Copyright © 2026 Oleksiy Babanskyy. The repository’s MIT license applies to the assurance software, not to this manuscript. This Hexagon version is distributed under the Creative Commons Attribution 4.0 International license (CC BY 4.0).
Introduction and exact results
Higher-order autocorrelations are translation invariants built from products of a function along several shifted copies. Their use in pattern recognition and periodic reconstruction has a long history [2 , 16 , 4 , 12 ] . On a finite cyclic group, the Fourier transform converts an order-k k k k k k [8 ] a problem about the integral generation of zero-sum relations.
The answer depends sharply on the exact Fourier support. For a support S ⊂ C n S \subset C_n S ⊂ C n L ≤ d ( S ) \mathcal{L}_{\le d}(S) L ≤ d ( S ) d d d K ( S ) K(S) K ( S )
δ ( S ) = min { d : L ≤ d ( S ) = K ( S ) } \delta(S) = \min\{d : \mathcal{L}_{\le d}(S) = K(S)\} δ ( S ) = min { d : L ≤ d ( S ) = K ( S )} the local closing degree. Section 2 identifies it with the maximum separating-atom length on S S S
We compute this object in two directions. The first is a signal-level theorem and complete ambiguity parametrization for the exact unit support of every even cyclic group. The second is a complete supportwise classification when the group order is 2 p q 2pq 2 pq
The orbit-stable classification
Let p < q p < q p < q C 2 p q C_{2pq} C 2 pq O d \mathcal{O}_d O d d ∣ 2 p q d \mid2pq d ∣ 2 pq 2 8 − 1 = 255 2^8 - 1 = 255 2 8 − 1 = 255
Theorem 1.1 (Orbit-stable profiles). For n = 2 p q n = 2pq n = 2 pq Aut ( C n ) \operatorname{Aut}(C_n) Aut ( C n ) 1 , … , 6 1,\ldots,6 1 , … , 6
( 1 , 2 , 154 , 80 , 4 , 14 ) if p = 3 , (1, 2, 154, 80, 4, 14) \qquad\text{if } p = 3, ( 1 , 2 , 154 , 80 , 4 , 14 ) if p = 3 , and
( 1 , 2 , 154 , 92 , 4 , 2 ) if p , q ≥ 5. (1, 2, 154, 92, 4, 2) \qquad\text{if } p,q \ge5. ( 1 , 2 , 154 , 92 , 4 , 2 ) if p , q ≥ 5. Equivalently, these are the distributions of the maximum separating-atom length on the stable supports. In both regimes the orbit-stable local maximum is six.
For A ∈ { Q , Z } A \in\{\mathbb{Q},\mathbb{Z}\} A ∈ { Q , Z } δ A ( S ) \delta_A(S) δ A ( S ) d ≥ 1 d \ge1 d ≥ 1 A A A S S S 1 , … , d 1,\ldots,d 1 , … , d δ Z ( S ) ≤ δ Q ( S ) ≤ δ ( S ) \delta_{\mathbb{Z}}(S) \le\delta_{\mathbb{Q}}(S) \le\delta(S) δ Z ( S ) ≤ δ Q ( S ) ≤ δ ( S )
Theorem 1.2 (Uniform rational and integer realization). Let n = 2 p q n = 2pq n = 2 pq p < q p < q p < q Aut ( C n ) \operatorname{Aut}(C_n) Aut ( C n ) S S S
δ Z ( S ) = δ Q ( S ) = δ ( S ) . (1) \delta_{\mathbb{Z}}(S) = \delta_{\mathbb{Q}}(S) = \delta(S). \tag*{(1)} δ Z ( S ) = δ Q ( S ) = δ ( S ) . (1) Thus the rational-signal and integer-signal reconstruction profiles are the two profiles in Theorem 1.1 . Equivalence is by translation only.
For a subset A ⊂ C n A \subset C_n A ⊂ C n k k k (12) with f = 1 A f=1_A f = 1 A r s e t ( C n ) r_{\mathrm{set}}(C_n) r set ( C n ) d d d d d d d d d d d d
Theorem 1.3 (Binary reconstruction in the 2 p q 2pq 2 pq p < q p<q p < q
r s e t ( C 2 p q ) = 4. (2) r_{\mathrm{set}}(C_{2pq})=4. \tag*{(2)} r set ( C 2 pq ) = 4. (2) The upper bound follows by combining the supportwise phase-lattice criterion with a binary exclusion of every depth-five and depth-six support. The lower bound is the published even-order three-deck obstruction of Keleti–Kolountzakis [13 ] .
Pebody determined the multiset reconstruction numbers of finite abelian groups and obtained the general cyclic set bound six [17 Theorems 2.1–2.2 and Lemma 9.3] . Combining his comparison of set and multiset reconstruction with the multiset classification and the even-order lower bound of Keleti–Kolountzakis already gives
r s e t ( C 2 a ) = 4 ( a ≥ 4 ) , r s e t ( C 2 ℓ k ) = 4 ( ℓ > 3 prime , 2 ℓ k > 10 ) . (3) r_{\mathrm{set}}(C_{2^a})=4 \quad(a\geq4), \qquad r_{\mathrm{set}}(C_{2\ell^k})=4 \quad(\ell>3\ \text{prime},\ 2\ell^k>10). \tag*{(3)} r set ( C 2 a ) = 4 ( a ≥ 4 ) , r set ( C 2 ℓ k ) = 4 ( ℓ > 3 prime , 2 ℓ k > 10 ) . (3) Pebody later conjectured value four for every even cyclic order greater than ten and reported that his method covered many even orders, including every one below 240 [18 ] . Theorem 1.3 adds the arithmetically disjoint squarefree family 2 p q 2pq 2 pq
The proof is uniform in the primes. It does not arise from interpolation of finite atlases. We first compute the cumulative degree-two, degree-three and degree-four presentations of seven nonzero CRT boxes. Exactly nineteen cubic box-count patterns can occur. Five propagation lemmas convert them into a ten-clause criterion selecting exactly 77 of the 127 nonempty seven-box masks; ten explicit defect families prove the converse. At order four, fourteen connector types reduce the generic branch to complete primary-coordinate graphs and the p = 3 p=3 p = 3
The ordinary separating Noether number is a different invariant: it maximizes over modules and supports, and its exact finite-abelian value belongs to the separating-invariant literature [10 ] . Theorem 1.1 is a supportwise distribution restricted to the eight order orbits of one cyclic group.
There is also a closer general framework. Domokos characterizes separating monomials by integral generation of zero-sum sequence monoids, first for diagonalizable actions and then, over non-closed fields, by imposing the criterion on every Galois-stable character subset [6 , 7 ] . That criterion explains why the stable supports considered here are the natural local objects. It does not give the least truncation degree for any prescribed support, the ten-clause Cubic Skeleton, either arithmetic profile, or the exact unit-support quotient. Theorem 1.1 computes precisely this supportwise layer for the regular representation of C 2 p q C_{2pq} C 2 pq
Exact unit supports
Write U n = ( Z / n Z ) × U_n=(\mathbb{Z}/n\mathbb{Z})^\times U n = ( Z / n Z ) ×
Theorem 1.4 (Even unit-support dichotomy and signal classification). Let n n n
The fourth-order quotient is
Z U n / L ≤ 4 ( U n ) ≅ { C n , 3 ∤ n , Z ⊕ C n / 6 , 6 ∣ n . (4) \mathbb{Z}^{U_n}/\mathcal{L}_{\leq4}(U_n)\cong
\begin{cases}
C_n, & 3\nmid n,\\
\mathbb{Z}\oplus C_{n/6}, & 6\mid n.
\end{cases} \tag*{(4)} Z U n / L ≤ 4 ( U n ) ≅ { C n , Z ⊕ C n /6 , 3 ∤ n , 6 ∣ n . (4) If 6 ∣ n 6\mid n 6 ∣ n L ≤ 6 ( U n ) = K ( U n ) \mathcal{L}_{\leq6}(U_n)=K(U_n) L ≤ 6 ( U n ) = K ( U n ) L ≤ 5 ( U n ) = L ≤ 4 ( U n ) \mathcal{L}_{\leq5}(U_n)=\mathcal{L}_{\leq4}(U_n) L ≤ 5 ( U n ) = L ≤ 4 ( U n )
If 3 ∤ n 3\nmid n 3 ∤ n U n U_n U n
If n = 6 m n=6m n = 6 m U n U_n U n
f ^ ( u ) = σ u ( α ) , g ^ ( u ) = σ u ( z α ) , (5) \widehat{f}(u)=\sigma_u(\alpha),\qquad\widehat{g}(u)=\sigma_u(z\alpha), \tag*{(5)} f ( u ) = σ u ( α ) , g ( u ) = σ u ( z α ) , (5) where α ∈ Q ( ζ n ) × \alpha\in\mathbb{Q}(\zeta_n)^\times α ∈ Q ( ζ n ) × z ∈ Q ( ζ 6 ) × z\in\mathbb{Q}(\zeta_6)^\times z ∈ Q ( ζ 6 ) × z z ‾ = 1 z\overline{z}=1 z z = 1 μ 6 \mu_6 μ 6 z 6 = 1 z^6=1 z 6 = 1
The integral calculation uses a universal ambient group for the sifted set
T q , c = { t ∈ C q : gcd ( 1 + c t , q ) = 1 } . T_{q,c}=\{t\in C_q:\gcd(1+ct,q)=1\}. T q , c = { t ∈ C q : g cd( 1 + c t , q ) = 1 } . It is a concrete use of the Freiman-homomorphism framework of Konyagin–Lev and Grynkiewicz [14 , 9 ] . The exceptional free direction is not an arbitrary cyclotomic effect: it is the second admissible class modulo three. After translation and rational descent, the relative phase lies in the quadratic subfield even though the common amplitude ranges over the full cyclotomic field.
