E l 0 ∩ I ∣ E l 0 ∩ I ∣ ≥ ∣ I ∣ / K d u r |E_l^0\cap I|\ge|I|/K_{\mathrm{dur}} ∣ E l 0 ∩ I ∣ ≥ ∣ I ∣/ K dur E l E_l E l d ρ H = 1 E l ( t ) d ν H ( l ) d t d\rho_H=1_{E_l}(t)d\nu_H(l)dt d ρ H = 1 E l ( t ) d ν H ( l ) d t 1 / K d u r 1/K_{\mathrm{dur}} 1/ K dur K p r e / K d u r K_{\mathrm{pre}}/K_{\mathrm{dur}} K pre / K dur
For fixed t ∈ I t\in I t ∈ I A t A_t A t t t t y y y r r r x ˉ ( t ) \bar{x}(t) x ˉ ( t ) w w w I I I
∣ I ∣ K d u r ν H ( A t ) ≤ ρ H ( I × enlarged constraints ) ≤ 2 δ H λ H ( I × enlarged constraints ) ≤ K ∣ I ∣ r 2 w d . \frac{|I|}{K_{\mathrm{dur}}}\nu_H(A_t)\le\rho_H(I\times\text{ enlarged constraints})\le\frac{2}{\delta_H}\lambda_H(I\times\text{ enlarged constraints})\le K|I|r^{2}w^{d}. K dur ∣ I ∣ ν H ( A t ) ≤ ρ H ( I × enlarged constraints ) ≤ δ H 2 λ H ( I × enlarged constraints ) ≤ K ∣ I ∣ r 2 w d . After division by ∣ I ∣ |I| ∣ I ∣
ν H { l : t ∈ E l , y r , x ˉ l ( t ) w in specified bins } ≤ K r 2 w d . \nu_H\{l:t\in E_l,\ y_r,\ \bar{x}_l(t)_w\text{ in specified bins}\}\le Kr^{2}w^{d}. ν H { l : t ∈ E l , y r , x ˉ l ( t ) w in specified bins } ≤ K r 2 w d . These are line–time-bin deletions. All subsequent selections restrict E l E_l E l ν H \nu_H ν H
We also have at each active t t t w = N − s ′ w=N^{-s'} w = N − s ′ 0 ≤ s ′ ≤ E 0\le s'\le E 0 ≤ s ′ ≤ E
N w { S ( t ) } ≤ K w − d . N_w\{S(t)\}\le Kw^{-d}. N w { S ( t )} ≤ K w − d . This support count uses the earlier full-domain pair witnesses in the horizontal law, frozen before conditioning to H 0 H_0 H 0 ρ H \rho_H ρ H Q s Q_s Q s s = s 0 + s ′ s=s_0+s' s = s 0 + s ′ Q s 0 , L s 0 Q_{s_0},L_{s_0} Q s 0 , L s 0 t t t G G G K D 0 KD_0 K D 0 h s 2 h_s^2 h s 2 K K K ≥ K − 1 h s 4 a s d \ge K^{-1}h_s^4a_s^d ≥ K − 1 h s 4 a s d L s 0 L_{s_0} L s 0 K h s 4 a 0 d Kh_s^4a_0^d K h s 4 a 0 d ≤ K ( a 0 / a s ) d \le K(a_0/a_s)^d ≤ K ( a 0 / a s ) d S S S K w Kw K w ( L s − L s 0 ) / a 0 (\mathcal L_s-\mathcal L_{s_0})/a_0 ( L s − L s 0 ) / a 0 F F F t t t K / h s 2 K/h_s^2 K / h s 2 ∥ L s − L s 0 ∥ ≤ K a 0 \|L_s-L_{s_0}\|\le Ka_0 ∥ L s − L s 0 ∥ ≤ K a 0 D 0 D_0 D 0 ≤ K w \le Kw ≤ K w (8.1) fit this estimate as well. This proves the support count.
At each tested width w w w ν H \nu_H ν H w d / K w^d/K w d / K K K K r r r S S S w ≤ r ≤ 1 w\le r\le1 w ≤ r ≤ 1 r r r y r y_r y r F ( t ) F(t) F ( t ) S S S K K K x ˉ \bar{x} x ˉ r r r K r 2 + d Kr^{2+d} K r 2 + d K r − 2 Kr^{-2} K r − 2 K r d Kr^d K r d K ( r / w ) d K(r/w)^d K ( r / w ) d
Prepare the true-chart and pair witnesses on fixed finite exponent grids before applying the baseline property to that dense input. After obtaining H 0 H_0 H 0
Regularizing the scalar-normal input. First we require the Lipschitz comparison for F F F ζ \zeta ζ r r r r r r ∣ t − t ′ ∣ ≤ r |t-t'|\le r ∣ t − t ′ ∣ ≤ r M = ∣ F ( t ) − F ( t ′ ) ∣ ≫ r M=|F(t)-F(t')|\gg r M = ∣ F ( t ) − F ( t ′ ) ∣ ≫ r ≤ K ( r / M ) 1 − d \le K(r/M)^{1-d} ≤ K ( r / M ) 1 − d
To prove this bound, given a scalar source S ( t ) r S(t)_r S ( t ) r S ( t ′ ) S(t') S ( t ′ ) K ( M + r ) K(M+r) K ( M + r ) x ˉ \bar{x} x ˉ K K K ≤ K ( M / r ) d \le K(M/r)^d ≤ K ( M / r ) d y y y K r / M Kr/M K r / M K / ( M r ) K/(Mr) K / ( M r ) r r r y y y M ≤ K M\le K M ≤ K ≤ K r 2 + d \le Kr^{2+d} ≤ K r 2 + d t t t ≤ K r − d \le Kr^{-d} ≤ K r − d
Consequently the pairs with M / r M/r M / r ∣ F − F a n c h o r ∣ ≤ K r |F-F_{\mathrm{anchor}}|\le Kr ∣ F − F anchor ∣ ≤ K r r r r K ( ∣ t − t ′ ∣ + ζ ) K(|t-t'|+\zeta) K ( ∣ t − t ′ ∣ + ζ )
Next prepare the slope comparison on spacetime ( t , y , x ˉ ) (t,y,\bar{x}) ( t , y , x ˉ ) ν H \nu_H ν H r r r y r y_r y r r 2 r^2 r 2 K K K K r − d Kr^{-d} K r − d x ˉ \bar{x} x ˉ S S S F F F
For each line and r r r I r I_r I r r / K r/K r / K K K K K r Kr K r r / K r/K r / K K r − 1 − d Kr^{-1-d} K r − 1 − d y r y_r y r 1 / K 1/K 1/ K L r \mathcal{L}_r L r ν H \nu_H ν H L r \mathcal{L}_r L r r 2 r^2 r 2 K K K K w d Kw^d K w d w ≥ r w\ge r w ≥ r ν H \nu_H ν H
Lemma 5.1 now permits retaining joint bins B r = ( y r , [ x ˉ coefficients ] r ) B_r=(y_r,[\bar{x}\text{ coefficients}]_r) B r = ( y r , [ x ˉ coefficients ] r ) ρ H \rho_H ρ H r 2 + d / K r^{2+d}/K r 2 + d / K
Next prune the distinct entries ( B r , I r ) (B_r,I_r) ( B r , I r ) I r I_r I r r r r ∣ I r ∣ r 2 + d / K |I_r|r^{2+d}/K ∣ I r ∣ r 2 + d / K r r r ( y , x ˉ ) (y,\bar{x}) ( y , x ˉ ) K r Kr K r O ( r ) O(r) O ( r ) B r B_r B r I r I_r I r K r Kr K r K K K
In each cube select one slope bin with dense aggregate success, then retain a separated residue class of cubes so that selected points within distance r r r ζ \zeta ζ y ′ = 0 y'=0 y ′ = 0
For each participating full parameter bin B r B_r B r ν H ( B r ) ≥ r 2 + d / K \nu_H(B_r) \ge r^{2+d}/K ν H ( B r ) ≥ r 2 + d / K
ν H { l ∈ B r : t ∈ E l , ( y l , x ˉ l ( t ) ) in a q -cube } ν H ( B r ) ≤ K q 2 + d r 2 + d , ζ ≤ q < r . \frac{\nu_H\{l \in B_r : t \in E_l,\ (y_l,\bar{x}_l(t))\text{ in a }q\text{-cube}\}}{\nu_H(B_r)}
\le K\frac{q^{2+d}}{r^{2+d}}, \qquad\zeta\le q < r. ν H ( B r ) ν H { l ∈ B r : t ∈ E l , ( y l , x ˉ l ( t )) in a q -cube } ≤ K r 2 + d q 2 + d , ζ ≤ q < r . Only static y y y x ˉ \bar{x} x ˉ ν H \nu_H ν H ν H \nu_H ν H E l E_l E l
One bootstrap step. Apply the first-jet conclusion of Proposition 5.4 to the normalized signals x ˉ , y \bar{x},y x ˉ , y A ( t ) A(t) A ( t ) K K K
H ( ( x ˉ ′ − A ( t ) y ) w ∣ t , S ( t ) w ) = o ( log N ) \mathsf{H}((\bar{x}'-A(t)y)_w\mid t,S(t)_w)=o(\log N) H (( x ˉ ′ − A ( t ) y ) w ∣ t , S ( t ) w ) = o ( log N ) for w = N − θ s 0 w=N^{-\theta s_0} w = N − θ s 0 θ = ( 1 − d ) ( u − 2 k ) / 10000 \theta=(1-d)(u-2k)/10000 θ = ( 1 − d ) ( u − 2 k ) /10000 E / 4 E/4 E /4 H ∗ = N − E / 4 H_*=N^{-E/4} H ∗ = N − E /4 ζ = N − E ≪ H ∗ 3 \zeta=N^{-E}\ll H_*^3 ζ = N − E ≪ H ∗ 3
θ s 0 = ( 1 − d ) E 1000 < ( 1 − d ) E 256 . \theta s_0=\frac{(1-d)E}{1000}<\frac{(1-d)E}{256}. θ s 0 = 1000 ( 1 − d ) E < 256 ( 1 − d ) E . These are the floor and reading conditions of the first-jet conclusion. Also D 0 / h s 0 + E 2 = N − ( 10 − 2 k ) E ≪ ζ D_0/h_{s_0+E}^2=N^{-(10-2k)E}\ll\zeta D 0 / h s 0 + E 2 = N − ( 10 − 2 k ) E ≪ ζ ≤ K w − d \le K w^{-d} ≤ K w − d t ∗ t_* t ∗
Assign these lines disjointly by the bin pair at t ∗ t_* t ∗ H 0 H_0 H 0 b , b ′ b,b' b , b ′
x ˉ ( t ) = b + F ( t ∗ ) y + ( t − t ∗ ) ( b ′ + A ( t ∗ ) y ) + O ( K w ) \bar{x}(t)=b+F(t_*)y+(t-t_*)(b'+A(t_*)y)+O(Kw) x ˉ ( t ) = b + F ( t ∗ ) y + ( t − t ∗ ) ( b ′ + A ( t ∗ ) y ) + O ( K w ) using exact affine evolution. Restoring H 0 H_0 H 0 K K K a 0 w = a s 1 a_0w=a_{s_1} a 0 w = a s 1 π ( d 0 ) D 0 2 a s 1 d / K \pi(d_0)D_0^2a_{s_1}^d/K π ( d 0 ) D 0 2 a s 1 d / K W G a d W_Ga^d W G a d K w − d Kw^{-d} K w − d G ( u / ( 1 + θ ) ) \mathcal{G}(u/(1+\theta)) G ( u / ( 1 + θ )) s 1 < s max s_1<s_{\max} s 1 < s m a x s max s_{\max} s m a x u − 2 k > 0 u-2k>0 u − 2 k > 0
A fixed reconstruction buffer. The accuracy requirements in the preceding step are uniform in the distance u − 2 k u-2k u − 2 k ζ = N − E \zeta=N^{-E} ζ = N − E E ≤ s max / 10 E\le s_{\max}/10 E ≤ s m a x /10 H 1 / 4 H^{1/4} H 1/4 H 1 / 8 H^{1/8} H 1/8 H ( 1 − d ) / 32 H^{(1-d)/32} H ( 1 − d ) /32 E E E
K D 0 a 0 ≤ K N 2 s max , 2 k < u ≤ 1. \frac{K}{D_0a_0}\le KN^{2s_{\max}},\qquad2k<u\le1. D 0 a 0 K ≤ K N 2 s m a x , 2 k < u ≤ 1. All further differentiated tests have fixed degree and order. Thus one absolute exponent C 0 s max C_0s_{\max} C 0 s m a x C 0 C_0 C 0 s max s_{\max} s m a x C 0 s max < e 0 / 2 C_0s_{\max}<e_0/2 C 0 s m a x < e 0 /2 C 0 C_0 C 0 N − e 0 N^{-e_0} N − e 0 1 / E 1/E 1/ E
The endpoint sequence and its actual labels. Put α = ( 1 − d ) / 10000 \alpha=(1-d)/10000 α = ( 1 − d ) /10000
u j + 1 = u j 1 + α ( u j − 2 k ) , u 0 = 1. u_{j+1}=\frac{u_j}{1+\alpha(u_j-2k)},\qquad u_0=1. u j + 1 = 1 + α ( u j − 2 k ) u j , u 0 = 1. It satisfies
u j + 1 − 2 k = ( u j − 2 k ) ( 1 − 2 k α ) 1 + α ( u j − 2 k ) > 0 , u_{j+1}-2k=\frac{(u_j-2k)(1-2k\alpha)}{1+\alpha(u_j-2k)}>0, u j + 1 − 2 k = 1 + α ( u j − 2 k ) ( u j − 2 k ) ( 1 − 2 k α ) > 0 , u j − u j + 1 = α u j ( u j − 2 k ) 1 + α ( u j − 2 k ) > 0. u_j-u_{j+1}=\frac{\alpha u_j(u_j-2k)}{1+\alpha(u_j-2k)}>0. u j − u j + 1 = 1 + α ( u j − 2 k ) α u j ( u j − 2 k ) > 0. The limit must therefore be 2 k 2k 2 k s < s max s<s_{\max} s < s m a x j j j K j ( N ) K_j(N) K j ( N ) N N N log K j ( N ) / log N < 1 / j \log K_j(N)/\log N<1/j log K j ( N ) / log N < 1/ j N − 1 / j N^{-1/j} N − 1/ j j = j ( N ) → ∞ j=j(N)\to\infty j = j ( N ) → ∞
Let D j = a s u j D_j=a_s^{u_j} D j = a s u j D ∗ = a s 2 k D_*=a_s^{2k} D ∗ = a s 2 k R j = D ∗ / D j R_j=D_*/D_j R j = D ∗ / D j log N R j = s ( u j − 2 k ) → 0 \log_N R_j=s(u_j-2k)\to0 log N R j = s ( u j − 2 k ) → 0 D ∗ D_* D ∗ C K j ( 1 + R j ) CK_j(1+R_j) C K j ( 1 + R j )
For each old label let T j = π ( d 0 ) D j 2 a s d T_j=\pi(d_0)D_j^2a_s^{d} T j = π ( d 0 ) D j 2 a s d ∑ T j ≤ K j \sum T_j\le K_j ∑ T j ≤ K j R j 2 T j R_j^2T_j R j 2 T j C K j R j 2 CK_jR_j^2 C K j R j 2 π ( d 0 ) D ∗ 2 a s d / K \pi(d_0)D_*^2a_s^d/K π ( d 0 ) D ∗ 2 a s d / K R j R_j R j
For several prescribed depths, apply this construction successively to the current dense input, preserving earlier assignments, predictors and witnesses. Each resulting family’s floor numerators have subpower sum. Include also the full-time chart labels, with floor numerators ν ♭ ( H ) \nu^\flat(H) ν ♭ ( H ) ν ♭ ( H ) / K \nu^\flat(H)/K ν ♭ ( H ) / K G ( 2 k ) \mathcal{G}(2k) G ( 2 k )
Returning to wide intervals away from the critical rate
Proposition 9.6 (Return to wide intervals). If 2 k ≠ 1 2k\ne1 2 k = 1 2 k < 1 2k< 1 2 k < 1 2 k > 1 2k> 1 2 k > 1