Corollary 1.5 (The common unit row). For n = 2 p q n=2pq n = 2 pq U 2 p q = O 2 p q U_{2pq}=O_{2pq} U 2 pq = O 2 pq
δ ( U 2 p q ) = { 6 , p = 3 , 4 , p , q ≥ 5. \delta(U_{2pq})=
\begin{cases}
6, & p=3,\\
4, & p,q\geq5.
\end{cases} δ ( U 2 pq ) = { 6 , 4 , p = 3 , p , q ≥ 5. Theorem 1.1 and Theorem 1.4 therefore agree on their literal common support.
Corollary 1.6 (U 30 U_{30} U 30 U 30 U_{30} U 30 Q ( ζ 30 ) \mathbb{Q}(\zeta_{30}) Q ( ζ 30 ) α \alpha α Q ( ζ 6 ) \mathbb{Q}(\zeta_6) Q ( ζ 6 )
This specializes to the question raised by Agulnick–Busick-Warner [1 ] ; § 5 gives the precise comparison. The Fourier monomial and Galois criteria used in passing from lattices to signals are due to Casper–Orozco [3 ] .
Claim levels and organization
The distinction between the two headline conclusions is binding. On the unit support, all surviving lattice characters are classified and rationally realized. On all 255 supports, we determine the exact rational and integer thresholds; whenever the depth exceeds one, we construct at least one Galois-equivariant lower-bound ratio for the failure immediately below closure. The sole depth-one support is handled directly. We do not classify the complete rational ambiguity torus on every atlas row.
Sections 2–5 establish the common dictionary and the unit-support theorem. Sections 6–9 prove the 2 p q 2pq 2 pq
Scope boundary. No absolute priority or firstness assertion is made. The binary phase/deck crosswalk is part of the Pebody–Grünbaum–Moore–Simon lineage, and the even three-deck lower bound is imported from Keleti–Kolountzakis. The generic critical support used later is the sharpness family of Casper–Orozco; the global sixth-order line and the ratio-character method have earlier sources. The general non-closed-field separating criterion is Domokos’s. The contribution certified here is the displayed fixed-support theorem chain, its exact supportwise classification, and the explicit Galois-equivariant realization of every nontrivial lower-bound layer.
Zero-sum lattices and local separating degree
Let G G G ∅ ≠ S ⊂ G \varnothing\ne S \subset G ∅ = S ⊂ G
B ( S ) = { a ∈ N 0 S : ∑ s ∈ S a s s = 0 } \mathcal{B}(S)=\left\{\boldsymbol{a}\in\mathbb{N}_0^S:\sum_{s\in S}a_s s=0\right\} B ( S ) = { a ∈ N 0 S : s ∈ S ∑ a s s = 0 } for the zero-sum sequence monoid. Its atoms are denoted by A ( S ) \mathcal{A}(S) A ( S ) d ≥ 1 d\ge1 d ≥ 1 B ( S ) ≤ d \mathcal{B}(S)_{\le d} B ( S ) ≤ d d d d
K ( S ) = ker ( Z S ⟶ ⟨ S ⟩ ) , ( a s ) ⟼ ∑ s a s s , K(S)=\ker\left(\mathbb{Z}^S\longrightarrow\langle S\rangle\right),\qquad(a_s)\longmapsto\sum_s a_s s, K ( S ) = ker ( Z S ⟶ ⟨ S ⟩ ) , ( a s ) ⟼ s ∑ a s s , and L ≤ d ( S ) L_{\le d}(S) L ≤ d ( S ) d d d
Using all rows of length at most d d d d d d
For reference, the objects used throughout are separated in Table 1 . In particular, A d ( S ) A_d(S) A d ( S )
Table 1. Notation and logical roles.
For a cancellative monoid M M M q ( M ) \mathrm{q}(M) q ( M )
Proposition 2.1 (Group-completion dictionary). For every S S S d d d
L ≤ d ( S ) = q ( B ( S ) ≤ d ) , K ( S ) = q ( B ( S ) ) , (6) \mathcal{L}_{\le d}(S)=q(\mathcal{B}(S)_{\le d}), \qquad K(S)=q(\mathcal{B}(S)), \tag*{(6)} L ≤ d ( S ) = q ( B ( S ) ≤ d ) , K ( S ) = q ( B ( S )) , (6) where both quotient groups are viewed in Z S \mathbb{Z}^{S} Z S
δ ( S ) = max { ∣ A ∣ : A ∈ A s e p ( S ) } , (7) \delta(S)=\max\{|A|:A\in\mathcal{A}_{\mathrm{sep}}(S)\}, \tag*{(7)} δ ( S ) = max { ∣ A ∣ : A ∈ A sep ( S )} , (7) the maximum length of a separating atom over S S S
Proof. Every nonnegative zero-sum vector of length at most d d d d d d d d d L ≤ d ( S ) \mathcal{L}_{\le d}(S) L ≤ d ( S ) (6) .
The inclusion q ( B ( S ) ) ⊂ K ( S ) q(\mathcal{B}(S))\subset K(S) q ( B ( S )) ⊂ K ( S ) x = x + − x − ∈ K ( S ) x=x^{+}-x^{-}\in K(S) x = x + − x − ∈ K ( S ) h ∈ ⟨ S ⟩ h\in\langle S\rangle h ∈ ⟨ S ⟩ ⟨ S ⟩ \langle S\rangle ⟨ S ⟩ S S S c ∈ N 0 S c\in\mathbb{N}_{0}^{S} c ∈ N 0 S − h -h − h x + + c x^{+}+c x + + c x − + c x^{-}+c x − + c B ( S ) \mathcal{B}(S) B ( S ) x x x
By Definition 2.5 of [SZZ26], a separating atom is an atom not contained in the quotient group generated by shorter atoms. The same source proves that separating atoms generate the full quotient group. Thus the least truncation degree generating K ( S ) K(S) K ( S ) (7) .
The proposition does not require S = − S S=-S S = − S S S S S = − S S=-S S = − S
Domokos’s lattice criterion for diagonalizable actions states the same generation principle at the level of coordinate subrepresentations [6 Theorem 1.3] . Its non-closed-field extension requires the criterion on every Galois-stable character subset of the regular representation [7 Theorem 5.4] . We use that result as the general field-theoretic boundary, not as a source of the numerical profiles: our local invariant asks for the least d d d C 2 2 p q C_{2}^{2p}q C 2 2 p q
Example 2.2 (The first two profile rows). For S = { 0 } S=\{0\} S = { 0 } e 0 e_0 e 0 K ( S ) = L ≤ 1 ( S ) K(S)=\mathcal{L}_{\leq1}(S) K ( S ) = L ≤ 1 ( S ) δ ( S ) = 1 \delta(S)=1 δ ( S ) = 1 S = O 2 S=O_2 S = O 2 Z e → C 2 \mathbb{Z}e\to C_2 Z e → C 2 2 Z e 2\mathbb{Z}e 2 Z e 2 e 2e 2 e δ ( S ) = 2 \delta(S)=2 δ ( S ) = 2
Ambient and obstruction quotients
Define
A d ( S ) = Z S / L ≤ d ( S ) . A_d(S)=\mathbb{Z}^{S}/\mathcal{L}_{\leq d}(S). A d ( S ) = Z S / L ≤ d ( S ) . There is an exact sequence
0 ⟶ K ( S ) / L ≤ d ( S ) ⟶ A d ( S ) ⟶ ⟨ S ⟩ ⟶ 0. (8) 0\longrightarrow K(S)/\mathcal{L}_{\leq d}(S)\longrightarrow A_d(S)\longrightarrow\langle S\rangle\longrightarrow0. \tag*{(8)} 0 ⟶ K ( S ) / L ≤ d ( S ) ⟶ A d ( S ) ⟶ ⟨ S ⟩ ⟶ 0. (8) The middle group is convenient for presentations. The left group is the obstruction to closing at degree d d d
If A d ( S ) A_d(S) A d ( S )
[ K ( S ) : L ≤ d ( S ) ] = ∣ A d ( S ) ∣ ∣ ⟨ S ⟩ ∣ . (9) [K(S):\mathcal{L}_{\leq d}(S)]=\frac{|A_d(S)|}{|\langle S\rangle|}. \tag*{(9)} [ K ( S ) : L ≤ d ( S )] = ∣ ⟨ S ⟩ ∣ ∣ A d ( S ) ∣ . (9) If A d ( S ) A_d(S) A d ( S )
For later comparison, define the support-restricted orbit-stable maximum
β o r b ( G ) = max { δ ( S ) : ∅ ≠ S ⊂ G , α S = S for every α ∈ Aut ( G ) } . (10) \beta_{\mathrm{orb}}(G)=\max\{\delta(S):\varnothing\ne S\subset G,\ \alpha S=S\text{ for every }\alpha\in\operatorname{Aut}(G)\}. \tag*{(10)} β orb ( G ) = max { δ ( S ) : ∅ = S ⊂ G , α S = S for every α ∈ Aut ( G )} . (10) This is only working notation. It is not the ordinary separating Noether number β s e p ( G ) \beta_{\mathrm{sep}}(G) β sep ( G ) C 2 p q C_{2pq} C 2 pq
The stability restriction is substantial. Without it one may choose small supports whose relation lattices close much later, but such supports cannot be exact Fourier supports of rational signals. Thus (2.4) is adapted to the rational Fourier problem and is not a new name for the unrestricted separating Noether number.
The zero-sum face picture
Let Δ S \Delta_S Δ S S S S Λ G \Lambda_G Λ G ∑ s a s s = 0 \sum_s a_s s=0 ∑ s a s s = 0 m m m S S S
m Δ S ∩ Λ G . (11) m\Delta_S\cap\Lambda_G. \tag*{(11)} m Δ S ∩ Λ G . (11) Indeed, the simplex equation is ∑ s a s = m \sum_s a_s=m ∑ s a s = m Λ G \Lambda_G Λ G [11 ] . Our question is not to count the points in (2.5), but to ask at which dilation their integral span generates the full relation lattice.
This is generation rather than enumeration. Ehrhart-type data count the zero-sum sequences at each length; the closing degree asks when their integral combinations span every relation. Many short rows may be redundant, while a single separating atom can delay closure. The two Skeleton arguments below replace large row lists by a small set of functional consequences whose span is proved uniformly.
There is also an equivariance distinction. A setwise Aut ( G ) \operatorname{Aut}(G) Aut ( G )
Autocorrelation phase lattices
Write C n = Z / n Z C_n = \mathbb{Z}/n\mathbb{Z} C n = Z / n Z ζ n = e 2 π i / n \zeta_n = e^{2\pi i/n} ζ n = e 2 π i / n f : C n → C f : C_n \to\mathbb{C} f : C n → C
f ^ ( s ) = ∑ x ∈ C n f ( x ) ζ n − s x . \widehat{f}(s) = \sum_{x \in C_n} f(x)\zeta_n^{-sx}. f ( s ) = x ∈ C n ∑ f ( x ) ζ n − s x . The order-k k k
M k ( f ; t 1 , … , t k − 1 ) = ∑ x ∈ C n f ( x ) f ( x + t 1 ) ⋯ f ( x + t k − 1 ) . (12) M_k(f;t_1,\ldots,t_{k-1}) = \sum_{x \in C_n} f(x)f(x+t_1)\cdots f(x+t_{k-1}). \tag*{(12)} M k ( f ; t 1 , … , t k − 1 ) = x ∈ C n ∑ f ( x ) f ( x + t 1 ) ⋯ f ( x + t k − 1 ) . (12) Our convention counts the number of signal factors: M 1 ( f ) = ∑ x f ( x ) M_1(f) = \sum_x f(x) M 1 ( f ) = ∑ x f ( x ) M 2 M_2 M 2 M 3 M_3 M 3
Taking the Fourier transform of (3.1) in all shift variables gives, up to the fixed normalization power of n n n
M k ^ ( f ; s 1 , … , s k − 1 ) = f ^ ( − s 1 − ⋯ − s k − 1 ) ∏ j = 1 k − 1 f ^ ( s j ) . (13) \widehat{M_k}(f;s_1,\ldots,s_{k-1}) = \widehat{f}(-s_1-\cdots-s_{k-1})\prod_{j=1}^{k-1}\widehat{f}(s_j). \tag*{(13)} M k ( f ; s 1 , … , s k − 1 ) = f ( − s 1 − ⋯ − s k − 1 ) j = 1 ∏ k − 1 f ( s j ) . (13) Thus the relevant monomials are indexed by ordered zero-sum k k k B ( S ) \mathcal{B}(S) B ( S )
We use two results of Casper–Orozco [3 ] . First, equality of (3.1) is equivalent to equality of Fourier products over zero-sum k k k f f f
f ^ ( a s ) = σ a ( f ^ ( s ) ) ( a ∈ ( Z / n Z ) × ) . (14) \widehat{f}(as) = \sigma_a(\widehat{f}(s)) \qquad(a \in(\mathbb{Z}/n\mathbb{Z})^\times). \tag*{(14)} f ( a s ) = σ a ( f ( s )) ( a ∈ ( Z / n Z ) × ) . (14) The earlier bispectral literature often numbers the order by one fewer Fourier factor; in particular the convention in [12 ] is shifted relative to (12) .
For clarity, the rationality assertion can also be seen directly. If f ( x ) ∈ Q f(x) \in\mathbb{Q} f ( x ) ∈ Q σ a \sigma_a σ a ζ n − s x \zeta_n^{-sx} ζ n − s x ζ n − a s x \zeta_n^{-asx} ζ n − a s x (14) . Conversely, if a Fourier vector is equivariant and is zero on a Galois-stable complement, the inverse transform
f ( x ) = 1 n ∑ s ∈ C n f ^ ( s ) ζ n s x (15) f(x) = \frac{1}{n}\sum_{s \in C_n}\widehat{f}(s)\zeta_n^{sx} \tag*{(15)} f ( x ) = n 1 s ∈ C n ∑ f ( s ) ζ n s x (15) is fixed by every σ a \sigma_a σ a Q \mathbb{Q} Q § 5 and § 10 .
Lemma 3.1 (Exact support from order two). If two real-valued signals have equal second autocorrelations, then their Fourier transforms have the same zero set.
Proof. Reality gives f ^ ( − s ) = f ^ ( s ) ‾ \widehat{f}(-s)=\overline{\widehat{f}(s)} f ( − s ) = f ( s ) ( s , − s ) (s,-s) ( s , − s ) ∣ f ^ ( s ) ∣ 2 |\widehat{f}(s)|^2 ∣ f ( s ) ∣ 2
Proposition 3.2 (Phase-ratio bridge). Let f , g : C n → Q f,g:\mathbb{C}_n\to\mathbb{Q} f , g : C n → Q S S S r s = g ^ ( s ) / f ^ ( s ) r_s=\widehat{g}(s)/\widehat{f}(s) r s = g ( s ) / f ( s ) d d d
r a = ∏ s ∈ S r s a s = 1 for every a ∈ L ≤ d ( S ) . (16) r^a=\prod_{s\in S}r_s^{a_s}=1 \qquad\text{for every }a\in\mathcal{L}_{\leq d}(S). \tag*{(16)} r a = s ∈ S ∏ r s a s = 1 for every a ∈ L ≤ d ( S ) . (16) Translation ratios are exactly the characters of Z S \mathbb{Z}^S Z S K ( S ) K(S) K ( S ) L ≤ d ( S ) = K ( S ) \mathcal{L}_{\leq d}(S)=K(S) L ≤ d ( S ) = K ( S ) d d d
Proof. By the Fourier monomial criterion, equality through order d d d d d d C n ∖ S \mathbb{C}_n\setminus S C n ∖ S S S S a ↦ r a a\mapsto r^a a ↦ r a Z S → C × \mathbb{Z}^S\to\mathbb{C}^{\times} Z S → C ×
If g ( x ) = f ( x + y ) g(x)=f(x+y) g ( x ) = f ( x + y ) r s = ζ n y s r_s=\zeta_n^{ys} r s = ζ n y s Z S → ⟨ S ⟩ \mathbb{Z}^S\to\langle S\rangle Z S → ⟨ S ⟩ K ( S ) K(S) K ( S ) K ( S ) K(S) K ( S ) ⟨ S ⟩ \langle S\rangle ⟨ S ⟩ ∣ ⟨ S ⟩ ∣ |\langle S\rangle| ∣ ⟨ S ⟩ ∣
Proposition 3.3 (Direct Galois realization). Let S ⊂ C n S\subset\mathbb{C}_n S ⊂ C n r : S → Q ( ζ n ) × r:S\to\mathbb{Q}(\zeta_n)^{\times} r : S → Q ( ζ n ) ×
r a s = σ a ( r s ) ( a ∈ U n ) , r c = 1 ( c ∈ L ≤ d ( S ) ) . (17) r_{as}=\sigma_a(r_s)\quad(a\in U_n),\qquad r^c=1\quad(c\in\mathcal{L}_{\leq d}(S)). \tag*{(17)} r a s = σ a ( r s ) ( a ∈ U n ) , r c = 1 ( c ∈ L ≤ d ( S )) . (17) If r r r S S S d d d
Proof. Set f ^ = 1 \widehat{f}=1 f = 1 g ^ = r \widehat{g}=r g = r S S S S S S S S S S S S d d d r r r
Choose one positive common denominator D D D f f f g g g D f , D g Df,Dg D f , D g k k k D k D^k D k D D D f , g f,g f , g
There is a useful exact-sequence version of the same argument. Applying Hom ( − , C × ) \operatorname{Hom}(-,\mathbb{C}^{\times}) Hom ( − , C × ) C × \mathbb{C}^{\times} C ×
1 ⟶ ⟨ S ⟩ ^ ⟶ Hom ( A d ( S ) , C × ) ⟶ Hom ( K ( S ) / L ≤ d ( S ) , C × ) ⟶ 1. (18) 1\longrightarrow\widehat{\langle S\rangle}\longrightarrow\operatorname{Hom}(A_d(S),\mathbb{C}^{\times})\longrightarrow\operatorname{Hom}(K(S)/\mathcal{L}_{\leq d}(S),\mathbb{C}^{\times})\longrightarrow1. \tag*{(18)} 1 ⟶ ⟨ S ⟩ ⟶ Hom ( A d ( S ) , C × ) ⟶ Hom ( K ( S ) / L ≤ d ( S ) , C × ) ⟶ 1. (18) The first term is the translation-character subgroup. The last term is the residual complex phase ambiguity. Formula (3.6) also explains why the integral obstruction itself and the phase group are dual objects, not the same object. A free summand in K / L K/\mathcal{L} K / L C × \mathbb{C}^{\times} C × C m C_m C m μ m \mu_m μ m
Example 3.4 (Why exact support is a hypothesis). Suppose a zero-sum tuple contains a frequency at which both Fourier coefficients vanish. Its monomial equality is then 0 = 0 0 = 0 0 = 0
The propositions separate two arithmetic tasks. The lattice calculation classifies all complex multiplicative ratios. On the unit support, Section 5 classifies the complete rational family. For all stable C 2 p q C_{2pq} C 2 pq
Remark 3.5 (Translations versus reflection). The equivalence throughout the paper is translation only. Reflection sends s s s − s -s − s
The ratio-character method and CRT patching already occur in the classical cyclic work of Grünbaum–Moore [8 ] . Our later use is a support-sensitive integral classification, not a claim of ownership of that method.