Proof. Parameter charts and their unchanged prior denominator. Keep the horizontal-model coordinates, relative depths and true assignment priors fixed. We will construct projected labels at three nearby depths whose common transverse width is D 0 = h s 0 2 D_0=h_{s_0}^2 D 0 = h s 0 2
D 0 = N − 2 k s 0 , P ~ = ( P − P c ) / D 0 D_0=N^{-2k s_0}, \qquad\widetilde{P}=(P-P_c)/D_0 D 0 = N − 2 k s 0 , P = ( P − P c ) / D 0 using groups at P D 0 P_{D_0} P D 0 d 0 = [ v ] χ d_0=[v]_\chi d 0 = [ v ] χ P c = ( p c , q c ) P_c=(p_c,q_c) P c = ( p c , q c ) χ ≤ N − e 0 \chi\le N^{-e_0} χ ≤ N − e 0 s 0 > 0 s_0>0 s 0 > 0 e 0 e_0 e 0 3 2 s 0 < s max \frac{3}{2}s_0<s_{\max} 2 3 s 0 < s m a x G ( 2 k ) \mathcal{G}(2k) G ( 2 k ) χ \chi χ
Prepare the earlier affine data (including (8.5) ) for the needed depths first. In particular for pairs used between s 0 s_0 s 0 s j s_j s j m Q s j / K m_{Q_{s_j}}/K m Q s j / K L s j , L s 0 L_{s_j},L_{s_0} L s j , L s 0 L s j − L s 0 L_{s_j}-L_{s_0} L s j − L s 0 K a s 0 K a_{s_0} K a s 0 x x x
If 2 k > 1 2k>1 2 k > 1 ( d 0 , P D 0 ) (d_0,P_{D_0}) ( d 0 , P D 0 ) H 0 = 0 H_0=0 H 0 = 0 A 0 = 1 A_0=1 A 0 = 1 2 k < 1 2k<1 2 k < 1 G ( 2 k ) \mathcal{G}(2k) G ( 2 k ) H 0 , H b H_0,H_b H 0 , H b s 0 , s b = 3 2 s 0 s_0,s_b=\frac{3}{2}s_0 s 0 , s b = 2 3 s 0 d 0 d_0 d 0 L s j , Q s j L_{s_j},Q_{s_j} L s j , Q s j L s j ( t , P ) = L s j ( t , p + t v d 0 , q 0 − t p ) \mathcal L_{s_j}(t,P)=L_{s_j}(t,p+tv_{d_0},q_0-tp) L s j ( t , P ) = L s j ( t , p + t v d 0 , q 0 − tp ) 2 k < 1 2k<1 2 k < 1 L s j \mathcal L_{s_j} L s j s j = s 0 , s b s_j=s_0,s_b s j = s 0 , s b K a s j K a_{s_j} K a s j N − 2 k s j N^{-2k s_j} N − 2 k s j K K K L s 0 , Q s 0 L_{s_0},Q_{s_0} L s 0 , Q s 0 H 0 H_0 H 0 H 0 H_0 H 0 A 0 = a s 0 A_0=a_{s_0} A 0 = a s 0
In either case keep typical such charts with active mass and prior mass denominator both of order up to K K K
W G A 0 d , W G = π ( d 0 ) D 0 2 . W_G A_0^d,\qquad W_G=\pi(d_0)D_0^2. W G A 0 d , W G = π ( d 0 ) D 0 2 . Indeed the thresholds sum to subpower by the earlier label floors (or ∑ W G ≤ K \sum W_G\le K ∑ W G ≤ K d 0 , x d_0,x d 0 , x H 0 H_0 H 0 A 0 A_0 A 0 χ + N − e 0 \chi+N^{-e_0} χ + N − e 0 ν ♭ \nu^\flat ν ♭ x x x
A scalar parameter and its fiber direction. The scalar projection will use
R ( t ) = P ~ 2 − 2 t P ~ 1 , W ( t ) = ( 1 , 2 t ) . R(t)=\widetilde{P}_2-2t\widetilde{P}_1,\qquad W(t)=(1,2t). R ( t ) = P 2 − 2 t P 1 , W ( t ) = ( 1 , 2 t ) . At fixed time the fibers of R R R W ( t ) W(t) W ( t ) P ~ = ( 0 , R ) \widetilde{P}=(0,R) P = ( 0 , R ) ( t , R ) (t,R) ( t , R )
Suppose first that 2 k < 1 2k<1 2 k < 1 K K K
G 0 ( t ) = ( D 0 / A 0 ) ∇ P ( L s 0 − H 0 ) ⋅ W ( t ) G_0(t)=(D_0/A_0)\nabla_P(\mathcal{L}_{s_0}-H_0)\cdot W(t) G 0 ( t ) = ( D 0 / A 0 ) ∇ P ( L s 0 − H 0 ) ⋅ W ( t ) agrees with ( D 0 / A 0 ) ∇ P ( H b − H 0 ) ⋅ W ( t ) (D_0/A_0)\nabla_P(H_b-H_0)\cdot W(t) ( D 0 / A 0 ) ∇ P ( H b − H 0 ) ⋅ W ( t )
K ( D 0 / h s b + ( D 0 / A 0 ) a s b / h s b 2 ) ≤ N − κ K\left(D_0/h_{s_b}+(D_0/A_0)a_{s_b}/h_{s_b}^{2}\right)\leq N^{-\kappa} K ( D 0 / h s b + ( D 0 / A 0 ) a s b / h s b 2 ) ≤ N − κ with, for example, κ = s 0 4 min { k , 1 − 2 k } > 0 \kappa=\frac{s_0}{4}\min\{k,1-2k\}>0 κ = 4 s 0 min { k , 1 − 2 k } > 0 ∇ P L j ⋅ W = ∂ Z L j + t ∂ Y L j \nabla_P\mathcal{L}_j\cdot W=\partial_ZL_j+t\partial_YL_j ∇ P L j ⋅ W = ∂ Z L j + t ∂ Y L j L s b L_{s_b} L s b L s 0 L_{s_0} L s 0 K A 0 KA_0 K A 0 K A 0 / h s b KA_0/h_{s_b} K A 0 / h s b K A 0 / h s b 2 KA_0/h_{s_b}^{2} K A 0 / h s b 2 ( 1 , t ) (1,t) ( 1 , t ) ( 1 , t c ) (1,t_c) ( 1 , t c ) K h s b Kh_{s_b} K h s b H b H_b H b N − k s 0 / 2 N^{-ks_0/2} N − k s 0 /2 N − ( 1 − 2 k ) s 0 / 2 N^{-(1-2k)s_0/2} N − ( 1 − 2 k ) s 0 /2
We now choose one such quadratic time polynomial while retaining mass from all paired trajectories. In a typical chart let λ p a i r , H \lambda_{\mathrm{pair},H} λ pair , H ν H \nu_H ν H λ p a i r , H ≤ K ν H ⊗ d t \lambda_{\mathrm{pair},H}\leq K\nu_H\otimes dt λ pair , H ≤ K ν H ⊗ d t ( λ p a i r , H ) t (\lambda_{\mathrm{pair},H})_t ( λ pair , H ) t
m H ( t ) = d ( λ p a i r , H ) t d t , M H = ∫ 0 1 m H ( t ) d t ≥ 1 / K . m_H(t)=\frac{d(\lambda_{\mathrm{pair},H})_t}{dt},\qquad M_H=\int_0^1m_H(t)\,dt\geq1/K. m H ( t ) = d t d ( λ pair , H ) t , M H = ∫ 0 1 m H ( t ) d t ≥ 1/ K . Retain T 0 = { t : m H ( t ) ≥ M H / 2 } T_0=\{t:m_H(t)\geq M_H/2\} T 0 = { t : m H ( t ) ≥ M H /2 } M H / 2 M_H/2 M H /2 K ν H ⊗ d t K\nu_H\otimes dt K ν H ⊗ d t T b ⊂ T 0 T_b\subset T_0 T b ⊂ T 0 ∣ T b ∣ ≥ M H / ( 2 K ) |T_b|\geq M_H/(2K) ∣ T b ∣ ≥ M H / ( 2 K ) H b H_b H b
For this fixed label, ( D 0 / A 0 ) ∇ P ( H b − H 0 ) ⋅ W ( t ) (D_0/A_0)\nabla_P(H_b-H_0)\cdot W(t) ( D 0 / A 0 ) ∇ P ( H b − H 0 ) ⋅ W ( t ) G 0 ( t ) G_0(t) G 0 ( t ) K N − κ KN^{-\kappa} K N − κ T b T_b T b G 0 G_0 G 0 K K K s 0 s_0 s 0 G 0 G_0 G 0
λ p a i r , H { t ∈ T b } = ∫ T b m H ( t ) d t ≥ ( M H / 2 ) ∣ T b ∣ ≥ M H 2 / ( 4 K ) . \lambda_{\mathrm{pair},H}\{t\in T_b\}=\int_{T_b}m_H(t)\,dt\geq(M_H/2)|T_b|\geq M_H^2/(4K). λ pair , H { t ∈ T b } = ∫ T b m H ( t ) d t ≥ ( M H /2 ) ∣ T b ∣ ≥ M H 2 / ( 4 K ) . It is therefore dense. This argument uses the selected line to find the time polynomial; the retained mass is that of all paired lines on its time set.
Choose H c o r r ( t , P ~ ) H_{\mathrm{corr}}(t,\widetilde{P}) H corr ( t , P ) K K K W ( t ) W(t) W ( t ) b 0 + b 1 t + b 2 t 2 b_0+b_1t+b_2t^2 b 0 + b 1 t + b 2 t 2 ( b 0 + b 1 t ) P ~ 1 + ( b 2 t / 2 ) P ~ 2 (b_0+b_1t)\widetilde{P}_1+(b_2t/2)\widetilde{P}_2 ( b 0 + b 1 t ) P 1 + ( b 2 t /2 ) P 2 2 k > 1 2k>1 2 k > 1 H c o r r = 0 H_{\mathrm{corr}}=0 H corr = 0
Choosing all three depths with positive margins. Let C d = 10 / ( 1 − d ) C_d = 10/(1-d) C d = 10/ ( 1 − d ) C = C d + 3 C = C_d + 3 C = C d + 3 s 0 s_0 s 0 100 s 0 < e 0 100s_0 < e_0 100 s 0 < e 0 3 s 0 / 2 < s max 3s_0/2 < s_{\max} 3 s 0 /2 < s m a x
B ∗ = { min { s 0 / 2 , κ , k s 0 / ( 1 + k ) } , 2 k < 1 , min { s 0 / 2 , ( 2 k − 1 ) s 0 } , 2 k > 1. B_* =
\begin{cases}
\min\{s_0/2,\kappa,ks_0/(1+k)\}, & 2k< 1,\\
\min\{s_0/2,(2k-1)s_0\}, & 2k> 1.
\end{cases} B ∗ = { min { s 0 /2 , κ , k s 0 / ( 1 + k )} , min { s 0 /2 , ( 2 k − 1 ) s 0 } , 2 k < 1 , 2 k > 1. Choose
δ = B ∗ / 4 , s = s 0 + δ , u = min { δ / 4 , k δ / ( 4 C ) } , c 2 = ( 1 − d ) u / 8 , s i ∈ { s , s + u , s + c 2 } . \delta= B_*/4,\quad s = s_0 + \delta,\quad u = \min\{\delta/4,k\delta/(4C)\},\quad c_2 = (1-d)u/8,\quad s_i \in\{s,s+u,s+c_2\}. δ = B ∗ /4 , s = s 0 + δ , u = min { δ /4 , k δ / ( 4 C )} , c 2 = ( 1 − d ) u /8 , s i ∈ { s , s + u , s + c 2 } . All these are fixed positive exponents. Writing Δ i = s i − s 0 \Delta_i = s_i-s_0 Δ i = s i − s 0 0 < Δ i ≤ 5 B ∗ / 16 < B ∗ 0 < \Delta_i \le5B_*/16 < B_* 0 < Δ i ≤ 5 B ∗ /16 < B ∗ s i < 3 s 0 / 2 s_i < 3s_0/2 s i < 3 s 0 /2 D 0 / h s i = N − k ( s 0 − Δ i ) ≪ 1 D_0/h_{s_i} = N^{-k(s_0-\Delta_i)} \ll1 D 0 / h s i = N − k ( s 0 − Δ i ) ≪ 1 2 k < 1 2k< 1 2 k < 1
D 0 / h s i a s i / A 0 = N − [ k s 0 − ( k + 1 ) Δ i ] ≪ 1 , \frac{D_0/h_{s_i}}{a_{s_i}/A_0}
= N^{-[ks_0-(k+1)\Delta_i]} \ll1, a s i / A 0 D 0 / h s i = N − [ k s 0 − ( k + 1 ) Δ i ] ≪ 1 , N − κ a s i / A 0 = N − ( κ − Δ i ) ≪ 1. \frac{N^{-\kappa}}{a_{s_i}/A_0}
= N^{-(\kappa-\Delta_i)} \ll1. a s i / A 0 N − κ = N − ( κ − Δ i ) ≪ 1. For 2 k > 1 2k> 1 2 k > 1
D 0 a s i = N − [ ( 2 k − 1 ) s 0 − Δ i ] ≪ 1. \frac{D_0}{a_{s_i}}
= N^{-[(2k-1)s_0-\Delta_i]} \ll1. a s i D 0 = N − [( 2 k − 1 ) s 0 − Δ i ] ≪ 1. These inequalities supply all the directional error margins. They also give 0 < c 2 < ( 1 − d ) u / 4 0 < c_2 < (1-d)u/4 0 < c 2 < ( 1 − d ) u /4
∣ ∇ P ~ ( ( L s i − H 0 ) / A 0 − H c o r r ) ⋅ W ( t ) ∣ ≤ K a s i / A 0 . (9.5) \left|\nabla_{\widetilde{P}}\left((L_{s_i}-H_0)/A_0-H_{\mathrm{corr}}\right)\cdot W(t)\right| \le K a_{s_i}/A_0.\tag{9.5} ∇ P ( ( L s i − H 0 ) / A 0 − H corr ) ⋅ W ( t ) ≤ K a s i / A 0 . ( 9.5 ) For 2 k > 1 2k> 1 2 k > 1 (8.5) times D 0 D_0 D 0 2 k < 1 2k< 1 2 k < 1 L s i , L s 0 L_{s_i}, L_{s_0} L s i , L s 0 K D 0 / h s i KD_0/h_{s_i} K D 0 / h s i
The scalar projection: measure ceilings and support. Apply Theorem 8.2 to the scalar parameter R R R
x ∘ = ( x − H 0 ( t , P ) ) / A 0 − H c o r r ( t , P ~ ) , w i = a s i / A 0 . x^\circ= (x-H_0(t,P))/A_0-H_{\mathrm{corr}}(t,\widetilde{P}),\qquad w_i = a_{s_i}/A_0. x ∘ = ( x − H 0 ( t , P )) / A 0 − H corr ( t , P ) , w i = a s i / A 0 . Both signals are affine on trajectories with coefficients ≤ K \le K ≤ K w i w_i w i ( t , R ) (t,R) ( t , R ) σ R \sigma_R σ R 20 s 0 20s_0 20 s 0 P ~ \widetilde{P} P ( R , R ′ ) (R,R') ( R , R ′ ) 2 2 2 ( t , R ) (t,R) ( t , R ) x ∘ x^\circ x ∘ 1 ≥ ϱ ≥ w i 1 \ge\varrho\ge w_i 1 ≥ ϱ ≥ w i R ′ R' R ′ N − v 0 N^{-v_0} N − v 0 0 < v 0 ≤ s 0 0 < v_0 \le s_0 0 < v 0 ≤ s 0
K π ( d 0 ) σ R 2 D 0 2 ( A 0 ϱ ) d K\pi(d_0)\sigma_R^2D_0^2(A_0\varrho)^d K π ( d 0 ) σ R 2 D 0 2 ( A 0 ϱ ) d times N − v 0 N^{-v_0} N − v 0 P P P ( Z , Y ) (Z,Y) ( Z , Y ) K ( N − e 0 + χ ) K(N^{-e_0}+\chi) K ( N − e 0 + χ ) P ↦ ( p + t v d 0 , q 0 − t p ) P \mapsto(p+tv_{d_0},q_0-tp) P ↦ ( p + t v d 0 , q 0 − tp ) D 0 D_0 D 0 a s i a_{s_i} a s i ( t , Z , Y ) (t,Z,Y) ( t , Z , Y ) d 0 d_0 d 0 K D 0 2 σ R KD_0^2\sigma_R K D 0 2 σ R R ′ R' R ′ x x x K A 0 ϱ KA_0\varrho K A 0 ϱ H 0 + A 0 H c o r r H_0+A_0H_{\mathrm{corr}} H 0 + A 0 H corr K K K K K K ∼ log , N W G A 0 d \sim_{\log,N}W_GA_0^d ∼ l o g , N W G A 0 d
At each ( t , R ) (t,R) ( t , R ) Q s i Q_{s_i} Q s i K K K R ∗ R_* R ∗