The unit-support lattice for every even modulus
We now compute the integral quotient in Theorem 1.4. The main input is a universal ambient group for a sifted arithmetic progression. For q ≥ 1 q \ge1 q ≥ 1 c ∈ Z c \in\mathbb{Z} c ∈ Z
T q , c = { t ∈ C q : gcd ( 1 + c t , q ) = 1 } . T_{q,c} = \{t \in C_q : \gcd(1 + ct,q) = 1\}. T q , c = { t ∈ C q : g cd( 1 + c t , q ) = 1 } . For T ⊂ C q T \subset C_q T ⊂ C q UAG ( T ) \operatorname{UAG}(T) UAG ( T ) Z T \mathbb{Z}^{T} Z T
e a + e b − e a ′ − e b ′ ( a + b = a ′ + b ′ in C q ) . e_a + e_b - e_{a'} - e_{b'} \qquad(a + b = a' + b' \text{ in } C_q). e a + e b − e a ′ − e b ′ ( a + b = a ′ + b ′ in C q ) . This is the universal target for Freiman homomorphisms on T T T [14 , 9 ] .
Call a prime divisor of q q q restricting if it does not divide c c c d d d d ∈ { 1 , 2 , 3 , 6 } d \in\{1, 2, 3, 6\} d ∈ { 1 , 2 , 3 , 6 }
Theorem 4.1 (Sifted universal ambient group). Assume every restricting prime not dividing d d d Then
UAG ( T q , c ) ≅ Z φ ( d ) ⊕ C q / d . (19) \operatorname{UAG}(T_{q,c}) \cong\mathbb{Z}^{\varphi(d)} \oplus C_{q/d}. \tag*{(19)} UAG ( T q , c ) ≅ Z φ ( d ) ⊕ C q / d . (19) Proof. Write T = T q , c T = T_{q,c} T = T q , c s , s ′ ∈ d C q s,s' \in dC_q s , s ′ ∈ d C q x x x
x , x + s ∈ T , x , x + s , x + s + s ′ ∈ T . (20) x, x+s \in T,\qquad x,x+s,x+s+s' \in T. \tag*{(20)} x , x + s ∈ T , x , x + s , x + s + s ′ ∈ T . (20) Indeed, at a prime dividing d d d
At a restricting prime power ℓ α ∥ q \ell^\alpha\mathbin{\|} q ℓ α ∥ q T T T ℓ \ell ℓ ℓ α \ell^\alpha ℓ α x x x ℓ \ell ℓ ℓ a \ell^{a} ℓ a ℓ ∣ c \ell\mid c ℓ ∣ c 1 + c x 1+cx 1 + c x ℓ \ell ℓ d d d
Let W W W d d d ∣ W ∣ = φ ( d ) |W|=\varphi(d) ∣ W ∣ = φ ( d ) W W W d = 2 , 6 d=2,6 d = 2 , 6 d d d UAG ( T ) \operatorname{UAG}(T) UAG ( T )
The multiset conclusion is stronger than preservation of the total class sum. Modulo three, translate the admissible classes to { 0 , 1 } \{0,1\} { 0 , 1 } { 0 , 0 } \{0,0\} { 0 , 0 } { 0 , 1 } \{0,1\} { 0 , 1 } { 1 , 1 } \{1,1\} { 1 , 1 } 0 , 1 , 2 0,1,2 0 , 1 , 2 d = 2 , 6 d=2,6 d = 2 , 6
For s ∈ d C q s\in dC_q s ∈ d C q
Δ ( s ) = [ e x + s ] − [ e x ] , \Delta(s)=[e_{x+s}]-[e_x], Δ ( s ) = [ e x + s ] − [ e x ] , where x , x + s ∈ T x,x+s\in T x , x + s ∈ T ( x + s ) + y = x + ( y + s ) (x+s)+y=x+(y+s) ( x + s ) + y = x + ( y + s ) x x x Δ ( s + s ′ ) = Δ ( s ) + Δ ( s ′ ) \Delta(s+s')=\Delta(s)+\Delta(s') Δ ( s + s ′ ) = Δ ( s ) + Δ ( s ′ ) D = Δ ( d ) D=\Delta(d) D = Δ ( d ) q / d q/d q / d
Choose t w ∈ T t_w\in T t w ∈ T w ∈ W w\in W w ∈ W
[ e t ] = [ e t w ] + Δ ( t − t w ) . [e_t]=[e_{t_w}]+\Delta(t-t_w). [ e t ] = [ e t w ] + Δ ( t − t w ) . Define
Ψ ( e t ) = ( e w , t − t w d ) ∈ Z W ⊕ C q / d . (21) \Psi(e_t)=\left(e_w,\frac{t-t_w}{d}\right)\in\mathbb{Z}^{W}\oplus C_{q/d}. \tag*{(21)} Ψ ( e t ) = ( e w , d t − t w ) ∈ Z W ⊕ C q / d . (21) Class-multiset preservation cancels the first coordinates of a Freiman row, and equality of the original sums cancels the second. Hence Ψ \Psi Ψ D D D 1 1 1 UAG ( T ) \operatorname{UAG}(T) UAG ( T ) Ψ \Psi Ψ
Equivalently, a Freiman homomorphism F : T q , c → A F:T_{q,c}\to A F : T q , c → A w w w d d d D D D q / d q/d q / d
F ( t ) = F ( t w ) + t − t w d D . (22) F(t)=F(t_w)+\frac{t-t_w}{d}D. \tag*{(22)} F ( t ) = F ( t w ) + d t − t w D . (22) There is no relation among the φ ( d ) \varphi(d) φ ( d ) d = 3 d=3 d = 3
Fourth-order dichotomy
Write n = 2 N n=2N n = 2 N t ↦ u ( t ) = 1 + 2 t t\mapsto u(t)=1+2t t ↦ u ( t ) = 1 + 2 t T N , 2 T_{N,2} T N , 2 U n U_n U n ι ( t ) = − 1 − t \iota(t)=-1-t ι ( t ) = − 1 − t U n U_n U n
u ( a ) + u ( b ) + u ( c ) + u ( d ) = 0 u(a) + u(b) + u(c) + u(d) = 0 u ( a ) + u ( b ) + u ( c ) + u ( d ) = 0 into a + b = ι ( c ) + ι ( d ) a+b=\iota(c)+\iota(d) a + b = ι ( c ) + ι ( d ) C N C_N C N
Z U n / L ≤ 4 ( U n ) ≅ UAG ( T N , 2 ) / ⟨ inverse rows ⟩ . (23) \mathbb{Z}^{U_n}/\mathcal{L}_{\leq4}(U_n) \cong\operatorname{UAG}(T_{N,2})/\langle\text{inverse rows}\rangle. \tag*{(23)} Z U n / L ≤ 4 ( U n ) ≅ UAG ( T N , 2 ) / ⟨ inverse rows ⟩ . (23) Proposition 4.2 (All-even unit quotient). For every even n n n
Z U n / L ≤ 4 ( U n ) ≅ { C n , 3 ∤ n , Z ⊕ C n / 6 , 6 ∣ n . (24) \mathbb{Z}^{U_n}/\mathcal{L}_{\leq4}(U_n) \cong
\begin{cases}
C_n, & 3\nmid n,\\
\mathbb{Z}\oplus C_{n/6}, & 6\mid n.
\end{cases} \tag*{(24)} Z U n / L ≤ 4 ( U n ) ≅ { C n , Z ⊕ C n /6 , 3 ∤ n , 6 ∣ n . (24) Moreover L ≤ 5 ( U n ) = L ≤ 4 ( U n ) \mathcal{L}_{\leq5}(U_n)=\mathcal{L}_{\leq4}(U_n) L ≤ 5 ( U n ) = L ≤ 4 ( U n )
Proof. If 3 ∤ n 3\nmid n 3 ∤ n Z ⊕ C N \mathbb{Z}\oplus C_N Z ⊕ C N e t ↦ ( 1 , t ) e_t\mapsto(1,t) e t ↦ ( 1 , t ) ( 2 , − 1 ) (2,-1) ( 2 , − 1 ) A A A B B B B = 2 A B=2A B = 2 A 2 N A = 0 2NA=0 2 N A = 0 C 2 N = C n C_{2N}=C_n C 2 N = C n
If n = 6 m n=6m n = 6 m Z 2 ⊕ C m \mathbb{Z}^2\oplus C_m Z 2 ⊕ C m 0 , 2 0,2 0 , 2 ( 1 , 1 , κ ) (1,1,\kappa) ( 1 , 1 , κ ) κ ∈ C m \kappa\in C_m κ ∈ C m ( 1 , 1 ) (1,1) ( 1 , 1 ) Z ⊕ C m \mathbb{Z}\oplus C_m Z ⊕ C m (23) excludes every odd row, proving the last assertion.
In the second branch, the torsion absorption can be made explicit. Let A 0 , A 2 A_0,A_2 A 0 , A 2 B B B C m C_m C m A 2 A_2 A 2 A 2 + κ B A_2+\kappa B A 2 + κ B Z 2 ⊕ C m \mathbb{Z}^2\oplus C_m Z 2 ⊕ C m A 0 + A 2 A_0+A_2 A 0 + A 2 Z ( A 0 − A 2 ) ⊕ C m \mathbb{Z}(A_0-A_2)\oplus C_m Z ( A 0 − A 2 ) ⊕ C m
The target map supplies a useful check. In the generic branch the surviving generator maps to 1 ∈ C n 1\in C_n 1 ∈ C n n n n C m C_m C m
Degree-six closure
Suppose n = 6 m n=6m n = 6 m
v t = 1 + 6 t , T m = { t ∈ C m : gcd ( 1 + 6 t , m ) = 1 } . v_t=1+6t,\qquad T_m=\{t\in C_m:\gcd(1+6t,m)=1\}. v t = 1 + 6 t , T m = { t ∈ C m : g cd( 1 + 6 t , m ) = 1 } . Then U 6 m = { ± v t : t ∈ T m } U_{6m}=\{\pm v_t:t\in T_m\} U 6 m = { ± v t : t ∈ T m }
e v t ⟼ ( 1 , t ) , e − v t ⟼ ( − 1 , − t ) in Z ⊕ C m . (25) e_{v_t}\longmapsto(1,t),\qquad e_{-v_t}\longmapsto(-1,-t)\quad\text{in }\mathbb{Z}\oplus C_m. \tag*{(25)} e v t ⟼ ( 1 , t ) , e − v t ⟼ ( − 1 , − t ) in Z ⊕ C m . (25) Every row of length at most five maps to zero. Its first coordinate is the difference between the numbers of residues congruent to 1 1 1 − 1 -1 − 1 (25) is generated by ( 6 , − 1 ) (6,-1) ( 6 , − 1 )
Lemma 4.3 (Sifted two-sum lemma). T m + T m = C m T_m+T_m=C_m T m + T m = C m
Proof. For prescribed y ∈ C m y\in C_m y ∈ C m a a a a a a y − a y-a y − a T m T_m T m a a a
Prime by prime, the two excluded classes are explicitly − 6 − 1 -6^{-1} − 6 − 1 y + 6 − 1 y+6^{-1} y + 6 − 1 ℓ ∤ 6 \ell\nmid6 ℓ ∤ 6 ℓ ≥ 5 \ell\geq5 ℓ ≥ 5 ℓ = 2 \ell=2 ℓ = 2 3 3 3 1 + 6 a 1+6a 1 + 6 a ℓ \ell ℓ ℓ \ell ℓ m m m
Choose three pairs in T m T_m T m − 1 , 0 , 0 -1,0,0 − 1 , 0 , 0 6 + 6 ( − 1 ) = 0 6+6(-1)=0 6 + 6 ( − 1 ) = 0 6 m 6m 6 m ( 6 , − 1 ) (6,-1) ( 6 , − 1 )
Every row of length at most five killed the entire presentation in (25) , whereas ( 6 , − 1 ) (6,-1) ( 6 , − 1 )
Corollary 4.4. If 6 ∣ n 6\mid n 6 ∣ n L ≤ 6 ( U n ) = K ( U n ) \mathcal{L}_{\leq6}(U_n)=K(U_n) L ≤ 6 ( U n ) = K ( U n ) (24) proves nonclosure through degree five.
Proposition 4.2 and Corollary 4.4 prove the lattice part of Theorem 1.4. The global order-six reconstruction theorem of Grünbaum–Moore [8 ] is compatible with this bound; the explicit support quotient and its closing row are the calculations above.
Rational signals on the exact unit support
We now add the Galois condition that is absent from the integral calculation. If 3 ∤ n 3\nmid n 3 ∤ n L ≤ 4 ( U n ) = K ( U n ) \mathcal{L}_{\leq4}(U_n)=K(U_n) L ≤ 4 ( U n ) = K ( U n )
Let n = 6 m n=6m n = 6 m L ≤ 5 ( U 6 m ) = L ≤ 4 ( U 6 m ) \mathcal{L}_{\leq5}(U_{6m})=\mathcal{L}_{\leq4}(U_{6m}) L ≤ 5 ( U 6 m ) = L ≤ 4 ( U 6 m ) (25) , such a character has
r ( v t ) = a b t , r ( − v t ) = a − 1 b − t , b ∈ μ m . (26) r(v_t)=ab^t,\qquad r(-v_t)=a^{-1}b^{-t},\qquad b\in\mu_m. \tag*{(26)} r ( v t ) = a b t , r ( − v t ) = a − 1 b − t , b ∈ μ m . (26) Indeed, a character of Z ⊕ C m \mathbb{Z}\oplus C_m Z ⊕ C m a a a ( 1 , 0 ) (1,0) ( 1 , 0 ) b b b ( 0 , 1 ) (0,1) ( 0 , 1 ) b m = 1 b^m=1 b m = 1
A translation by y y y ( a , b ) (a,b) ( a , b ) ( ζ n y , ζ m y ) (\zeta_n^y,\zeta_m^y) ( ζ n y , ζ m y ) y y y m m m b = 1 b=1 b = 1
ε ( u ) = { 1 , u ≡ 1 ( m o d 6 ) , − 1 , u ≡ − 1 ( m o d 6 ) . \varepsilon(u)=
\begin{cases}
1, & u\equiv1\pmod{6},\\
-1, & u\equiv-1\pmod{6}.