Z r e f = p c + t v d 0 , Y r e f = q c + D 0 R ∗ − t p c . Z_{\mathrm{ref}}=p_c+tv_{d_0},\qquad Y_{\mathrm{ref}}=q_c+D_0R_*-tp_c. Z ref = p c + t v d 0 , Y ref = q c + D 0 R ∗ − t p c . At contact their spatial discrepancies are D 0 P ~ 1 ( 1 , t ) D_0\widetilde P_1(1,t) D 0 P 1 ( 1 , t ) K ( D 0 σ R + N − e 0 + χ ) K(D_0\sigma_R+N^{-e_0}+\chi) K ( D 0 σ R + N − e 0 + χ ) D 0 ≪ h s i 2 D_0\ll h_{s_i}^2 D 0 ≪ h s i 2 h s i 2 h_{s_i}^2 h s i 2 K h s i Kh_{s_i} K h s i ( 1 , t ) (1,t) ( 1 , t ) K h s i Kh_{s_i} K h s i K h s i 2 Kh_{s_i}^2 K h s i 2 K h s i Kh_{s_i} K h s i K h s i 4 Kh_{s_i}^4 K h s i 4
The common box bounds the number of contributing true labels, including their affine entries, by K w i − d Kw_i^{-d} K w i − d 2 k > 1 2k>1 2 k > 1 m Q s i / K m_{Q_{s_i}}/K m Q s i / K x x x 2 k < 1 2k<1 2 k < 1 L s 0 L_{s_0} L s 0 s 0 s_0 s 0 ≤ K A 0 d h s i 4 \le KA_0^d h_{s_i}^4 ≤ K A 0 d h s i 4 x x x
Each affine entry yields only K K K x ∘ x^\circ x ∘ w i w_i w i ( L s i − H 0 ) / A 0 − H c o r r (L_{s_i}-H_0)/A_0-H_{\mathrm{corr}} ( L s i − H 0 ) / A 0 − H corr K w i Kw_i K w i (8.1) and (8.5) . Moreover one can evaluate this expression at P ~ = ( 0 , R ( t ) ) \widetilde P=(0,R(t)) P = ( 0 , R ( t )) (9.5) and bounded P ~ 1 \widetilde P_1 P 1 (8.5) and an active position: the ordinary slopes of L s i L_{s_i} L s i K ( 1 + a s i / h s i 2 ) ≤ K N ( 2 k − 1 ) + s i K(1+a_{s_i}/h_{s_i}^2)\le KN^{(2k-1)+s_i} K ( 1 + a s i / h s i 2 ) ≤ K N ( 2 k − 1 ) + s i H 0 , H c o r r H_0,H_{\mathrm{corr}} H 0 , H corr K K K A 0 A_0 A 0 R R R K N 5 s 0 / 2 KN^{5s_0/2} K N 5 s 0 /2 s i < 3 s 0 / 2 s_i<3s_0/2 s i < 3 s 0 /2 0 < k ≤ 1 0<k\le1 0 < k ≤ 1 σ R \sigma_R σ R K N − 35 s 0 / 2 ≪ w i KN^{-35s_0/2}\ll w_i K N − 35 s 0 /2 ≪ w i w i ≥ N − 3 s 0 / 2 w_i\ge N^{-3s_0/2} w i ≥ N − 3 s 0 /2
D 0 σ R ≪ h s i 2 , N − e 0 + χ D 0 ≪ σ R . D_0\sigma_R\ll h_{s_i}^2,\qquad\frac{N^{-e_0}+\chi}{D_0}\ll\sigma_R. D 0 σ R ≪ h s i 2 , D 0 N − e 0 + χ ≪ σ R . when 100 s 0 < e 0 100s_0<e_0 100 s 0 < e 0 A 0 A_0 A 0 K σ R − 2 K\sigma_R^{-2} K σ R − 2 K w i − d Kw_i^{-d} K w i − d
Theorem 8.2, with its angular gain, gives disjoint full-time trajectory labels of active mass ≥ K − 1 w i d \ge K^{-1}w_i^d ≥ K − 1 w i d x ∘ x^\circ x ∘ K w i Kw_i K w i F i ( t , R ) F_i(t,R) F i ( t , R ) ≤ K \le K ≤ K K K K K K K d 0 d_0 d 0
Returning the predictors and their masses to true coordinates. Split p = Z ( 0 ) p=Z(0) p = Z ( 0 ) χ \chi χ α \alpha α p α p_\alpha p α P # = ( p α , Y + t p α ) P^\#=(p_\alpha,Y+tp_\alpha) P # = ( p α , Y + t p α ) P P P P ~ # \widetilde P^\# P # R # R^\# R #
H 0 + A 0 H c o r r + A 0 F i ( t , R ) . H_0+A_0H_{\mathrm{corr}}+A_0F_i(t,R). H 0 + A 0 H corr + A 0 F i ( t , R ) . The resulting predictor for x x x t , Y t,Y t , Y P # P^\# P # P ~ # \widetilde P^\# P # R # R^\# R # K K K K D 0 KD_0 K D 0 q c − t p α q_c-tp_\alpha q c − t p α K a s i Ka_{s_i} K a s i (8.1) holds throughout participating lines and ∥ P ~ # − P ~ ∥ ≤ K ( N − e 0 + χ ) / D 0 \lVert\widetilde{P}^{\#}-\widetilde{P}\rVert\le K(N^{-e_0}+\chi)/D_0 ∥ P # − P ∥ ≤ K ( N − e 0 + χ ) / D 0 Y Y Y D 0 D_0 D 0
Their typical sliced weights are ≥ K − 1 a s i d w α D 0 \ge K^{-1}a_{s_i}^{d}w_{\alpha}D_0 ≥ K − 1 a s i d w α D 0 w α = π ( d 0 ) χ w_{\alpha}=\pi(d_0)\chi w α = π ( d 0 ) χ π ( d 0 ) D 0 2 A d 0 d w i d / K \pi(d_0)D_0^2A_{d_0}^{d}w_i^d/K π ( d 0 ) D 0 2 A d 0 d w i d / K ≤ K D 0 / χ \le KD_0/\chi ≤ K D 0 / χ
T i = π ( d 0 ) D 0 2 A d 0 d w i d = π ( d 0 ) D 0 2 a s i d . T_i=\pi(d_0)D_0^2A_{d_0}^{d}w_i^d=\pi(d_0)D_0^2a_{s_i}^d. T i = π ( d 0 ) D 0 2 A d 0 d w i d = π ( d 0 ) D 0 2 a s i d . The numerator required for one slice is t i = π ( d 0 ) χ D 0 a s i d t_i=\pi(d_0)\chi D_0a_{s_i}^d t i = π ( d 0 ) χ D 0 a s i d K D 0 / χ KD_0/\chi K D 0 / χ K T i KT_i K T i T i T_i T i
Before the final common pruning, prepare the conditioning states needed for Proposition 8.4. Each state consists of a true Q s Q_s Q s x x x K K K e e e R e R_e R e K π ( b ) R e K\pi(b)R_e K π ( b ) R e Q s Q_s Q s K m Q s Km_{Q_s} K m Q s K ∑ Q s m Q s ≤ K K\sum_{Q_s}m_{Q_s}\le K K ∑ Q s m Q s ≤ K
Apply Lemma 9.1 to these entries and all three sliced projected layers together. This restores the projected floors while giving each retained history–state entry mass at least R e / K R_e/K R e / K K π K\pi K π
Width D 0 D_0 D 0 b s / D 0 ∼ log , N N − 2 k ( s − s 0 ) b_s/D_0\sim_{\log,N}N^{-2k(s-s_0)} b s / D 0 ∼ l o g , N N − 2 k ( s − s 0 ) D 0 ≪ h s = g s D_0\ll h_s=g_s D 0 ≪ h s = g s
( C d + 3 ) u ≤ k δ / 4 < min { 2 k ( s − s 0 ) , k s } . (C_d+3)u\le k\delta/4<\min\{2k(s-s_0),ks\}. ( C d + 3 ) u ≤ k δ /4 < min { 2 k ( s − s 0 ) , k s } . Consequently both wide inequalities hold with a fixed power of slack: b s / D 0 b_s/D_0 b s / D 0 2 k ( s − s 0 ) 2k(s-s_0) 2 k ( s − s 0 ) h s β max / β min h_s\beta_{\max}/\beta_{\min} h s β m a x / β m i n k s ks k s β i \beta_i β i D 0 D_0 D 0
We finish by checking the wide-interval input and returning its output to the initial tangent parent.
First, use H s H_s H s q s q_s q s v v v (8.3) . The common pruning just performed gives conditional velocity domination at the fine true-base and x x x Q s Q_s Q s
Second, each true label has only K K K L s L_s L s x x x K a s K a_s K a s Q s Q_s Q s K K K x x x (8.5) multiplied by h s h_s h s h s 2 h_s^2 h s 2 Φ \Phi Φ
Third, Proposition 8.4 gives a quadratic prediction on a true relative descendant Q s Q_s Q s H s ν ♭ ( Q s ) N − d c 2 / K H_s\nu^\flat(Q_s)N^{-dc_2}/K H s ν ♭ ( Q s ) N − d c 2 / K q d e s c q_{\mathrm{desc}} q desc ν d e s c \nu_{\mathrm{desc}} ν desc
q d e s c = q r H s , ν d e s c = ν Q r ν ♭ ( Q s ) . q_{\mathrm{desc}}=q_rH_s,\qquad\nu_{\mathrm{desc}}=\nu_{Q_r}\nu^\flat(Q_s). q desc = q r H s , ν desc = ν Q r ν ♭ ( Q s ) . The additional restrictions and bounded-trajectory normalizations already used affect comparisons of retained incidence mass by only K K K q r ν Q r q_r\nu_{Q_r} q r ν Q r
q r ν Q r H s ν ♭ ( Q s ) N − d c 2 K = q d e s c ν d e s c N − d c 2 K . \frac{q_r\nu_{Q_r}\,H_s\nu^\flat(Q_s)N^{-dc_2}}{K}=\frac{q_{\mathrm{desc}}\nu_{\mathrm{desc}}N^{-dc_2}}{K}. K q r ν Q r H s ν ♭ ( Q s ) N − d c 2 = K q desc ν desc N − d c 2 . Finally restore w = L + a r x w=L+a_rx w = L + a r x K a r a s + c 2 Ka_ra_{s+c_2} K a r a s + c 2
This is a candidate of the precise kind used in Proposition 4.11. Its mass is charged to the original residual and its fit is on the full assigned block. The argument works anew on each substantial residual, so the candidate-upgrade criterion applies.
Together Propositions 9.2 and 9.6 finish the tangent contradiction for 0 < ℓ < ∞ 0<\ell<\infty 0 < ℓ < ∞
Unbounded optimized narrowness and completion of the proof
The remaining case of the second model is excluded by the following proposition. Here ℓ \ell ℓ
Proposition 10.1 (Exclusion of unbounded optimized narrowness). In the second model’s extremal setup of Proposition 4.3, the global optimized narrowness ℓ \ell ℓ
Assume, for a contradiction, that ℓ = ∞ \ell=\infty ℓ = ∞ k k k
Throughout this section, K 0 = N 0 o ( 1 ) K_0=N^{o(1)}_0 K 0 = N 0 o ( 1 ) K = N o ( 1 ) K=N^{o(1)} K = N o ( 1 )
Nested true systems
Take a true intermediate system on a maximizing critical path, normalize through it as in Lemma 2.13, and rescale the remaining depth interval to length one. Denote the resulting maximizing sequence by E i E_i E i original coordinates means these coordinates of E i E_i E i
n i ( 0 ) ∼ log 1. n_i(0)\sim_{\log}1. n i ( 0 ) ∼ l o g 1. Consequently the relative-profile conclusion of Proposition 4.3 gives
f i ( r ) ⟶ d r uniformly for 0 ≤ r ≤ 1 , d i ⟶ d = d ∗ < 1 , H ( E i ) ⟶ k > 0. (10.1) f_i(r)\longrightarrow dr\quad\text{uniformly for }0\le r\le1,\qquad d_i\longrightarrow d=d_*<1,\qquad H(E_i)\longrightarrow k>0.\tag{10.1} f i ( r ) ⟶ d r uniformly for 0 ≤ r ≤ 1 , d i ⟶ d = d ∗ < 1 , H ( E i ) ⟶ k > 0. ( 10.1 ) Here n i ( r ) = N − f i ( r ) n_i(r)=N^{-f_i(r)} n i ( r ) = N − f i ( r ) E i ∈ C ( d i ) E_i\in C(d_i) E i ∈ C ( d i ) S , η > 0 S,\eta>0 S , η > 0
The original trajectories satisfy ∣ y ∣ + ∣ V ∣ ≤ K 0 |y| + |V| \le K_0 ∣ y ∣ + ∣ V ∣ ≤ K 0 X = P ∗ ( t , y , w ) X = P_*(t,y,w) X = P ∗ ( t , y , w ) B B B ( t , y ) (t,y) ( t , y )
∣ P ∗ ∣ ≤ K 0 , ∣ ( P ∗ ) w ∣ ≥ B / K 0 |P_*| \le K_0,\qquad|(P_*)_w| \ge B/K_0 ∣ P ∗ ∣ ≤ K 0 , ∣ ( P ∗ ) w ∣ ≥ B / K 0 on incidences require only K 0 K_0 K 0 1 / B 1/B 1/ B O ( K 0 ) O(K_0) O ( K 0 ) O ( K 0 2 / B ) O(K_0^2/B) O ( K 0 2 / B ) (10.1) , this gives a base-density ceiling K 0 K_0 K 0
Choose parameters in the order
0 < u < η , 0 < s 1 < s 2 < 1 , ( d + 2 k ) s 2 < S u 4 . (10.2) 0 < u < \eta,\qquad0 < s_1 < s_2 < 1,\qquad(d+2k)s_2 < \frac{Su}{4}.\tag{10.2} 0 < u < η , 0 < s 1 < s 2 < 1 , ( d + 2 k ) s 2 < 4 S u . ( 10.2 ) The segment construction in Proposition 4.3 supplies nested true nodes Q 1 , Q 2 Q_1,Q_2 Q 1 , Q 2 r j , i → s j r_{j,i} \to s_j r j , i → s j q j q_j q j
q j = N 0 − h j , i + o ( 1 ) , h j , i → k s j , g j = N 0 − α j , i , α 2 , i − α 1 , i → ∞ . (10.3) q_j = N_0^{-h_{j,i}+o(1)},\qquad h_{j,i} \to ks_j,\qquad g_j = N_0^{-\alpha_{j,i}},\qquad\alpha_{2,i} - \alpha_{1,i} \to\infty.\tag{10.3} q j = N 0 − h j , i + o ( 1 ) , h j , i → k s j , g j = N 0 − α j , i , α 2 , i − α 1 , i → ∞. ( 10.3 ) Here q j 2 g j q_j^2g_j q j 2 g j ℓ = ∞ \ell= \infty ℓ = ∞ k k k s 2 − s 1 > 0 s_2-s_1>0 s 2 − s 1 > 0
All labels include their required path histories. We retain their full-layer trajectory assignments, so that, within each original world γ \gamma γ
ν γ ( Q j ) ∼ log n i ( r j , i ) q j 2 g j . (10.4) \nu_\gamma(Q_j) \sim_{\log} n_i(r_{j,i})q_j^2g_j.\tag{10.4} ν γ ( Q j ) ∼ l o g n i ( r j , i ) q j 2 g j . ( 10.4 ) The subsequent path incidences are subsets of these assignments. Assignments at one node and in one time block are disjoint; an earlier floor is not asserted for every later restriction.