\end{cases} ε ( u ) = { 1 , − 1 , u ≡ 1 ( mod 6 ) , u ≡ − 1 ( mod 6 ) . The normalized ratio is then
r u = z ε ( u ) . (27) r_u=z^{\varepsilon(u)}. \tag*{(27)} r u = z ε ( u ) . (27) Theorem 5.1 (Exact-unit rational descent). Let f , g : C 6 m → Q f,g:C_{6m}\to\mathbb{Q} f , g : C 6 m → Q U 6 m U_{6m} U 6 m g g g α ∈ Q ( ζ 6 m ) × \alpha\in\mathbb{Q}(\zeta_{6m})^{\times} α ∈ Q ( ζ 6 m ) × z ∈ Q ( ζ 6 ) × z\in\mathbb{Q}(\zeta_6)^{\times} z ∈ Q ( ζ 6 ) ×
f ^ ( u ) = σ u ( α ) , g ^ ( u ) = σ u ( z α ) , z ‾ z = 1. (28) \widehat{f}(u)=\sigma_u(\alpha),\qquad\widehat{g}(u)=\sigma_u(z\alpha),\qquad\overline{z}z=1. \tag*{(28)} f ( u ) = σ u ( α ) , g ( u ) = σ u ( z α ) , z z = 1. (28) Conversely every such pair, extended by zero off the unit support, is rational-valued and agrees through order five, hence also through order four. Two normalized parameters are translation-equivalent exactly modulo μ 6 \mu_6 μ 6 z 6 = 1 z^6=1 z 6 = 1
Proof. Rationality gives Galois covariance (14) . Applying it to the ratio at u = 1 u=1 u = 1
σ a ( z ) = z ε ( a ) ( a ∈ U 6 m ) . (29) \sigma_a(z)=z^{\varepsilon(a)}\qquad(a\in U_{6m}). \tag*{(29)} σ a ( z ) = z ε ( a ) ( a ∈ U 6 m ) . (29) The kernel of ε : U 6 m → { ± 1 } \varepsilon:U_{6m}\to\{\pm1\} ε : U 6 m → { ± 1 } ζ 6 \zeta_6 ζ 6 Q ( ζ 6 ) \mathbb{Q}(\zeta_6) Q ( ζ 6 ) z ∈ Q ( ζ 6 ) × z\in\mathbb{Q}(\zeta_6)^\times z ∈ Q ( ζ 6 ) × a = − 1 a=-1 a = − 1 (29) gives z ‾ = z − 1 \overline{z}=z^{-1} z = z − 1
To justify the fixed field statement without a degree shortcut, restriction of automorphisms gives the surjection
Gal ( Q ( ζ 6 m ) / Q ) ⟶ Gal ( Q ( ζ 6 ) / Q ) ≅ { ± 1 } . \operatorname{Gal}(\mathbb{Q}(\zeta_{6m})/\mathbb{Q})\longrightarrow\operatorname{Gal}(\mathbb{Q}(\zeta_6)/\mathbb{Q})\cong\{\pm1\}. Gal ( Q ( ζ 6 m ) / Q ) ⟶ Gal ( Q ( ζ 6 ) / Q ) ≅ { ± 1 } . It is exactly a ↦ ε ( a ) a\mapsto\varepsilon(a) a ↦ ε ( a ) Q ( ζ 6 ) \mathbb{Q}(\zeta_6) Q ( ζ 6 ) (29) says that every element of this kernel fixes z z z
Put α = f ^ ( 1 ) \alpha=\widehat{f}(1) α = f ( 1 ) (28) , and (27) –(29) give the second. Conversely, for arbitrary nonzero α \alpha α z z z (28) is Galois equivariant and therefore defines rational signals with exact support U 6 m U_{6m} U 6 m ε \varepsilon ε (25) , so its ratio product is one. Proposition 3.2 proves equality of the autocorrelations.
The exact-support assertion in the converse is immediate but worth recording: every conjugate σ u ( α ) \sigma_u(\alpha) σ u ( α ) σ u ( z α ) \sigma_u(z\alpha) σ u ( z α ) (15) then gives rational time-domain values, and the automorphism u = − 1 u=-1 u = − 1
Translations preserving the normalization b = 1 b=1 b = 1 m ∣ y m\mid y m ∣ y ζ 6 m y \zeta_{6m}^{y} ζ 6 m y μ 6 \mu_6 μ 6 ε \varepsilon ε 0 0 0 ± 6 \pm6 ± 6 Lemma 4.3 has sum six. Hence the sixth-order data read exactly z 6 z^6 z 6 z ∈ μ 6 z\in\mu_6 z ∈ μ 6
The norm-one parameter is a one-dimensional rational torus. Since Q ( ζ 6 ) = Q ( − 3 ) \mathbb{Q}(\zeta_6)=\mathbb{Q}(\sqrt{-3}) Q ( ζ 6 ) = Q ( − 3 ) − 1 -1 − 1
z = a + b − 3 a − b − 3 ( a , b ∈ Q ) , (30) z=\frac{a+b\sqrt{-3}}{a-b\sqrt{-3}}\qquad(a,b\in\mathbb{Q}), \tag*{(30)} z = a − b − 3 a + b − 3 ( a , b ∈ Q ) , (30) with − 1 -1 − 1 a = 0 a=0 a = 0 μ 6 \mu_6 μ 6
This proves all signal-level assertions in Theorem 1.4 . The exact-support hypothesis is essential: it licenses every Fourier division in Proposition 3.2 and in (26) .
The C 30 C_{30} C 30
At n = 30 n=30 n = 30 Theorem 5.1 gives the order-four and order-five classification stated in Corollary 1.6. The displayed construction of Agulnick–Busick-Warner [1 ] uses a common amplitude in a quadratic subfield. The complete classification permits α \alpha α Q ( ζ 30 ) × \mathbb{Q}(\zeta_{30})^{\times} Q ( ζ 30 ) × Q ( ζ 6 ) \mathbb{Q}(\zeta_{6}) Q ( ζ 6 )
For example, choose any primitive α ∈ Q ( ζ 30 ) × \alpha\in\mathbb{Q}(\zeta_{30})^{\times} α ∈ Q ( ζ 30 ) × (30) with z 6 ≠ 1 z^{6} \ne1 z 6 = 1 (28) form a rational pair agreeing through order five, and therefore through order four, but not order six. Changing α \alpha α z z z
Scope boundary. Theorem 5.1 classifies rational signal pairs only on the exact unit support. Later sections use the same lattice formalism on all 255 stable supports, but do not claim that every integral obstruction there has a rational signal realization.
Local CRT box presentations for C 2 p q C_{2pq} C 2 pq
Fix distinct odd primes p < q p<q p < q C 2 p q = C 2 × C p × C q C_{2pq}=C_{2}\times C_{p}\times C_{q} C 2 pq = C 2 × C p × C q
( E , P , Q , P 1 , Q 1 , F , G ) = ( O 2 , O p , O q , O 2 p , O 2 q , O p q , O 2 p q ) (31) (E,P,Q,P1,Q1,F,G)=\left(\mathcal{O}_{2},\mathcal{O}_{p},\mathcal{O}_{q},\mathcal{O}_{2p},\mathcal{O}_{2q},\mathcal{O}_{pq},\mathcal{O}_{2pq}\right) \tag*{(31)} ( E , P , Q , P 1 , Q 1 , F , G ) = ( O 2 , O p , O q , O 2 p , O 2 q , O pq , O 2 pq ) (31) these are the seven nonzero automorphism orbits. The orbit O 1 = { 0 } \mathcal{O}_{1}=\{0\} O 1 = { 0 }
The name “box” records the nonzero CRT coordinates. With a ∈ C p × a\in C_{p}^{\times} a ∈ C p × b ∈ C q × b\in C_{q}^{\times} b ∈ C q ×
Table 6.1a.