Both labels are known by p L = p 1 p_L=p_1 p L = p 1 q j q_j q j D j D_j D j y j ( t ) y_j(t) y j ( t ) n j n_j n j
∣ n j ⋅ ( y − y j ( t ) ) ∣ ≤ K 0 q j g j throughout the Q j -block , ∣ n j ⋅ ( V − y j ′ ) ∣ ≤ K 0 g j . (10.5) \begin{aligned}
|n_j \cdot(y-y_j(t))| &\le K_0q_jg_j &&\text{throughout the }Q_j\text{-block},\\
|n_j \cdot(V-y'_j)| &\le K_0g_j.
\end{aligned}\tag{10.5} ∣ n j ⋅ ( y − y j ( t )) ∣ ∣ n j ⋅ ( V − y j ′ ) ∣ ≤ K 0 q j g j ≤ K 0 g j . throughout the Q j -block , ( 10.5 ) The full center velocities are bounded by K 0 K_0 K 0
At a common path incidence,
∣ sin ∠ ( n 1 , n 2 ) ∣ ≤ K 0 g 1 . (10.6) |\sin\angle(n_1,n_2)| \le K_0g_1.\tag{10.6} ∣ sin ∠ ( n 1 , n 2 ) ∣ ≤ K 0 g 1 . ( 10.6 ) In fact D 2 = D 1 R D_2=D_1R D 2 = D 1 R ∥ R ∥ ≤ K 0 q 2 / q 1 \lVert R\rVert\le K_0q_2/q_1 ∥ R ∥ ≤ K 0 q 2 / q 1 ∥ n 1 T D 2 ∥ ≤ K 0 q 2 g 1 \lVert n_1^{\mathsf T}D_2\rVert\le K_0q_2g_1 ∥ n 1 T D 2 ∥ ≤ K 0 q 2 g 1 D 2 D_2 D 2 q 2 / K 0 q_2/K_0 q 2 / K 0 (10.6) .
Localization for the whole time interval
The true charts control each trajectory only on their own blocks. We first compare two occurrences of the later chart on the same trajectory. Their nearly parallel normal strips will determine one thin chart valid for the entire original time interval.
Proposition 10.2 (Full-time localization). In the setting of (10.2) and (10.3), a permitted further selection admits full-time charts C C C
g C = g 2 N 0 2 u . (10.7) g_C = g_2N_0^{2u}.\tag{10.7} g C = g 2 N 0 2 u . ( 10.7 ) Their labels are known by the original p 1 p_1 p 1 P ∗ P_* P ∗ n C n_C n C y C ( t ) y_C(t) y C ( t ) C C C Q 1 Q_1 Q 1
∣ n 1 − n C ∣ ≤ K 0 g 1 , ∣ n C ⋅ ( y 1 ′ − y C ′ ) ∣ ≤ K 0 g 1 . (10.8) \lvert n_1-n_C\rvert\le K_0g_1,\qquad\lvert n_C\cdot(y'_1-y'_C)\rvert\le K_0g_1.\tag{10.8} ∣ n 1 − n C ∣ ≤ K 0 g 1 , ∣ n C ⋅ ( y 1 ′ − y C ′ )∣ ≤ K 0 g 1 . ( 10.8 ) All assignments, geometric tests and coordinate changes have tempered bounds within each fixed E i E_i E i
Proof. Sample two independent times t 0 , t t_0,t t 0 , t p p p p 2 p^2 p 2 1 / K 0 1/K_0 1/ K 0
We first prove that, outside negligible pair mass,
∣ sin ∠ ( n 2 ( t 0 ) , n 2 ( t ) ) ∣ ≤ g 2 N 0 u . (10.9) \lvert\sin\angle(n_2(t_0),n_2(t))\rvert\le g_2N_0^u.\tag{10.9} ∣ sin ∠ ( n 2 ( t 0 ) , n 2 ( t ))∣ ≤ g 2 N 0 u . ( 10.9 ) Before counting labels, remove from the t t t u / 2 u/2 u /2
For fixed t 0 t_0 t 0 γ \gamma γ
K 0 n i ( r 2 , i ) q 2 2 g 2 (10.10) \frac{K_0}{n_i(r_{2,i})q_2^2g_2}\tag{10.10} n i ( r 2 , i ) q 2 2 g 2 K 0 ( 10.10 ) possible first labels Q 2 ( t 0 ) Q_2(t_0) Q 2 ( t 0 ) (10.9) fails, the two velocity strips in (10.5) intersect in a ball of radius at most K 0 N 0 − u K_0N_0^{-u} K 0 N 0 − u p 1 ( t ) p_1(t) p 1 ( t ) K 0 K_0 K 0 K 0 K_0 K 0 N 0 − u / 2 N_0^{-u/2} N 0 − u /2 K 0 N 0 − S u / 2 K_0N_0^{-Su/2} K 0 N 0 − S u /2 p 1 p_1 p 1
To integrate this denominator, sum only over its permitted good zero-depth ancestors. At each exact base only K 0 K_0 K 0 K 0 K_0 K 0 K 0 g 2 K_0g_2 K 0 g 2 t t t K 0 g 2 K_0g_2 K 0 g 2 (10.10) , and then integrating the times and weighting the worlds, bounds the failure mass by
K 0 N 0 − S u / 2 q 2 2 n i ( r 2 , i ) = N 0 − S u / 2 + ( d + 2 k ) s 2 + o i ( 1 ) + o N 0 ( 1 ) . (10.11) \frac{K_0N_0^{-Su/2}}{q_2^2n_i(r_{2,i})}
=
N_0^{-Su/2+(d+2k)s_2+o_i(1)+o_{N_0}(1)}.\tag{10.11} q 2 2 n i ( r 2 , i ) K 0 N 0 − S u /2 = N 0 − S u /2 + ( d + 2 k ) s 2 + o i ( 1 ) + o N 0 ( 1 ) . ( 10.11 ) This is power-small by (10.2) . The exceptional density entries were removed before the sum, so their mass is not multiplied by the number of first labels.
Fix t 0 t_0 t 0 t 0 t_0 t 0 Q 2 ( t 0 ) Q_2(t_0) Q 2 ( t 0 ) g C g_C g C Q 2 ( t 0 ) Q_2(t_0) Q 2 ( t 0 ) C C C
For each bin choose a representative unit normal and a representative affine normal motion. The transverse bound in (10.5) extrapolates to K 0 g 2 K_0g_2 K 0 g 2 K 0 g C K_0g_C K 0 g C K 0 K_0 K 0 g C g_C g C P ∗ P_* P ∗
We verify terminal knowledge of the whole label, including its center motion. A later Q 2 ( t ) Q_2(t) Q 2 ( t ) p 1 ( t ) p_1(t) p 1 ( t ) K 0 g 2 N 0 u = o ( g C ) K_0g_2N_0^u=o(g_C) K 0 g 2 N 0 u = o ( g C ) K 0 g 2 N 0 u K_0g_2N_0^u K 0 g 2 N 0 u (10.5) , and then account for the change of normal. Each later label thus determines boundedly many neighboring bins for all the C C C Q 2 ( t ) Q_2(t) Q 2 ( t ) C C C p 1 ( t ) p_1(t) p 1 ( t )
Since
g C g 1 = N 0 − ( α 2 , i − α 1 , i ) + 2 u , \frac{g_C}{g_1}=N_0^{-(\alpha_{2,i}-\alpha_{1,i})+2u}, g 1 g C = N 0 − ( α 2 , i − α 1 , i ) + 2 u , the binning error is much smaller than g 1 g_1 g 1 i i i (10.6) gives the first inequality in (10.8) . At a retained occurrence, compare both center velocities with the actual V V V Q 1 Q_1 Q 1 K 0 g 1 K_0g_1 K 0 g 1 C C C K 0 g C K_0g_C K 0 g C K 0 g 1 K_0g_1 K 0 g 1 V − y 1 ′ V-y'_1 V − y 1 ′
Slicing at a fixed real t 0 t_0 t 0 i i i E i E_i E i
Balanced localized packets
The full-time charts give a common thin direction, but their masses need not yet be comparable to their spatial areas. The next normalization produces this comparison in the two-limit sense needed for scalar projection. Its time cost tends to zero, leaving room for the fixed positive depth gain constructed below.
Lemma 10.3 (Normalization and relative density cancellation). After a further permitted selection and passage to subsequences, the charts C C C A A A p 1 p_1 p 1
q A = N 0 − h A , i + o ( 1 ) , h A , i ⟶ 0. q_A=N_0^{-h_{A,i}+o(1)},\qquad h_{A,i}\longrightarrow0. q A = N 0 − h A , i + o ( 1 ) , h A , i ⟶ 0. Write D A D_A D A J A = ∣ det D A ∣ J_A=|\det D_A| J A = ∣ det D A ∣
ν γ ( A ) J A = N 0 − d b i + ε i + o N 0 ( 1 ) = N 0 o i → ∞ ( 1 ) + o N 0 → ∞ ( 1 ) , b i ⟶ 0 , ε i ⟶ 0. (10.12) \frac{\nu_\gamma(A)}{J_A}
=N_0^{-db_i+\varepsilon_i+o_{N_0}(1)}
=N_0^{o_{i\to\infty}(1)+o_{N_0\to\infty}(1)},
\qquad b_i\longrightarrow0,\quad\varepsilon_i\longrightarrow0.\tag{10.12} J A ν γ ( A ) = N 0 − d b i + ε i + o N 0 ( 1 ) = N 0 o i → ∞ ( 1 ) + o N 0 → ∞ ( 1 ) , b i ⟶ 0 , ε i ⟶ 0. ( 10.12 ) The second equality is a two-limit statement. It does not assert an actual subpower bound for fixed i i i e Z e_Z e Z C C C
∥ D A T e Z ∥ ≥ q A / K 0 , ∥ D A T n C ∥ ≤ K 0 g C q A . (10.13) \lVert D_A^{\mathsf T}e_Z\rVert\ge q_A/K_0,\qquad
\lVert D_A^{\mathsf T}n_C\rVert\le K_0g_Cq_A.\tag{10.13} ∥ D A T e Z ∥ ≥ q A / K 0 , ∥ D A T n C ∥ ≤ K 0 g C q A . ( 10.13 ) Actual lists, native derivative bounds and geometric slacks remain subpower within each fixed exact problem. Proof. Normalize spatially inside C C C P ∗ P_\ast P ∗ ( γ , C ) (\gamma,C) ( γ , C ) ν γ ( ⋅ ∣ C ) \nu_\gamma(\cdot\mid C) ν γ ( ⋅ ∣ C )
W = ∑ γ , C ω γ ν γ ( C ) . W=\sum_{\gamma,C}\omega_\gamma\nu_\gamma(C). W = γ , C ∑ ω γ ν γ ( C ) . Then K 0 − 1 ≤ W ≤ 1 K_0^{-1}\le W\le1 K 0 − 1 ≤ W ≤ 1 ω γ ν γ ( C ) / W \omega_\gamma\nu_\gamma(C)/W ω γ ν γ ( C ) / W W W W
n i ′ ( 1 ) ∼ log n i ( 1 ) g C ν γ ( C ) . (10.14) n'_i(1)\sim_{\log} n_i(1)\frac{g_C}{\nu_\gamma(C)}.\tag{10.14} n i ′ ( 1 ) ∼ l o g n i ( 1 ) ν γ ( C ) g C . ( 10.14 ) At intermediate depths only the one-sided comparison of Lemma 2.13 is needed. To see the cap inheritance explicitly, homogenize g C / ν γ ( C ) = N 0 t i + o ( 1 ) g_C/\nu_\gamma(C)=N_0^{t_i+o(1)} g C / ν γ ( C ) = N 0 t i + o ( 1 ) f i ′ ( 1 ) = f i ( 1 ) − t i f'_i(1)=f_i(1)-t_i f i ′ ( 1 ) = f i ( 1 ) − t i f i ′ ( r ) ≥ f i ( r ) − t i f'_i(r)\ge f_i(r)-t_i f i ′ ( r ) ≥ f i ( r ) − t i
f i ′ ( 1 ) − f i ′ ( r ) ≤ f i ( 1 ) − f i ( r ) ≤ d i ( 1 − r ) . f'_i(1)-f'_i(r)\le f_i(1)-f_i(r)\le d_i(1-r). f i ′ ( 1 ) − f i ′ ( r ) ≤ f i ( 1 ) − f i ( r ) ≤ d i ( 1 − r ) . A primed velocity bin maps into an old velocity ball of its radius times K 0 K_0 K 0 C C C K 0 K_0 K 0 E i ′ E'_i E i ′ C ( d i ) C(d_i) C ( d i ) E i ′ E'_i E i ′ C C C
H ( E i ) ≤ H ( E i ′ ) ≤ k ( d i ) . H(E_i)\le H(E'_i)\le k(d_i). H ( E i ) ≤ H ( E i ′ ) ≤ k ( d i ) . Since the outer sequence is maximizing, both endpoints tend to k k k E i ′ E'_i E i ′ f i ′ ( 0 ) f'_i(0) f i ′ ( 0 )
Choose small positive depths b i → 0 b_i\to0 b i → 0 b = 1 / m b=1/m b = 1/ m i i i b , 1 b,1 b , 1 1 / m 1/m 1/ m b b b b + 1 / m b+1/m b + 1/ m H ≤ L H\le L H ≤ L m = m ( i ) m=m(i) m = m ( i )
f i ′ ( 1 ) − f i ′ ( b i ) = d ( 1 − b i ) + o i ( 1 ) , (10.15) f'_i(1)-f'_i(b_i)=d(1-b_i)+o_i(1),\tag{10.15} f i ′ ( 1 ) − f i ′ ( b i ) = d ( 1 − b i ) + o i ( 1 ) , ( 10.15 ) and the packet horizon tends to zero. Saturate those primed true packets and compose them with C C C C C C A A A
The primed spatial determinant of A A A J A / g C J_A/g_C J A / g C
ν γ ( A ) ν γ ( C ) ∼ log n i ′ ( b i ) J A g C . \frac{\nu_\gamma(A)}{\nu_\gamma(C)}\sim_{\log} n'_i(b_i)\frac{J_A}{g_C}. ν γ ( C ) ν γ ( A ) ∼ l o g n i ′ ( b i ) g C J A . Combining with Equation (10.14) yields
ν γ ( A ) J A ∼ log n i ( 1 ) n i ′ ( b i ) n i ′ ( 1 ) . (10.16) \frac{\nu_\gamma(A)}{J_A}\sim_{\log} n_i(1)\frac{n'_i(b_i)}{n'_i(1)}.\tag{10.16} J A ν γ ( A ) ∼ l o g n i ( 1 ) n i ′ ( 1 ) n i ′ ( b i ) . ( 10.16 ) The logarithmic exponent of the right-hand side is
− f i ( 1 ) + f i ′ ( 1 ) − f i ′ ( b i ) = − d b i + o i ( 1 ) , -f_i(1)+f'_i(1)-f'_i(b_i)=-d b_i+o_i(1), − f i ( 1 ) + f i ′ ( 1 ) − f i ′ ( b i ) = − d b i + o i ( 1 ) , by Equations (10.1) and (10.15). This proves (10.12) . In particular, an arbitrarily large absolute primed density shift cancels in the quotient in (10.16) .