Thus a stable support is a choice of boxes and possibly O 1 \mathcal{O}_{1} O 1 2 7 − 1 = 127 2^{7}-1=127 2 7 − 1 = 127 2 ( 127 ) + 1 = 255 2(127)+1=255 2 ( 127 ) + 1 = 255
The quotient A d ( B ) A_{d}(B) A d ( B ) h : B → H h:B\to H h : B → H d d d A d ( B ) A_{d}(B) A d ( B )
For a single box B B B p = 3 p=3 p = 3
Equivalently, at degree four a homomorphism from a selected box quotient to an arbitrary abelian target has the universal normal forms
E ∣ c , 2 c = 0 , P ∣ a u , p u = 0 , Q ∣ b v , q v = 0 , P 1 ( p ≥ 5 ) ∣ c + a u , 2 c = p u = 0 , P 1 ( p = 3 ) ∣ a u , u free , Q 1 ∣ c + b v , 2 c = q v = 0 , F ∣ a u + b v , p u = q v = 0 , G ( p ≥ 5 ) ∣ c + a u + b v , 2 c = p u = q v = 0 , G ( p = 3 ) ∣ a u + b v , u free , q v = 0. (32) \begin{array}{rcl}
E &\mid& c,\ 2c=0,\\
P &\mid& au,\ pu=0,\\
Q &\mid& bv,\ qv=0,\\
P1\ (p\geq5) &\mid& c+au,\ 2c=pu=0,\\
P1\ (p=3) &\mid& au,\ u\ \text{free},\\
Q1 &\mid& c+bv,\ 2c=qv=0,\\
F &\mid& au+bv,\ pu=qv=0,\\
G\ (p\geq5) &\mid& c+au+bv,\ 2c=pu=qv=0,\\
G\ (p=3) &\mid& au+bv,\ u\ \text{free},\ qv=0. \tag*{(32)}
\end{array} E P Q P 1 ( p ≥ 5 ) P 1 ( p = 3 ) Q 1 F G ( p ≥ 5 ) G ( p = 3 ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ c , 2 c = 0 , a u , p u = 0 , b v , q v = 0 , c + a u , 2 c = p u = 0 , a u , u free , c + b v , 2 c = q v = 0 , a u + b v , p u = q v = 0 , c + a u + b v , 2 c = p u = q v = 0 , a u + b v , u free , q v = 0. (32) Proposition 6.1 (Local box theorem). Table 2 and the normal forms in (6.2) hold uniformly for all admissible p , q p,q p , q
Proof. Every nontrivial box is stable under negation without fixed points. Pair rows identify opposite generators and give the free ranks in the A 2 A_2 A 2 E E E 2 e E 2e_E 2 e E C 2 C_2 C 2
On a parity-zero punctured prime line, pairs and triples give partial additivity f ( x + y ) = f ( x ) + f ( y ) f(x+y)=f(x)+f(y) f ( x + y ) = f ( x ) + f ( y ) f ( x ) = x u f(x)=xu f ( x ) = xu r u = 0 ru=0 r u = 0 P , Q P,Q P , Q
To see the punctured-line step explicitly, fix x ≠ 0 x\neq0 x = 0 f ( k x ) = k f ( x ) f(kx)=kf(x) f ( k x ) = k f ( x ) r f ( x ) = 0 rf(x)=0 r f ( x ) = 0 x x x u = x − 1 f ( x ) u=x^{-1}f(x) u = x − 1 f ( x ) f ( a ) = a u f(a)=au f ( a ) = a u C r C_r C r
On a parity-one punctured line there are no odd rows. For r ≥ 5 r\geq5 r ≥ 5 ( x , 1 , 1 , − x − 2 ) (x,1,1,-x-2) ( x , 1 , 1 , − x − 2 ) f ( x + 2 ) = f ( x ) + 2 f ( 1 ) f(x+2)=f(x)+2f(1) f ( x + 2 ) = f ( x ) + 2 f ( 1 ) f ( x ) = c + x u f(x)=c+xu f ( x ) = c + xu 2 c = 0 2c=0 2 c = 0 r ≥ 7 r\geq7 r ≥ 7 ( 2 , 2 , 2 , r − 6 ) (2,2,2,r-6) ( 2 , 2 , 2 , r − 6 ) 4 c + r u = 0 4c+ru=0 4 c + r u = 0 r u = 0 ru=0 r u = 0 r = 5 r=5 r = 5 ( 1 , 1 , 1 , 2 ) (1,1,1,2) ( 1 , 1 , 1 , 2 ) 4 c + 5 u = 0 4c+5u=0 4 c + 5 u = 0 5 u = 0 5u=0 5 u = 0 r = 3 r=3 r = 3 P 1 P1 P 1 Q 1 Q1 Q 1 q = 5 q=5 q = 5
The exception at three is structural. When r ≥ 5 r\geq5 r ≥ 5 r = 3 r=3 r = 3 C 2 C_2 C 2 C 3 C_3 C 3
For F F F p , q ≥ 5 p,q\ge5 p , q ≥ 5 s s s
( x + s , y ) + ( c , d ) = ( x , y ) + ( c + s , d ) . (33) (x+s,y)+(c,d)=(x,y)+(c+s,d). \tag*{(33)} ( x + s , y ) + ( c , d ) = ( x , y ) + ( c + s , d ) . (33) An auxiliary ( c , d ) (c,d) ( c , d ) p p p q q q p = 5 p=5 p = 5 f ( a , b ) = c + a u + b v f(a,b)=c+au+bv f ( a , b ) = c + a u + b v p u = q v = 0 pu=qv=0 p u = q v = 0 2 c = 0 2c=0 2 c = 0 3 c = 0 3c=0 3 c = 0 c = 0 c=0 c = 0
When p = 3 p=3 p = 3 h ( b ) = f ( 1 , b ) h(b)=f(1,b) h ( b ) = f ( 1 , b ) h h h q q q q = 5 q=5 q = 5 h ( b ) = u + b v h(b)=u+bv h ( b ) = u + b v 3 u = 0 3u=0 3 u = 0 q v = 0 qv=0 q v = 0 A 3 ( F ) = C 3 ⊕ C q A_3(F)=C_3\oplus C_q A 3 ( F ) = C 3 ⊕ C q
For G G G p , q ≥ 5 p,q\ge5 p , q ≥ 5 f ( a , b ) = c + a u + b v f(a,b)=c+au+bv f ( a , b ) = c + a u + b v 2 c = 0 2c=0 2 c = 0 C 2 , C p , C q C_2,C_p,C_q C 2 , C p , C q
Finally let p = 3 p=3 p = 3 h ( b ) = f ( 1 , b ) h(b)=f(1,b) h ( b ) = f ( 1 , b ) C q × C_q^\times C q × h ( b ) = u + b v h(b)=u+bv h ( b ) = u + b v u u u q v = 0 qv=0 q v = 0 A 4 ( G ) = Z ⊕ C q A_4(G)=\mathbb{Z}\oplus C_q A 4 ( G ) = Z ⊕ C q
The avoidance arguments are uniform at the small endpoints. The two points being compared are fixed before the auxiliary point is selected. At most four p p p q q q p = 5 p=5 p = 5 p − 3 p-3 p − 3 p = 3 p=3 p = 3
Finally, the three CRT coordinate characters restrict to the parity constant, the p p p q q q P 1 , G P1,G P 1 , G
The free sign in the last row is the unique local mechanism capable of surviving through degree five. It is the local form of the exceptional unit support in § 4 .
Corollary 6.2 (Compatibility on the unit box). For G = Z 2 p q = U 2 p q G=\mathbb{Z}_{2pq}=U_{2pq} G = Z 2 pq = U 2 pq
δ ( G ) = 6 ( p = 3 ) , δ ( G ) = 4 ( p , q ≥ 5 ) . \delta(G)=6 \quad(p=3),\qquad\delta(G)=4 \quad(p,q\ge5). δ ( G ) = 6 ( p = 3 ) , δ ( G ) = 4 ( p , q ≥ 5 ) . Proof. If p = 3 p=3 p = 3 Z ⊕ C q \mathbb{Z}\oplus C_q Z ⊕ C q § 9 shows exact depth six. Independently, Proposition 4.2 and Corollary 4.4 give the same conclusion for n = 6 q n=6q n = 6 q p , q ≥ 5 p,q\ge5 p , q ≥ 5 3 ∤ 2 p q 3\nmid2pq 3 ∤ 2 pq
The Cubic Skeleton
Represent a nonempty union of the seven boxes (6.1) by its Boolean mask in the order ( E , P , Q , P 1 , Q 1 , F , G ) (E,P,Q,P1,Q1,F,G) ( E , P , Q , P 1 , Q 1 , F , G ) E E E
A homomorphism A 3 ( S ) → H A_3(S) \to H A 3 ( S ) → H S S S ( η , α , β ) (\eta,\alpha,\beta) ( η , α , β )
2 η = 0 , p α = 0 , q β = 0 , (34) 2\eta=0,\qquad p\alpha=0,\qquad q\beta=0, \tag*{(34)} 2 η = 0 , p α = 0 , q β = 0 , (34) and the seven restrictions
E η P [ a ] a α Q [ b ] b β P 1 [ a ] η + a α Q 1 [ b ] η + b β F [ a , b ] a α + b β G [ a , b ] η + a α + b β \begin{array}{c|c c|c}
E & \eta& P[a] & a\alpha\\
Q[b] & b\beta& P1[a] & \eta+a\alpha\\
Q1[b] & \eta+b\beta& F[a,b] & a\alpha+b\beta\\
G[a,b] & \eta+a\alpha+b\beta& &
\end{array} E Q [ b ] Q 1 [ b ] G [ a , b ] η b β η + b β η + a α + b β P [ a ] P 1 [ a ] F [ a , b ] a α η + a α a α + b β The local cubic equations already force the displayed linear forms on P , Q , F P,Q,F P , Q , F x ( a ) , y ( b ) , z ( a , b ) x(a),y(b),z(a,b) x ( a ) , y ( b ) , z ( a , b )
T 3 = ( F ∧ G ) ∨ ( P ∧ P 1 ∧ F ) ∨ ( Q ∧ Q 1 ∧ F ) ∨ ( E ∧ P ∧ Q ∧ P 1 ) ∨ ( E ∧ P ∧ Q ∧ Q 1 ) ∨ ( P ∧ ¬ Q 1 ∧ ¬ G ) ∨ ( Q ∧ ¬ P 1 ∧ ¬ G ) ∨ ( F ∧ ¬ P 1 ∧ ¬ Q 1 ) ∨ ( P ∧ Q ∧ G ∧ ¬ E ) ∨ ( P 1 ∧ Q 1 ∧ F ∧ ¬ E ) . (35) \begin{aligned}
T_3={}&(F\land G)\lor(P\land P1\land F)\lor(Q\land Q1\land F)\\
&\lor(E\land P\land Q\land P1)\lor(E\land P\land Q\land Q1)\\
&\lor(P\land\neg Q1\land\neg G)\lor(Q\land\neg P1\land\neg G)\\
&\lor(F\land\neg P1\land\neg Q1)\lor(P\land Q\land G\land\neg E)\\
&\lor(P1\land Q1\land F\land\neg E). \tag*{(35)}
\end{aligned} T 3 = ( F ∧ G ) ∨ ( P ∧ P 1 ∧ F ) ∨ ( Q ∧ Q 1 ∧ F ) ∨ ( E ∧ P ∧ Q ∧ P 1 ) ∨ ( E ∧ P ∧ Q ∧ Q 1 ) ∨ ( P ∧ ¬ Q 1 ∧ ¬ G ) ∨ ( Q ∧ ¬ P 1 ∧ ¬ G ) ∨ ( F ∧ ¬ P 1 ∧ ¬ Q 1 ) ∨ ( P ∧ Q ∧ G ∧ ¬ E ) ∨ ( P 1 ∧ Q 1 ∧ F ∧ ¬ E ) . (35) Theorem 7.1 (Cubic iff theorem). A nonempty seven-box mask other than E E E
Exhaustive triple types
A zero-sum triple contains zero or two parity-one points. Coordinate feasibility gives exactly the following nineteen box-count patterns:
P P P , Q Q Q , F F F , P F F , Q F F , P Q F , F G G , Q 1 F G , P 1 F G , PPP,\ QQQ,\ FFF,\ PFF,\ QFF,\ PQF,\ FGG,\ Q1FG,\ P1FG, P P P , QQQ , F F F , P F F , QF F , P QF , F GG , Q 1 F G , P 1 F G , P 1 Q 1 F , Q G G , Q Q 1 Q 1 , Q P 1 G , P G G , P Q 1 G , P P 1 P 1 , P1Q1F,\ QGG,\ QQ1Q1,\ QP1G,\ PGG,\ PQ1G,\ PP1P1, P 1 Q 1 F , QGG , QQ 1 Q 1 , QP 1 G , P GG , P Q 1 G , P P 1 P 1 , E F G , E Q Q 1 , E P P 1. (36) EFG,\ EQQ1,\ EPP1. \tag*{(36)} E F G , E QQ 1 , E P P 1. (36) Up to exchanging p p p q q q
P + 2 P 1 , P + 2 F , P + Q + F , P + 2 G , F + 2 G , P+2P1,\quad P+2F,\quad P+Q+F,\quad P+2G,\quad F+2G, P + 2 P 1 , P + 2 F , P + Q + F , P + 2 G , F + 2 G , E + P + P 1 , E + F + G , Q + P 1 + G , P 1 + F + G , P 1 + Q 1 + F . (37) E+P+P1,\quad E+F+G,\quad Q+P1+G,\quad P1+F+G,\quad P1+Q1+F. \tag*{(37)} E + P + P 1 , E + F + G , Q + P 1 + G , P 1 + F + G , P 1 + Q 1 + F . (37) There is no parity-compatible twentieth pattern.