We record the mass bookkeeping for subsequent restrictions. At the C C C ν γ ( C ) \nu_\gamma(C) ν γ ( C ) W − 1 W^{-1} W − 1 A A A
∑ γ , I , A ω γ q A ν γ ( A ) ≤ 1. (10.17) \sum_{\gamma,I,A}\omega_\gamma q_A\nu_\gamma(A)\le1.\tag{10.17} γ , I , A ∑ ω γ q A ν γ ( A ) ≤ 1. ( 10.17 ) Here the sum includes the time blocks, and the inequality follows from disjointness within each block. Every later homogeneous selection has subpower total success in its fixed exact problem. An original exception of fixed power-small mass is therefore still negligible relative to that success. Worlds or labels with too small a conditional success fraction may be discarded by summing (10.17) . We never require an exception bound uniformly after conditioning on every individual tiny C C C A A A
The label knowledge is also actual. At p 1 p_1 p 1 C C C C C C B B B N 0 − 1 / B N_0^{-1}/B N 0 − 1 / B b i b_i b i L C , L A ′ L_C,L_{A'} L C , L A ′ A A A 2 L C L A ′ = N 0 o ( 1 ) 2L_CL_{A'}=N_0^{o(1)} 2 L C L A ′ = N 0 o ( 1 ) i i i (10.14) .
Composition gives the second inequality of (10.13) and also an original largest-axis upper bound K 0 q A K_0q_A K 0 q A N 0 − ε N_0^{-\varepsilon} N 0 − ε A A A K 0 ( N 0 − ε + g C ) K_0(N_0^{-\varepsilon}+g_C) K 0 ( N 0 − ε + g C ) v > 0 v>0 v > 0 ε \varepsilon ε g C g_C g C A A A p 1 p_1 p 1 K 0 K_0 K 0 K 0 N 0 − S v K_0N_0^{-Sv} K 0 N 0 − S v (10.13) . □
The depth gain and the diagonal
Retain the final tempered path in original worlds as the reference incidence measure μ r e f \mu_{\mathrm{ref}} μ ref μ r e f , γ \mu_{\mathrm{ref},\gamma} μ ref , γ ω γ \omega_\gamma ω γ
μ r e f = ∑ γ ω γ μ r e f , γ , ∥ μ r e f ∥ = P p a t h ≥ K 0 − 1 . \mu_{\mathrm{ref}}=\sum_\gamma\omega_\gamma\mu_{\mathrm{ref},\gamma},\qquad\lVert\mu_{\mathrm{ref}}\rVert=\mathbb{P}_{\mathrm{path}}\ge K_0^{-1}. μ ref = γ ∑ ω γ μ ref , γ , ∥ μ ref ∥ = P path ≥ K 0 − 1 . Here ∥ μ ∥ \lVert\mu\rVert ∥ μ ∥ γ \gamma γ
We fix the depth gain and the losses allowed in coefficient matching before selecting diagonal samples. Fix d < d 0 < 1 d<d_0<1 d < d 0 < 1 ε ∗ \varepsilon_* ε ∗ d 0 d_0 d 0
0 < θ < min { 1 , ε ∗ / 4 } , 2 θ < ε < ε ∗ . 0<\theta<\min\{1,\varepsilon_*/4\},\qquad2\theta<\varepsilon<\varepsilon_*. 0 < θ < min { 1 , ε ∗ /4 } , 2 θ < ε < ε ∗ . Let C m a t c h ≥ 1 C_{\mathrm{match}}\ge1 C match ≥ 1
0 < δ < min { θ s 1 4 , ε s 1 2 C m a t c h } . 0<\delta<\min\left\{\frac{\theta s_1}{4},\frac{\varepsilon s_1}{2C_{\mathrm{match}}}\right\}. 0 < δ < min { 4 θ s 1 , 2 C match ε s 1 } . and put
c ′ = θ s 1 2 ∈ ( 0 , 1 ) . (10.18) c' = \frac{\theta s_1}{2} \in(0,1).\tag{10.18} c ′ = 2 θ s 1 ∈ ( 0 , 1 ) . ( 10.18 ) For sufficiently large i i i E i E_i E i c ′ c' c ′ q A q_A q A h A , i < k c ′ / 2 h_{A,i}<kc'/2 h A , i < k c ′ /2
H ( E i ) ≤ k c ′ 2 + k ( d i ) ( 1 − c ′ ) . (10.19) H(E_i) \le\frac{kc'}{2}+k(d_i)(1-c').\tag{10.19} H ( E i ) ≤ 2 k c ′ + k ( d i ) ( 1 − c ′ ) . ( 10.19 ) The right-hand side tends to k − k c ′ / 2 < k k-kc'/2<k k − k c ′ /2 < k (10.1) .
Use the finite-sample exclusion argument of Lemma 4.6 in the following precise form. Within fixed i i i K p a t h log N 0 K_{\mathrm{path}}\log N_0 K path log N 0 K p a t h K_{\mathrm{path}} K path
p p a t h / log N 0 . (10.20) \mathfrak{p}_{\mathrm{path}}/\log N_0.\tag{10.20} p path / log N 0 . ( 10.20 ) If infinitely many samples in this fixed exact problem admitted such outputs, their bounded complexity and actual subpower controls would supply an exact known atlas on a subsequence, contradicting (10.19) . Thus one may sample N 0 N_0 N 0
We take a diagonal N = N 0 → ∞ N=N_0\to\infty N = N 0 → ∞ i → ∞ i\to\infty i → ∞ (10.12) are absorbed in K = N o ( 1 ) K=N^{o(1)} K = N o ( 1 ) u , s 1 , s 2 , θ , δ u,s_1,s_2,\theta,\delta u , s 1 , s 2 , θ , δ K K K
Prepare analytical caps before sparse searches. Remove atypical pure densities at depths through s 1 s_1 s 1 0 , r 1 , i 0,r_{1,i} 0 , r 1 , i p 1 p_1 p 1 o ( p p a t h ) o(\mathfrak{p}_{\mathrm{path}}) o ( p path ) i i i s 1 s_1 s 1 K N − r d KN^{-r d} K N − r d r 1 , i r_{1,i} r 1 , i K n i ( r 1 , i ) K n_i(r_{1,i}) K n i ( r 1 , i ) μ p r e p , γ ≤ μ r e f , γ \mu_{\mathrm{prep},\gamma}\le\mu_{\mathrm{ref},\gamma} μ prep , γ ≤ μ ref , γ
μ p r e p = ∑ γ ω γ μ p r e p , γ . \mu_{\mathrm{prep}}=\sum_{\gamma}\omega_\gamma\mu_{\mathrm{prep},\gamma}. μ prep = γ ∑ ω γ μ prep , γ . The next lemma makes the remaining light-label and light-entry deletions and then fixes μ p r e p \mu_{\mathrm{prep}} μ prep
Stable graphs on the earlier charts
The scalar projection keeps time and the tangential coordinate e Z ⋅ y e_Z\cdot y e Z ⋅ y X X X Q 1 Q_1 Q 1 A A A
Lemma 10.4 (Graph descriptions on the earlier charts). After analytical deletions of mass o ( p p a t h ) o(\mathfrak p_{\mathrm{path}}) o ( p path ) each Q 1 Q_1 Q 1 label still meeting μ p r e p \mu_{\mathrm{prep}} μ prep has the following properties.
It has a frozen earlier witness W γ , I , Q 1 W_{\gamma,I,Q_1} W γ , I , Q 1 q 1 ν γ ( Q 1 ) / K q_1\nu_\gamma(Q_1)/K q 1 ν γ ( Q 1 ) / K
In its normalized base coordinates u ^ = ( ( t − t 1 ) / q 1 , D − 1 ( y − y 1 ( t ) ) ) \hat u=((t-t_1)/q_1,D^{-1}(y-y_1(t))) u ^ = (( t − t 1 ) / q 1 , D − 1 ( y − y 1 ( t ))) t 1 t_1 t 1 K K K ( U , G ) (U,G) ( U , G ) U U U G G G U U U u ^ ∈ U \hat u\in U u ^ ∈ U
∣ X − G ( u ^ ) ∣ ≤ K a , ∣ G ∣ + ∣ ∇ G ∣ ≤ K , a = N − s 1 , |X-G(\hat u)|\le K a,\qquad|G|+|\nabla G|\le K,\qquad a=N^{-s_1}, ∣ X − G ( u ^ ) ∣ ≤ K a , ∣ G ∣ + ∣∇ G ∣ ≤ K , a = N − s 1 , where the value and derivative bounds for G G G U U U
The final prepared measure has mass ∥ μ p r e p ∥ = ( 1 − o ( 1 ) ) p p a t h \|\mu_{\mathrm{prep}}\|=(1-o(1))\mathfrak p_{\mathrm{path}} ∥ μ prep ∥ = ( 1 − o ( 1 )) p path
Proof. Delete Q 1 Q_1 Q 1 μ p r e p , γ \mu_{\mathrm{prep},\gamma} μ prep , γ
q 1 ν γ ( Q 1 ) / K . (10.21) q_1\nu_\gamma(Q_1)/K.\tag{10.21} q 1 ν γ ( Q 1 ) / K . ( 10.21 ) Choose the reciprocal-subpower threshold small enough relative to the reference mass. The loss is negligible because
∑ γ , I , Q 1 ω γ q 1 ν γ ( Q 1 ) ≤ 1. \sum_{\gamma,I,Q_1}\omega_\gamma q_1\nu_\gamma(Q_1)\le1. γ , I , Q 1 ∑ ω γ q 1 ν γ ( Q 1 ) ≤ 1. Write W γ , I , Q 1 W_{\gamma,I,Q_1} W γ , I , Q 1 Q 1 Q_1 Q 1 μ p r e p \mu_{\mathrm{prep}} μ prep ν γ ( Q 1 ) \nu_\gamma(Q_1) ν γ ( Q 1 ) (10.21) is the original full-layer prior in (10.4) . The floor is global within Q 1 Q_1 Q 1 A A A
We next describe the scalar signal by stable graphs in the normalized coordinates of each surviving earlier chart. Write z = B w z=Bw z = B w P ( u ^ , z ) P(\hat u,z) P ( u ^ , z ) Q 1 Q_1 Q 1 a 1 f i t = N − r 1 , i ∼ log , N a a^{\mathrm{fit}}_1=N^{-r_{1,i}}\sim_{\log,N}a a 1 fit = N − r 1 , i ∼ l o g , N a
∣ P ∣ ≤ K 0 ( a 1 f i t ) , K 0 − 1 ≤ ∣ P z ∣ ≤ K 0 , ∣ P z z ∣ ≤ K 0 / a 1 f i t . (10.22) |P|\le K_0(a^{\mathrm{fit}}_1),\qquad K_0^{-1}\le|P_z|\le K_0,\qquad|P_{zz}|\le K_0/a^{\mathrm{fit}}_1.\tag{10.22} ∣ P ∣ ≤ K 0 ( a 1 fit ) , K 0 − 1 ≤ ∣ P z ∣ ≤ K 0 , ∣ P z z ∣ ≤ K 0 / a 1 fit . ( 10.22 ) The depth-r 1 , i r_{1,i} r 1 , i K 0 K_0 K 0 K q 1 ν γ ( Q 1 ) Kq_1\nu_\gamma(Q_1) K q 1 ν γ ( Q 1 ) u ^ \hat u u ^ (10.21) therefore implies a real base projection of volume at least 1 / K 1/K 1/ K K 0 K_0 K 0
The discriminant
D ( u ^ ) = P z 2 − 2 P z z P \mathcal{D}(\hat u)=P_z^2-2P_{zz}P D ( u ^ ) = P z 2 − 2 P z z P is independent of z z z u ^ \hat u u ^ (10.22) . Polynomial Remez and fixed-degree coefficient bounds consequently bound D \mathcal{D} D K K K
Choose a sufficiently large subpower cutoff K ′ K' K ′ ∣ P z z ∣ ≥ ( a 1 f i t K ′ ) − 1 |P_{zz}|\ge(a^{\mathrm{fit}}_1K')^{-1} ∣ P z z ∣ ≥ ( a 1 fit K ′ ) − 1
z c ( u ^ ) = − coefficient of z in P P z z z_c(\hat u)=-\frac{\text{coefficient of }z\text{ in }P}{P_{zz}} z c ( u ^ ) = − P z z coefficient of z in P approximates the observed z z z K a 1 f i t K a^{\mathrm{fit}}_1 K a 1 fit P z = P z z ( z − z c ) P_z=P_{zz}(z-z_c) P z = P z z ( z − z c ) K ′ K' K ′ 2 ∣ P z z P ∣ ≪ ∣ P z ∣ 2 2|P_{zz}P|\ll|P_z|^2 2∣ P z z P ∣ ≪ ∣ P z ∣ 2 1 / K 1/K 1/ K D \mathcal{D} D z c z_c z c z z z K a 1 f i t K a_1^{\mathrm{fit}} K a 1 fit (10.22) and the preceding center estimate.
There are K K K q 1 ν γ ( Q 1 ) q_1\nu_\gamma(Q_1) q 1 ν γ ( Q 1 ) 1 / K 1/K 1/ K P ∗ P_* P ∗ w = z / B w=z/B w = z / B K a 1 f i t K a_1^{\mathrm{fit}} K a 1 fit X X X K K K ∣ X ∣ ≤ K 0 |X|\le K_0 ∣ X ∣ ≤ K 0 K K K
The real cores lie strictly inside those interiors. Boundary points may be assigned to a neighboring core, or to a bounded overlapping cover, without changing the K K K
The core/branch norm bounds use the incidence witnesses retained after the second deletion. The earlier witnesses W γ , I , Q 1 W_{\gamma,I,Q_1} W γ , I , Q 1 μ p r e p \mu_{\mathrm{prep}} μ prep
The scalar projection: caps and support
Scalar mesh projection requires two distinct estimates: a mass ceiling for each scalar bin and a bound on the number of occupied bins. A mass ceiling alone does not bound that number. The graph description just prepared will supply the support bound: we count the earlier charts compatible with a projected cell, and then the scalar bins met by their graphs.
Fix a world γ \gamma γ A A A I I I τ ∈ [ 0 , 1 ] \tau\in[0,1] τ ∈ [ 0 , 1 ] A A A
Z = e Z ⋅ y , ζ = Z − Z ∗ q A , (10.23) Z=e_Z\cdot y,\qquad\zeta=\frac{Z-Z_*}{q_A},\tag{10.23} Z = e Z ⋅ y , ζ = q A Z − Z ∗ , ( 10.23 ) where Z ∗ Z_* Z ∗ ζ \zeta ζ d ζ / d τ = e Z ⋅ V \mathrm{d}\zeta/\mathrm{d}\tau=e_Z\cdot V d ζ / d τ = e Z ⋅ V X = P ∗ X=P_* X = P ∗ T A T_A T A
λ A = ( T A ) ∗ ( μ p r e p , γ ∣ I , A ) q A B γ ( A ) , R A = ω γ q A ν γ ( A ) . (10.24) \lambda_A=\frac{(T_A)_*(\mu_{\mathrm{prep},\gamma\mid I,A})}{q_A B_\gamma(A)},\qquad R_A=\omega_\gamma q_A\nu_\gamma(A).\tag{10.24} λ A = q A B γ ( A ) ( T A ) ∗ ( μ prep , γ ∣ I , A ) , R A = ω γ q A ν γ ( A ) . ( 10.24 ) The denominator is the fixed prior-times-block budget. Thus λ A \lambda_A λ A ν γ ( ⋅ ∣ A ) \nu_\gamma(\,\cdot\mid A) ν γ ( ⋅ ∣ A ) d τ \mathrm{d}\tau d τ
∥ μ p r e p ∥ = ∑ A R A ∥ λ A ∥ , ∑ A R A ≤ 1. (10.25) \lVert\mu_{\mathrm{prep}}\rVert=\sum_A R_A\lVert\lambda_A\rVert,\qquad\sum_A R_A\le1.\tag{10.25} ∥ μ prep ∥ = A ∑ R A ∥ λ A ∥ , A ∑ R A ≤ 1. ( 10.25 ) The projected law means the pushforward of λ A \lambda_A λ A ( τ , ζ , X , ζ ′ ) (\tau,\zeta,X,\zeta') ( τ , ζ , X , ζ ′ )
Lemma 10.5 (Projected cap and support bounds ). Put a = N − s 1 a=N^{-s_1} a = N − s 1 λ A \lambda_A λ A every used A A A
An X X X p = N − r p=N^{-r} p = N − r 0 ≤ r ≤ s 1 0\le r\le s_1 0 ≤ r ≤ s 1 K p d Kp^d K p d ( τ , ζ ) (\tau,\zeta) ( τ , ζ ) p = a p=a p = a ζ ′ \zeta' ζ ′ N − v N^{-v} N − v 0 < v ≤ u 1 0<v\le u_1 0 < v ≤ u 1 K N − S v KN^{-S v} K N − S v 0 < u 1 < u 0<u_1<u 0 < u 1 < u
For a fixed sufficiently large M M M σ ≍ N − M \sigma\asymp N^{-M} σ ≍ N − M ( τ , ζ ) (\tau,\zeta) ( τ , ζ ) σ \sigma σ K a − d Ka^{-d} K a − d X X X a a a ( σ , σ , a ) (\sigma,\sigma,a) ( σ , σ , a ) K a − d σ − 2 Ka^{-d}\sigma^{-2} K a − d σ − 2
Proof. Projected densities. At fixed original time, an interval of length b b b ζ \zeta ζ K 0 J A b K_0J_A b K 0 J A b A A A A A A ∥ D A T e Z ∥ / q A ≥ 1 / K 0 \lVert D_A^{\mathsf T}e_Z\rVert/q_A\ge1/K_0 ∥ D A T e Z ∥ / q A ≥ 1/ K 0 (10.13) . Thus this estimate is also a density bound for the area pushforward.
At each exact original base, an X X X p p p K 0 K_0 K 0 p / B p/B p / B A A A d t = q A d τ dt=q_A\,d\tau d t = q A d τ q A ν γ ( A ) q_A\nu_\gamma(A) q A ν γ ( A )
K p d J A ν γ ( A ) ≤ K p d , Kp^d\frac{J_A}{\nu_\gamma(A)}\le Kp^d, K p d ν γ ( A ) J A ≤ K p d , where the last K K K (10.12) only on the chosen diagonal.
For the angular refinement, the normal component of original V V V C C C K 0 g C K_0g_C K 0 g C ζ ′ \zeta' ζ ′ N − v N^{-v} N − v V V V K 0 K_0 K 0 0 < v ≤ u 1 < u 0<v\le u_1<u 0 < v ≤ u 1 < u i i i K N − S v KN^{-Sv} K N − S v s 1 s_1 s 1
Counting whole compatible domains. Fix A A A Q 1 Q_1 Q 1 q A σ ≪ q 1 q_A\sigma\ll q_1 q A σ ≪ q 1 Q 1 Q_1 Q 1 K 0 q 1 2 g 1 K_0q_1^2g_1 K 0 q 1 2 g 1
At a compatible occurrence, its center has tangential distance at most K 0 q 1 K_0q_1 K 0 q 1 Z Z Z C C C K 0 q 1 g 1 K_0q_1g_1 K 0 q 1 g 1 (10.8) , the true position bounds, and g C ≪ q 1 g 1 g_C\ll q_1g_1 g C ≪ q 1 g 1 K 0 g 1 K_0g_1 K 0 g 1 (10.8) . The tangential center difference changes at rate at most K 0 K_0 K 0 Q 1 Q_1 Q 1
The frames obey the same constant angle comparison throughout that block. Projecting a major axis of length K 0 q 1 K_0q_1 K 0 q 1 n C n_C n C K 0 q 1 g 1 K_0q_1g_1 K 0 q 1 g 1 whole domain of each compatible label, not only its intersection with A A A W γ , I , Q 1 W_{\gamma,I,Q_1} W γ , I , Q 1 (10.21) , can consequently be charged there.
Those witnesses are disjoint in the original block and obey the original zero-depth base cap. If there are L 1 L_1 L 1 (10.4) and (10.21) give
L 1 K q 1 n i ( r 1 , i ) q 1 2 g 1 2 ≤ K q 1 3 g 1 . \frac{L_1}{K}q_1n_i(r_{1,i})q_1^2g_1^2 \le Kq_1^3g_1. K L 1 q 1 n i ( r 1 , i ) q 1 2 g 1 2 ≤ K q 1 3 g 1 . It follows that
L 1 ≤ K n i ( r 1 , i ) ≤ K N d s 1 . (10.26) L_1 \le\frac{K}{n_i(r_{1,i})} \le KN^{d s_1}.\tag{10.26} L 1 ≤ n i ( r 1 , i ) K ≤ K N d s 1 . ( 10.26 ) This is a separate bound for the fixed A A A C C C A A A
Coordinate precision without an absolute-width loss. Fix one compatible Q 1 Q_1 Q 1 u ^ \hat{u} u ^ Δ t = t ′ − t \Delta t=t'-t Δ t = t ′ − t Δ y = y ′ − y \Delta y=y'-y Δ y = y ′ − y
∣ Δ t ∣ + ∣ Δ Z ∣ ≤ K 0 q A σ . |\Delta t|+|\Delta Z|\le K_0q_A\sigma. ∣Δ t ∣ + ∣Δ Z ∣ ≤ K 0 q A σ . Since both are in the same C C C
n C ⋅ Δ y = n C ⋅ y C ′ Δ t + O ( K 0 g C ) . n_C\cdot\Delta y=n_C\cdot y'_C\Delta t+O(K_0g_C). n C ⋅ Δ y = n C ⋅ y C ′ Δ t + O ( K 0 g C ) . Decompose n 1 = c 1 n C + c 2 e Z n_1=c_1n_C+c_2e_Z n 1 = c 1 n C + c 2 e Z ∣ c 2 ∣ ≤ K 0 g 1 |c_2|\le K_0g_1 ∣ c 2 ∣ ≤ K 0 g 1 Q 1 Q_1 Q 1 (10.8) gives
∣ n 1 ⋅ ( Δ y − y 1 ′ Δ t ) ∣ ≤ K 0 ( g 1 q A σ + g C ) , (10.27) |n_1\cdot(\Delta y-y'_1\Delta t)|\le K_0(g_1q_A\sigma+g_C),\tag{10.27} ∣ n 1 ⋅ ( Δ y − y 1 ′ Δ t ) ∣ ≤ K 0 ( g 1 q A σ + g C ) , ( 10.27 ) ∣ major component of Δ y − y 1 ′ Δ t ∣ ≤ K 0 ( q A σ + g C ) . |\text{major component of }\Delta y-y'_1\Delta t|\le K_0(q_A\sigma+g_C). ∣ major component of Δ y − y 1 ′ Δ t ∣ ≤ K 0 ( q A σ + g C ) . Divide by the respective widths q 1 g 1 q_1g_1 q 1 g 1 q 1 q_1 q 1
∣ Δ u ^ ∣ ≤ K 0 ( q A σ q 1 + g C q 1 g 1 ) . (10.28) |\Delta\hat{u}|\le K_0\left(\frac{q_A\sigma}{q_1}+\frac{g_C}{q_1g_1}\right).\tag{10.28} ∣Δ u ^ ∣ ≤ K 0 ( q 1 q A σ + q 1 g 1 g C ) . ( 10.28 ) The factor g 1 g_1 g 1 (10.27) cancels before the narrow width is inverted.
For clarity, write Δ i = α 2 , i − α 1 , i \Delta_i=\alpha_{2,i}-\alpha_{1,i} Δ i = α 2 , i − α 1 , i (10.28) , including subpower factors, are bounded on the diagonal by
N − M + k s 1 + o ( 1 ) + N − Δ i + 2 u + k s 1 + o ( 1 ) . N^{-M+ks_1+o(1)}+N^{-\Delta_i+2u+ks_1+o(1)}. N − M + k s 1 + o ( 1 ) + N − Δ i + 2 u + k s 1 + o ( 1 ) . Choose M > 1 + s 1 + k s 1 M>1+s_1+ks_1 M > 1 + s 1 + k s 1 Δ i → ∞ \Delta_i\to\infty Δ i → ∞ a = N − s 1 a=N^{-s_1} a = N − s 1 g 1 g_1 g 1
Combine Equation (10.28) with the derivative and approximation bounds in Lemma 10.4. Points of one projected cell using one graph vary in X X X K a Ka K a K X a KX_a K X a K K K (10.26) to prove the cellwise support bound. Finally the ( τ , ζ ) (\tau,\zeta) ( τ , ζ ) K K K K a − d σ − 2 Ka^{-d}\sigma^{-2} K a − d σ − 2 λ A \lambda_A λ A
Scalar candidates and an actual improvement
We now construct a known atlas at the fixed positive depth c ′ c' c ′ (10.18) , with time length q A q_A q A
Proof of Proposition 10.1. We use the fixed prepared measure μ p r e p \mu_{\mathrm{prep}} μ prep A A A (10.24) .
λ A , r e s = ( T A ) ∗ ( μ r e s , γ ∣ I , A ) q A ν γ ( A ) ≤ λ A . \lambda_{A,\mathrm{res}} = \frac{(T_A)_*(\mu_{\mathrm{res},\gamma|I,A})}{q_A\nu_\gamma(A)} \leq\lambda_A. λ A , res = q A ν γ ( A ) ( T A ) ∗ ( μ res , γ ∣ I , A ) ≤ λ A . The identity in (10.25) , applied to this residual, shows that some A A A ∥ λ A , r e s ∥ ≥ 1 / K \|\lambda_{A,\mathrm{res}}\| \geq1/K ∥ λ A , res ∥ ≥ 1/ K ν γ ( ⋅ ∣ A ) \nu_\gamma(\mathord{\cdot}\mid A) ν γ ( ⋅ ∣ A ) λ A , r e s \lambda_{A,\mathrm{res}} λ A , res c = 1 c=1 c = 1
σ 2 # { occupied ( σ , σ , a ) -cells } ≤ K a − d , \sigma^{2}\#\{\text{occupied }(\sigma,\sigma,a)\text{-cells}\} \leq Ka^{-d}, σ 2 # { occupied ( σ , σ , a ) -cells } ≤ K a − d , together with the pure caps and a positive terminal angular gain. The upper caps and occupied-cell bound persist under this restriction, and the displayed mass floor supplies the required density relative to the fixed prior.
On a further subsequence the theorem gives a quadratic F ( τ , ζ ) F(\tau,\zeta) F ( τ , ζ ) X X X K a Ka K a
q A ν γ ( A ) a d K . (10.29) \frac{q_A\nu_\gamma(A)a^d}{K}.\tag{10.29} K q A ν γ ( A ) a d . ( 10.29 ) Indeed the angular conclusion makes its moving interval scale β ∼ log , N 1 \beta\sim_{\log,N} 1 β ∼ l o g , N 1 F F F K K K K K K
Greedy coverage in original mass. In every A A A N δ N^\delta N δ a N δ aN^\delta a N δ
q A ν γ ( A ) a d N − δ . (10.30) q_A\nu_\gamma(A)a^dN^{-\delta}.\tag{10.30} q A ν γ ( A ) a d N − δ . ( 10.30 ) Let μ c ≤ μ p r e p \mu_c \leq\mu_{\mathrm{prep}} μ c ≤ μ prep μ p r e p − μ c \mu_{\mathrm{prep}}-\mu_c μ prep − μ c A A A a d N − δ ν γ ( A ) a^dN^{-\delta}\nu_\gamma(A) a d N − δ ν γ ( A ) q A q_A q A N δ a − d N^\delta a^{-d} N δ a − d A A A
Maximality forces substantial coverage on a subsequence. If the covered fraction ∥ μ c ∥ / p p a t h \|\mu_c\|/\mathfrak{p}_{\mathrm{path}} ∥ μ c ∥/ p path K ≤ N δ K \leq N^\delta K ≤ N δ (10.30) , contradicting termination. Thus for some fixed κ > 0 \kappa>0 κ > 0
∥ μ c ∥ ≥ κ p p a t h . \|\mu_c\| \geq\kappa \mathfrak{p}_{\mathrm{path}}. ∥ μ c ∥ ≥ κ p path . An individual candidate’s reciprocal-subpower floor is used to bound the insertion count, not as the eventual coverage.
Posterior comparison. Keep μ c \mu_c μ c O ( ξ ) = ( p 1 ( ξ ) , A ( ξ ) ) O(\xi)=(p_1(\xi),A(\xi)) O ( ξ ) = ( p 1 ( ξ ) , A ( ξ )) ξ \xi ξ
m c = O ∗ μ c m_c = O_*\mu_c m c = O ∗ μ c m c m_c m c o o o π o \pi_o π o
Π = ∫ ( π o ⊗ π o ) d m c ( o ) . (10.31) \Pi= \int(\pi_o \otimes\pi_o)\,d m_c(o).\tag{10.31} Π = ∫ ( π o ⊗ π o ) d m c ( o ) . ( 10.31 ) The base measure in this integral is the unnormalized observation pushforward. Consequently ∥ Π ∥ = ∥ μ c ∥ \lVert\Pi\rVert= \lVert\mu_c\rVert ∥ Π ∥ = ∥ μ c ∥ Π \Pi Π μ c \mu_c μ c F , G F,G F , G ( τ , ζ ) (\tau,\zeta) ( τ , ζ )
∣ X F − X G ∣ ≤ K 0 N − 1 . \lvert X_F-X_G\rvert\le K_0N^{-1}. ∣ X F − X G ∣ ≤ K 0 N − 1 . Since s 1 < 1 s_1 < 1 s 1 < 1
∣ F ( τ , ζ ) − G ( τ , ζ ) ∣ ≤ a N O ( δ ) . (10.32) \lvert F(\tau,\zeta)-G(\tau,\zeta)\rvert\le aN^{O(\delta)}.\tag{10.32} ∣ F ( τ , ζ ) − G ( τ , ζ )∣ ≤ a N O ( δ ) . ( 10.32 ) Use the maximum norm on the six quadratic coefficients in the original ( τ , ζ ) (\tau,\zeta) ( τ , ζ )
B = { ( ξ 1 , ξ 2 ) : ∥ coeff F − coeff G ∥ ∞ > a 2 θ } , Π b a d = 1 B Π . \mathcal{B}=\{(\xi_1,\xi_2):\lVert\operatorname{coeff} F-\operatorname{coeff} G\rVert_\infty>a^{2\theta}\},\qquad\Pi_{\mathrm{bad}}=1_{\mathcal{B}}\Pi. B = {( ξ 1 , ξ 2 ) : ∥ coeff F − coeff G ∥ ∞ > a 2 θ } , Π bad = 1 B Π. We claim that ∥ Π b a d ∥ = o ( ∥ Π ∥ ) \lVert\Pi_{\mathrm{bad}}\rVert=o(\lVert\Pi\rVert) ∥ Π bad ∥ = o (∥ Π ∥) (10.17) then give a world and an A A A Π b a d , γ , A = ω γ − 1 Π b a d ∣ γ , I , A \Pi_{\mathrm{bad},\gamma,A}=\omega_\gamma^{-1}\Pi_{\mathrm{bad}}\vert_{\gamma,I,A} Π bad , γ , A = ω γ − 1 Π bad ∣ γ , I , A
b A : = ∥ Π b a d , γ , A ∥ ≥ q A ν γ ( A ) K . b_A:=\lVert\Pi_{\mathrm{bad},\gamma,A}\rVert\ge\frac{q_A\nu_\gamma(A)}{K}. b A := ∥ Π bad , γ , A ∥ ≥ K q A ν γ ( A ) . Normalize only these selected pairs, after the coordinate change:
Π ^ A = ( T A × T A ) ∗ Π b a d , γ , A b A , ( Π ^ A ) j ≤ q A ν γ ( A ) b A λ A ≤ K λ A ( j = 1 , 2 ) . (10.33) \widehat{\Pi}_A=\frac{(T_A\times T_A)_*\Pi_{\mathrm{bad},\gamma,A}}{b_A},\qquad
(\widehat{\Pi}_A)_j\le\frac{q_A\nu_\gamma(A)}{b_A}\lambda_A\le K\lambda_A\qquad(j=1,2).\tag{10.33} Π A = b A ( T A × T A ) ∗ Π bad , γ , A , ( Π A ) j ≤ b A q A ν γ ( A ) λ A ≤ K λ A ( j = 1 , 2 ) . ( 10.33 ) Here ( Π ^ A ) j (\widehat{\Pi}_A)_j ( Π A ) j j j j μ c , γ ∣ I , A \mu_{c,\gamma\vert I,A} μ c , γ ∣ I , A μ p r e p , γ ∣ I , A \mu_{\mathrm{prep},\gamma\vert I,A} μ prep , γ ∣ I , A
For this probability law each individual polynomial index has marginal base density at most
K N O ( δ ) a d in ( τ , ζ ) . (10.34) KN^{O(\delta)}a^d\quad\text{in }(\tau,\zeta).\tag{10.34} K N O ( δ ) a d in ( τ , ζ ) . ( 10.34 ) Indeed, at a fixed base the fit of one assigned polynomial permits only N O ( δ ) X a N^{O(\delta)}X_a N O ( δ ) X a λ A \lambda_A λ A (10.33) , give this ceiling. Each color has a list of at most N δ a − d N^\delta a^{-d} N δ a − d ρ j , f \rho_{j,f} ρ j , f a d ≤ N δ ρ j , f a^d\le N^\delta\rho_{j,f} a d ≤ N δ ρ j , f K N O ( δ ) ρ j , f d τ d ζ KN^{O(\delta)}\rho_{j,f}\,d\tau\,d\zeta K N O ( δ ) ρ j , f d τ d ζ
Enlarge and rescale the ζ \zeta ζ K K K K N δ KN^\delta K N δ a 2 θ a^{2\theta} a 2 θ M = 1 M=1 M = 1 Δ = a \Delta=a Δ = a R = a − 1 R=a^{-1} R = a − 1 ε \varepsilon ε (10.18) . For every fixed loss exponent C ≤ C m a t c h C\le C_{\mathrm{match}} C ≤ C match
N C δ = R C δ / s 1 , C δ / s 1 < ε / 2. N^{C\delta}=R^{C\delta/s_1},\qquad C\delta/s_1<\varepsilon/2. N C δ = R C δ / s 1 , C δ / s 1 < ε /2. This accounts for the list count, polynomial norm, value discrepancy and index-base density. The range rescaling and the bad-pair normalization contribute only factors K = R o ( 1 ) K = R^{o(1)} K = R o ( 1 ) ε \varepsilon ε R − 2 θ / K R^{-2\theta}/K R − 2 θ / K R − ε R^{-\varepsilon} R − ε 2 θ < ε 2\theta< \varepsilon 2 θ < ε Π ^ A \widehat{\Pi}_A Π A d < d 0 < 1 d < d_0 < 1 d < d 0 < 1 Π b a d \Pi_{\mathrm{bad}} Π bad
Bin the six coefficients of a quadratic in ( τ , ζ ) (\tau,\zeta) ( τ , ζ ) A A A a θ a^\theta a θ a 2 θ ≪ a θ a^{2\theta} \ll a^\theta a 2 θ ≪ a θ o = ( p 1 , A ) o=(p_1,A) o = ( p 1 , A ) β o \beta_o β o π o \pi_o π o L ( b ) L(b) L ( b ) b b b B 1 , B 2 B_1,B_2 B 1 , B 2 β o \beta_o β o B 2 ∈ L ( B 1 ) B_2 \in L(B_1) B 2 ∈ L ( B 1 )
∫ β o ( L ( b ) ) d β o ( b ) = P { B 2 ∈ L ( B 1 ) ∣ o } . \int\beta_o(L(b))\,\mathrm{d}\beta_o(b)=\mathbb{P}\{B_2\in L(B_1)\mid o\}. ∫ β o ( L ( b )) d β o ( b ) = P { B 2 ∈ L ( B 1 ) ∣ o } . Choose a maximizing center b b b o o o m c m_c m c A A A p 1 p_1 p 1
The new test and its actual bounds. Within each A A A b b b F ~ A , b \widetilde{F}_{A,b} F A , b ( A , b ) (A,b) ( A , b )
P n e w ( t , y , w ) = P ∗ ( t , y , w ) − F ~ A , b ( t − t A q A , e Z ⋅ y − Z ∗ q A ) , (10.35) P_{\mathrm{new}}(t,y,w)=P_*(t,y,w)-\widetilde{F}_{A,b}\left(\frac{t-t_A}{q_A},\frac{e_Z\cdot y-Z_*}{q_A}\right),\tag{10.35} P new ( t , y , w ) = P ∗ ( t , y , w ) − F A , b ( q A t − t A , q A e Z ⋅ y − Z ∗ ) , ( 10.35 ) where t A t_A t A
a N δ + K a θ . aN^\delta+Ka^\theta. a N δ + K a θ . Our fixed choices give δ < θ s 1 / 4 \delta<\theta s_1/4 δ < θ s 1 /4 θ < 1 \theta<1 θ < 1 c ′ = θ s 1 / 2 c'=\theta s_1/2 c ′ = θ s 1 /2 s 1 − δ s_1-\delta s 1 − δ θ s 1 \theta s_1 θ s 1 c ′ c' c ′ N 0 − c ′ N_0^{-c'} N 0 − c ′ K K K
The polynomial subtracted in (10.35) does not involve w w w
( P n e w ) w = ( P ∗ ) w , ( P n e w ) w w = ( P ∗ ) w w . (P_{\mathrm{new}})_w=(P_*)_w,\qquad(P_{\mathrm{new}})_{ww}=(P_*)_{ww}. ( P new ) w = ( P ∗ ) w , ( P new ) w w = ( P ∗ ) w w . The original native bounds give the actual full-block derivative upper bound and, on retained original path incidences, the actual lower bound. The native curvature bound K 0 B 2 K_0B^2 K 0 B 2 K 0 B 2 N 0 c ′ K_0B^2N_0^{c'} K 0 B 2 N 0 c ′ A A A
Assignments and lists on the original law. Store the ordered quadratic tests for each A A A N δ a − d N^\delta a^{-d} N δ a − d A A A i i i
Attach this geometric label map to the original tempered path μ r e f \mu_{\mathrm{ref}} μ ref μ c \mu_c μ c A A A p 1 p_1 p 1
μ c ≤ μ p r e p ≤ μ r e f . \mu_c\leq\mu_{\mathrm{prep}}\leq\mu_{\mathrm{ref}}. μ c ≤ μ prep ≤ μ ref . Thus Lemma 4.12 applies with original law μ r e f \mu_{\mathrm{ref}} μ ref μ c \mu_{\mathrm{c}} μ c ( A , b ) (A,b) ( A , b ) p 1 p_1 p 1
All predicates entering this application are the original path predicates and the finitely many geometric fit tests. The analytical cap masks and residual searches are absent. The polynomial bounds for these geometric data give a fixed mesh exponent and list table of the complexity allowed by the prechosen finite-sample exclusion. The table loses only O ( N 0 − 1 ) O(N_0^{-1}) O ( N 0 − 1 )
A fixed fraction of κ p p a t h \kappa \mathfrak{p}_{\mathrm{path}} κ p path p p a t h ≥ N 0 − o ( 1 ) \mathfrak{p}_{\mathrm{path}} \ge N_0^{-o(1)} p path ≥ N 0 − o ( 1 ) (10.20) for all sufficiently late samples. Together with the geometric and derivative bounds above, this is precisely the excluded known atlas at depth c ′ c' c ′ q A q_A q A (10.19) and proves Proposition 10.1.
Completion of the proof
Proof of Theorem 1.1. Let K ⊂ R 4 K \subset\mathbb{R}^4 K ⊂ R 4 dim H K < 4 \dim_{\mathrm{H}} K < 4 dim H K < 4 σ > 0 \sigma> 0 σ > 0 dim H K < 4 − σ \dim_{\mathrm{H}} K < 4-\sigma dim H K < 4 − σ C ( d 0 ) C(d_0) C ( d 0 ) d 0 < 1 d_0 < 1 d 0 < 1 H > 0 H > 0 H > 0 K K K
Optimize cap dimension and horizon as in Proposition 4.3. The resulting critical dimension is less than one and the maximizing time rate satisfies k > 0 k > 0 k > 0
For the second model, consider the global optimized narrowness ℓ \ell ℓ ℓ = 0 \ell= 0 ℓ = 0 0 < ℓ < ∞ 0 < \ell< \infty 0 < ℓ < ∞ k k k 2 k = 1 2k = 1 2 k = 1 ℓ = ∞ \ell= \infty ℓ = ∞
Every possible optimized narrowness has therefore been excluded, contradicting the positive-horizon problem supplied by Theorem 2.17. It follows that dim H K ≥ 4 \dim_{\mathrm{H}} K \ge4 dim H K ≥ 4 dim H K ≤ 4 \dim_{\mathrm{H}} K \le4 dim H K ≤ 4 dim H K = 4 \dim_{\mathrm{H}} K = 4 dim H K = 4 K K K
Geometric consequences
Theorem 1.1 has consequences for other notions of dimension, for unions of segments with prescribed directions, and for several curved incidence problems. The direction-set and curved conclusions use the transfer theorems of Keleti and Máthé, Gao, Liu, and Xi, and Nadjimzadah.
Packing and Minkowski dimensions
Every Kakeya set K ⊂ R 4 K \subset\mathbb{R}^4 K ⊂ R 4 log N δ ( K ) / log ( 1 / δ ) \log N_{\delta}(K)/\log(1/\delta) log N δ ( K ) / log ( 1/ δ ) N δ ( K ) N_{\delta}(K) N δ ( K ) δ \delta δ K K K δ ↓ 0 \delta\downarrow0 δ ↓ 0
Projection to four dimensions
Theorem 1.1 also implies that every Kakeya set E ⊂ R d E \subset\mathbb{R}^{d} E ⊂ R d d ≥ 4 d \ge4 d ≥ 4 dim H E ≥ 4 \dim_{\mathrm{H}} E \ge4 dim H E ≥ 4 V V V P V P_{V} P V V V V v ∈ V v \in V v ∈ V a + [ 0 , 1 ] v ⊂ E a + [0,1]v \subset E a + [ 0 , 1 ] v ⊂ E P V a + [ 0 , 1 ] v P_{V}a + [0,1]v P V a + [ 0 , 1 ] v P V E P_{V}E P V E V ≃ R 4 V \simeq\mathbb{R}^{4} V ≃ R 4 4 = dim H ( P V E ) ≤ dim H E 4 = \dim_{\mathrm{H}}(P_{V}E) \le\dim_{\mathrm{H}} E 4 = dim H ( P V E ) ≤ dim H E R d \mathbb{R}^{d} R d d d d
Arbitrary directions and extension of segments
Keleti and Máthé transfer the full-direction assertion to arbitrary direction sets. We identify unoriented directions in R 4 \mathbb{R}^{4} R 4 R P 3 \mathbb{RP}^{3} RP 3
Corollary 11.1 (General direction-set bound). Let D ⊂ R P 3 D \subset\mathbb{RP}^{3} D ⊂ RP 3 be nonempty and let E ⊂ R 4 E \subset\mathbb{R}^{4} E ⊂ R 4 If E E E contains a nondegenerate line segment in every direction in D D D then
dim H E ≥ 1 + dim H D . \dim_{\mathrm{H}} E \ge1 + \dim_{\mathrm{H}} D. dim H E ≥ 1 + dim H D . Proof. By Keleti and Máthé [20 ] , Theorem 1.4, there is a compact Besicovitch set B ⊂ R 4 B \subset\mathbb{R}^{4} B ⊂ R 4 dim H E ≥ dim H B − ( 3 − dim H D ) \dim_{\mathrm{H}} E \ge\dim_{\mathrm{H}} B - (3 - \dim_{\mathrm{H}} D) dim H E ≥ dim H B − ( 3 − dim H D ) dim H B = 4 \dim_{\mathrm{H}} B = 4 dim H B = 4
In particular, a direction set of Hausdorff dimension three already forces full spatial dimension. We will use this below when the directions form a nonempty open patch. Neither E E E D D D
The same conclusion gives a line-segment extension principle by the implication proved in Keleti and Máthé [20 ] , Section 1.2. For any nonempty family S \mathcal{S} S R 4 \mathbb{R}^{4} R 4 L \mathcal{L} L
dim H ( ⋃ S ∈ S S ) = dim H ( ⋃ L ∈ L L ) . \dim_{\mathrm{H}}\left(\bigcup_{S\in\mathcal{S}} S\right) = \dim_{\mathrm{H}}\left(\bigcup_{L\in\mathcal{L}} L\right). dim H ( S ∈ S ⋃ S ) = dim H ( L ∈ L ⋃ L ) . Thus extending every segment in a fixed family to its whole line preserves the Hausdorff dimension of the union. This application uses the authors’ same-dimensional implication from the general direction-set bound.
Nikodym sets on constant-curvature manifolds
The Nikodym formulation selects geodesics through a positive-volume family of base points, rather than selecting a segment for every direction. Theorem 1.1 controls this different incidence requirement through the established Kakeya-to-Nikodym implication of Gao, Liu, and Xi. We use their normalized local geodesic formulation [11 ] , Definition 1.3 and Theorem 2.1; the metric is normalized so that the injectivity radius is at least 10, and the geodesic segments below have unit length. For a Borel set Ω ⊂ M \Omega\subset M Ω ⊂ M 0 < λ < 1 0 < \lambda< 1 0 < λ < 1
Ω λ ∗ = { x ∈ M : there is a unit geodesic segment γ x through x such that ∣ γ x ∩ Ω ∣ g ≥ λ ∣ γ x ∣ g } . \Omega_\lambda^\ast= \left\{ x \in M : \begin{array}{l}
\text{there is a unit geodesic segment } \gamma_x \text{ through } x\\
\text{such that } \lvert\gamma_x \cap\Omega\rvert_g \ge\lambda\lvert\gamma_x \rvert_g
\end{array} \right\}. Ω λ ∗ = { x ∈ M : there is a unit geodesic segment γ x through x such that ∣ γ x ∩ Ω ∣ g ≥ λ ∣ γ x ∣ g } . where the bars denote geodesic length. The set Ω \Omega Ω Nikodym set if Ω λ ∗ \Omega_\lambda^\ast Ω λ ∗ λ \lambda λ
Corollary 11.2 (Four-dimensional Nikodym dimension). Let ( M 4 , g ) (M^4,g) ( M 4 , g ) be a Riemannian manifold of constant sectional curvature in the preceding normalized setting. Every Nikodym set Ω ⊂ M \Omega\subset M Ω ⊂ M has dim H Ω = 4 \dim_{\mathrm H}\Omega= 4 dim H Ω = 4
Proof. Theorem 1.1 applies in particular to the compact Euclidean Kakeya sets in the convention of Gao et al. [11 ] , Definition 1.1. It therefore supplies their Kakeya Hausdorff lower bound with d = α = 4 d=\alpha=4 d = α = 4 [11 ] , Theorem 1.5 transfers this bound to Nikodym sets on constant-sectional-curvature d d d dim H Ω ≥ 4 \dim_{\mathrm H}\Omega\ge4 dim H Ω ≥ 4
This resolves Conjecture 1.4 of Gao et al. [11 ] in dimension four and includes Euclidean R 4 \mathbb{R}^4 R 4
Curved Kakeya sets for a fixed translation-invariant phase
A separate consequence concerns the structured curved-Kakeya class of Gao, Liu, and Xi. Unlike the preceding Nikodym corollary, it uses a direction-indexed family from one phase in a Euclidean chart. We use their phase and core conventions [11 ] , Definitions 1.7–1.11, with the active support made explicit. One common local change of coordinates straightens a direction patch of this family. This lets the Euclidean dimension theorem control the original curved set through local bi-Lipschitz invariance.
Write X = ( x , t ) ∈ R 3 × R X=(x,t)\in\mathbb{R}^3\times\mathbb{R} X = ( x , t ) ∈ R 3 × R
U = B ρ 3 × ( − ρ , ρ ) , Y = B ρ 3 . U=B_\rho^3\times(-\rho,\rho), \qquad Y=B_\rho^3. U = B ρ 3 × ( − ρ , ρ ) , Y = B ρ 3 . where B ρ 3 B_\rho^3 B ρ 3 U ‾ × Y ‾ \overline{U}\times\overline{Y} U × Y
ϕ ( x , t ; y ) = x ⋅ y + ψ ( t ; y ) , ψ ( 0 ; y ) = 0. \phi(x,t;y)=x\cdot y+\psi(t;y), \qquad\psi(0;y)=0. ϕ ( x , t ; y ) = x ⋅ y + ψ ( t ; y ) , ψ ( 0 ; y ) = 0. For ( X ; y ) (X;y) ( X ; y )
G 0 ( X ; y ) = ⋀ j = 1 3 ∂ y j ∇ X ϕ ( X ; y ) , D y = G 0 ( X ; y ) ⋅ ∇ X , G_0(X;y)=\bigwedge_{j=1}^{3}\partial_{y_j}\nabla_X\phi(X;y), \qquad \mathcal{D}_y=G_0(X;y)\cdot\nabla_X, G 0 ( X ; y ) = j = 1 ⋀ 3 ∂ y j ∇ X ϕ ( X ; y ) , D y = G 0 ( X ; y ) ⋅ ∇ X , where the wedge is identified with its normal vector in R 4 \mathbb{R}^4 R 4 D y \mathcal{D}_y D y X X X y y y
rank ∇ X ∇ y ϕ ( X ; y ) = 3 , \operatorname{rank}\nabla_X\nabla_y\phi(X;y)=3, rank ∇ X ∇ y ϕ ( X ; y ) = 3 , det ( ∇ y 2 ⟨ ∇ X ϕ ( X ; y ) , G 0 ( X ; y 0 ) ⟩ ∣ y = y 0 ) ≠ 0. \det\left(\nabla_y^2\left\langle\nabla_X\phi(X;y),G_0(X;y_0)\right\rangle\bigg|_{y=y_0}\right)\ne0. det ( ∇ y 2 ⟨ ∇ X ϕ ( X ; y ) , G 0 ( X ; y 0 ) ⟩ y = y 0 ) = 0. and Bourgain’s condition in the invariant formulation of Guo et al. [12 ] , Theorem 2.1,
D y 2 ∂ y i y j 2 ϕ ( X ; y ) = C ( X ; y ) D y ∂ y i y j 2 ϕ ( X ; y ) ( 1 ≤ i , j ≤ 3 ) . \mathcal{D}_y^2\partial_{y_i y_j}^2\phi(X;y)=C(X;y)\mathcal{D}_y\partial_{y_i y_j}^2\phi(X;y)\quad(1\le i,j\le3). D y 2 ∂ y i y j 2 ϕ ( X ; y ) = C ( X ; y ) D y ∂ y i y j 2 ϕ ( X ; y ) ( 1 ≤ i , j ≤ 3 ) . with the same scalar C ( X ; y ) C(X;y) C ( X ; y ) i , j i,j i , j U ‾ × Y ‾ \overline{U}\times\overline{Y} U × Y y ∈ Y y\in Y y ∈ Y z ∈ U z\in U z ∈ U
Γ y ϕ ( z ) = { X ∈ U : ∇ y ϕ ( X ; y ) = ∇ y ϕ ( z ; y ) } . \Gamma_y^\phi(z)=\{X\in U:\nabla_y\phi(X;y)=\nabla_y\phi(z;y)\}. Γ y ϕ ( z ) = { X ∈ U : ∇ y ϕ ( X ; y ) = ∇ y ϕ ( z ; y )} . Corollary 11.3 (Fixed-chart curved Kakeya dimension). Under the preceding fixed-chart phase hypotheses , let E ⊂ R 4 E\subset\mathbb{R}^4 E ⊂ R 4 Suppose that for every y ∈ Y y\in Y y ∈ Y ω y ∈ B ρ 3 \omega_y\in B_\rho^3 ω y ∈ B ρ 3
Γ y ϕ ( ( ω y , 0 ) ) ⊂ E . \Gamma_y^\phi((\omega_y,0))\subset E. Γ y ϕ (( ω y , 0 )) ⊂ E . Then dim H E = 4 \dim_H E=4 dim H E = 4
Proof. For this phase, G 0 = ( − ∂ t ∇ y ψ , 1 ) G_0=(-\partial_t\nabla_y\psi,1) G 0 = ( − ∂ t ∇ y ψ , 1 )
det ∇ y 2 ∂ t ψ ( t ; y ) ≠ 0 , ∇ y 2 ∂ t 2 ψ ( t ; y ) = c ( t , y ) ∇ y 2 ∂ t ψ ( t ; y ) . \det\nabla_y^2\partial_t\psi(t;y)\ne0,\qquad\nabla_y^2\partial_t^2\psi(t;y)=c(t,y)\nabla_y^2\partial_t\psi(t;y). det ∇ y 2 ∂ t ψ ( t ; y ) = 0 , ∇ y 2 ∂ t 2 ψ ( t ; y ) = c ( t , y ) ∇ y 2 ∂ t ψ ( t ; y ) . The nonsingular matrix determines this scalar smoothly and independently of x x x [11 ] , Corollary 4.2 and Theorem 1.12, show that c c c t t t I I I 0 0 0 Y 0 ⊂ Y Y_0\subset Y Y 0 ⊂ Y
κ ( u , s ) = ( u + B ( α ( s ) ) , α ( s ) ) , α ( 0 ) = 0 , α ′ ( s ) ≠ 0 , \kappa(u,s)=(u+B(\alpha(s)),\alpha(s)),\qquad\alpha(0)=0,\quad\alpha'(s)\ne0, κ ( u , s ) = ( u + B ( α ( s )) , α ( s )) , α ( 0 ) = 0 , α ′ ( s ) = 0 , ϕ ( κ ( u , s ) ; y ) = u ⋅ y + s h ( y ) + q ( y ) + f ( s ) , det ∇ y 2 h ( y ) ≠ 0 , \phi(\kappa(u,s);y)=u\cdot y+sh(y)+q(y)+f(s),\qquad\det\nabla_y^2h(y)\ne0, ϕ ( κ ( u , s ) ; y ) = u ⋅ y + s h ( y ) + q ( y ) + f ( s ) , det ∇ y 2 h ( y ) = 0 , whenever s ∈ I s\in I s ∈ I y ∈ Y 0 y\in Y_0 y ∈ Y 0 κ ( u , s ) ∈ U \kappa(u,s)\in U κ ( u , s ) ∈ U ψ ( 0 ; y ) = 0 \psi(0;y)=0 ψ ( 0 ; y ) = 0 ( ω y , 0 ) (\omega_y,0) ( ω y , 0 )
Γ y ϕ ( ( ω y , 0 ) ) = { ( ω y − ∇ y ψ ( t ; y ) , t ) ∈ U } . \Gamma_y^\phi((\omega_y,0))=\{(\omega_y-\nabla_y\psi(t;y),t)\in U\}. Γ y ϕ (( ω y , 0 )) = {( ω y − ∇ y ψ ( t ; y ) , t ) ∈ U } . Put V = U ∩ ( R 3 × α ( I ) ) V=U\cap(\mathbb{R}^3\times\alpha(I)) V = U ∩ ( R 3 × α ( I )) F = κ − 1 ( E ∩ V ) F=\kappa^{-1}(E\cap V) F = κ − 1 ( E ∩ V ) ( ω y , 0 ) (\omega_y,0) ( ω y , 0 ) V V V V V V
u = ( ω y − B ( 0 ) ) − s ∇ h ( y ) . u=(\omega_y-B(0))-s\nabla h(y). u = ( ω y − B ( 0 )) − s ∇ h ( y ) . Indeed, the ∇ q ( y ) \nabla q(y) ∇ q ( y ) f ( s ) f(s) f ( s ) y y y F F F
ν ( y ) = ( − ∇ h ( y ) , 1 ) 1 + ∣ ∇ h ( y ) ∣ 2 \nu(y)=\frac{(-\nabla h(y),1)}{\sqrt{1+\lvert\nabla h(y)\rvert^2}} ν ( y ) = 1 + ∣ ∇ h ( y ) ∣ 2 ( − ∇ h ( y ) , 1 ) for every y ∈ Y 0 y\in Y_0 y ∈ Y 0 h h h ν \nu ν S 3 S^3 S 3 D ⊂ R P 3 D\subset\mathbb{RP}^3 D ⊂ RP 3 dim H D = 3 \dim_H D=3 dim H D = 3 dim H F = 4 \dim_H F=4 dim H F = 4 y y y
A smooth local diffeomorphism preserves Hausdorff dimension of arbitrary subsets, using a countable cover by coordinate neighborhoods on which it and its inverse are Lipschitz. Consequently
4 = dim H F = dim H ( E ∩ V ) ≤ dim H E ≤ 4. 4 = \dim_H F = \dim_H(E \cap V) \leq\dim_H E \leq4. 4 = dim H F = dim H ( E ∩ V ) ≤ dim H E ≤ 4. No compactness or measurability of E E E y ↦ ω y y \mapsto\omega_y y ↦ ω y
The straightening and general-dimensional set transfer are due to Gao et al. [11 ] [Theorem 1.12 and Section 4.2]. The corollary applies their transfer to full gradient-defined cores in one fixed active chart.
The translation-invariance assumption specifies the class being straightened here. Bourgain’s condition alone need not allow one coordinate change to straighten the curves, as the examples of Nadjimzadah [22 ] [Example 1.24 and Proposition 1.25] show.
A three-dimensional quadratic curved consequence
A further application uses Nadjimzadah’s lifting theorem to pass from the four-dimensional linear result to curved sets in dimension three. For a real 2 × 2 2 \times2 2 × 2 B B B I 2 I_2 I 2 W B ⊂ R 2 W_B \subset\mathbb{R}^2 W B ⊂ R 2 I B = [ − τ B , τ B ] I_B = [-\tau_B,\tau_B] I B = [ − τ B , τ B ] τ B > 0 \tau_B > 0 τ B > 0 [23 ] .
Corollary 11.4 (Rank-one quadratic curved Kakeya sets). Let B B B be a real 2 × 2 2 \times2 2 × 2 matrix with rank B = rank ( B 2 ) = 1 \operatorname{rank} B = \operatorname{rank}(B^2) = 1 rank B = rank ( B 2 ) = 1 and use the fixed domains just specified. Suppose that a compact set K ⊂ R 3 K \subset\mathbb{R}^3 K ⊂ R 3 contains, for every y ∈ [ − 1 , 1 ] 2 y \in[-1,1]^2 y ∈ [ − 1 , 1 ] 2 the full curve
{ ( w y + ( t I 2 + t 2 B ) y , t ) : t ∈ I B } \{(w_y + (tI_2 + t^2B)y,t) : t \in I_B\} {( w y + ( t I 2 + t 2 B ) y , t ) : t ∈ I B } for some w y ∈ W B w_y \in W_B w y ∈ W B Then dim H K = 3 \dim_H K = 3 dim H K = 3
Proof. Corollary 11.1 supplies the linear Kakeya hypothesis in the convention of Nadjimzadah [23 ] [Section 1.2, equations (1.8)–(1.13)]. Indeed, a compact linear-chart set F ⊂ R 4 F \subset\mathbb{R}^4 F ⊂ R 4
{ ( b η + t η , t ) : t ∈ J } \{(b_\eta+ t\eta,t) : t \in J\} {( b η + t η , t ) : t ∈ J } for every slope η \eta η J J J [ η : 1 ] ∈ R P 3 [\eta: 1] \in\mathbb{RP}^3 [ η : 1 ] ∈ RP 3 dim H F = 4 \dim_H F = 4 dim H F = 4
The geometry of Nadjimzadah’s quadratic lift explains the two rank hypotheses. Write B = U V B = UV B = U V U : R → R 2 U : \mathbb{R} \to\mathbb{R}^2 U : R → R 2 V : R 2 → R V : \mathbb{R}^2 \to\mathbb{R} V : R 2 → R
π : R 2 × R × R ⟶ R 2 × R , π ( x , z , t ) = ( x + t U z , t ) . \pi: \mathbb{R}^2 \times\mathbb{R} \times\mathbb{R} \longrightarrow\mathbb{R}^2 \times\mathbb{R}, \qquad\pi(x,z,t) = (x + tUz,t). π : R 2 × R × R ⟶ R 2 × R , π ( x , z , t ) = ( x + t U z , t ) . The lifted segments in Nadjimzadah [23 ] [equations (4.1)–(4.9)] are
L y , w , r ( t ) = ( ( w , r ) + t ( y − U r , V y ) , t ) , t ∈ I B , L_{y,w,r}(t) = ((w,r) + t(y-Ur,Vy),t), \qquad t \in I_B, L y , w , r ( t ) = (( w , r ) + t ( y − U r , V y ) , t ) , t ∈ I B , with r r r
π ( L y , w , r ( t ) ) = ( w + t y + t 2 U V y , t ) = ( w + ( t I 2 + t 2 B ) y , t ) . \pi(L_{y,w,r}(t)) = (w + ty + t^2UVy,t) = (w + (tI_2 + t^2B)y,t). π ( L y , w , r ( t )) = ( w + t y + t 2 U V y , t ) = ( w + ( t I 2 + t 2 B ) y , t ) . Since B 2 = U ( V U ) V B^2 = U(VU)V B 2 = U ( V U ) V V U ≠ 0 VU \ne0 V U = 0 Θ ( y , r ) = ( y − U r , V y ) \Theta(y,r) = (y - Ur, Vy) Θ ( y , r ) = ( y − U r , V y )
det D Θ = det ( I 2 − U V 0 ) = V U ≠ 0. \det D\Theta= \det\begin{pmatrix} I_2 & -U \\ V & 0 \end{pmatrix} = VU \ne0. det D Θ = det ( I 2 V − U 0 ) = V U = 0. Thus varying y y y r r r [23 ] retains every t ∈ I B t \in I_B t ∈ I B 2 τ B 2\tau_B 2 τ B
With the four-dimensional linear hypothesis established, Nadjimzadah [23 ] , Theorem 1.3 applies with 3 + rank ( B 2 ) = 4 3 + \operatorname{rank}(B^2) = 4 3 + rank ( B 2 ) = 4 3 − rank B + rank ( B 2 ) = 3 3 - \operatorname{rank} B + \operatorname{rank}(B^2) = 3 3 − rank B + rank ( B 2 ) = 3 dim H K ≥ 3 \dim_H K \ge3 dim H K ≥ 3
This is a lower-dimensional application of the headline result: the prescribed quadratic curves lift to linear Kakeya sets in one additional dimension. Compactness and containment of the full curves over the fixed interval are part of this application. No regularity of the anchor choice y ↦ w y y \mapsto w_y y ↦ w y