Here is the coordinate check behind exhaustiveness. The parity coordinate of a triple is zero only when it contains zero or two points from P 1 , Q 1 , G P1,Q1,G P 1 , Q 1 , G P , Q , F P,Q,F P , Q , F F G G FGG F GG Q G G QGG QGG P G G PGG P GG Q Q 1 Q 1 QQ1Q1 QQ 1 Q 1 P P 1 P 1 PP1P1 P P 1 P 1 Q 1 F G Q1FG Q 1 F G P 1 F G P1FG P 1 F G P 1 Q 1 F P1Q1F P 1 Q 1 F Q P 1 G QP1G QP 1 G P Q 1 G PQ1G P Q 1 G E F G EFG E F G E Q Q 1 EQQ1 E QQ 1 E P P 1 EPP1 E P P 1
Sufficiency
We use five propagation facts, all valid with values in an arbitrary abelian group.
Lemma 7.2 (Cubic propagation). The equations in (7.3) have the following consequences.
If P P P P 1 P1 P 1 P + 2 P 1 P + 2P1 P + 2 P 1 x ( a ) = η + a α x(a) = \eta+ a\alpha x ( a ) = η + a α 2 η = 0 2\eta= 0 2 η = 0
P + 2 F P + 2F P + 2 F P P P F F F p p p
If P P P G G G z ( a , b ) = a α + v ( b ) z(a,b) = a\alpha+ v(b) z ( a , b ) = a α + v ( b ) v v v Q Q Q G G G z ( a , b ) = η + a α + b β z(a,b) = \eta+ a\alpha+ b\beta z ( a , b ) = η + a α + b β 2 η = 0 2\eta= 0 2 η = 0
If F F F G G G z ( a , b ) = η + a ϕ + b ψ z(a,b) = \eta+ a\phi+ b\psi z ( a , b ) = η + a ϕ + b ψ 2 η = 0 2\eta= 0 2 η = 0
The rows E + P + P 1 E + P + P1 E + P + P 1 E + Q + Q 1 E + Q + Q1 E + Q + Q 1 E + F + G E + F + G E + F + G E E E Q + P 1 + G Q + P1 + G Q + P 1 + G P + Q 1 + G P + Q1 + G P + Q 1 + G z ( a , b ) = x ( a ) + b β z(a,b) = x(a) + b\beta z ( a , b ) = x ( a ) + b β z ( a , b ) = a α + y ( b ) z(a,b) = a\alpha+ y(b) z ( a , b ) = a α + y ( b ) x ( a ) = η + a ϕ x(a) = \eta+ a\phi x ( a ) = η + a ϕ P 1 + F + G P1 + F + G P 1 + F + G z ( a , b ) = η + a ϕ + b ψ z(a,b) = \eta+ a\phi+ b\psi z ( a , b ) = η + a ϕ + b ψ P 1 + Q 1 + F P1 + Q1 + F P 1 + Q 1 + F x ( a ) − a ϕ = − ( y ( b ) − b ψ ) = η x(a) - a\phi= -(y(b) - b\psi) = \eta x ( a ) − a ϕ = − ( y ( b ) − b ψ ) = η 2 η = 0 2\eta= 0 2 η = 0
Proof. For (1), P + 2 P 1 P + 2P1 P + 2 P 1 x ( a ) + x ( c ) = ( a + c ) α x(a) + x(c) = (a + c)\alpha x ( a ) + x ( c ) = ( a + c ) α h ( a ) = x ( a ) − a α h(a) = x(a) - a\alpha h ( a ) = x ( a ) − a α c = a c = a c = a 2 h ( a ) = 0 2h(a) = 0 2 h ( a ) = 0 c c c h h h p = 3 p = 3 p = 3
For (2), P + 2 F P + 2F P + 2 F 2 a ( ϕ − α ) = 0 2a(\phi- \alpha) = 0 2 a ( ϕ − α ) = 0 p p p p p p ϕ = α \phi= \alpha ϕ = α
For (3), fixing one coordinate in P + 2 G P + 2G P + 2 G p p p q q q a α + v ( b ) a\alpha+ v(b) a α + v ( b ) v v v p = 3 p = 3 p = 3
For (4), subtract the F F F h ( a , b ) = z ( a , b ) − a ϕ − b ψ h(a,b) = z(a,b) - a\phi- b\psi h ( a , b ) = z ( a , b ) − a ϕ − b ψ F + 2 G F + 2G F + 2 G h h h 2 h = 0 2h = 0 2 h = 0 p ≥ 5 p \ge5 p ≥ 5 p = 3 p = 3 p = 3 q q q
For (5), the three rows containing E E E Q + P 1 + G Q + P1 + G Q + P 1 + G P + Q 1 + G P + Q1 + G P + Q 1 + G P 1 + F + G P1 + F + G P 1 + F + G G G G P 1 + Q 1 + F P1 + Q1 + F P 1 + Q 1 + F p p p q q q x ( a ) − a ϕ x(a) - a\phi x ( a ) − a ϕ − ( y ( b ) − b ψ ) -(y(b) - b\psi) − ( y ( b ) − b ψ )
For each of the ten clauses in (35) , Appendix § 14 gives one uniform routing scheme with five explicit fields: the optional selected box, the mixed row used, the part of Lemma 7.2 applied, and the resulting normal form. The table includes the cases in which no mixed row is needed because a selected primary coordinate or the sole parity box is an independent character factor. Thus every bit not fixed by a clause is accounted for. Reading any row of that matrix shows that every selected box carries the restriction of one common character (η , α , β \eta,\alpha,\beta η , α , β Theorem 7.1 without a finite mask enumeration.
Necessity
The Boolean complement of (35) , together with the special mask E E E
N 1 = G ¬ F ¬ P , N 2 = G ¬ F ¬ Q , N 3 = P 1 ¬ F ¬ P , N 4 = Q 1 ¬ F ¬ Q , N 5 = P 1 ¬ G ¬ P ¬ Q 1 , N 6 = Q 1 ¬ G ¬ P 1 ¬ Q , N 7 = ¬ E ¬ F ¬ P ¬ Q , N 8 = E G ¬ F ¬ P 1 ¬ Q 1 , N 9 = E Q 1 ¬ G ¬ P ¬ Q , N 10 = P 1 Q 1 ¬ E ¬ F ¬ G . (38) \begin{aligned}
N_1 &= G\neg F\neg P, & N_2 &= G\neg F\neg Q, \\
N_3 &= P1\neg F\neg P, & N_4 &= Q1\neg F\neg Q, \\
N_5 &= P1\neg G\neg P\neg Q1, & N_6 &= Q1\neg G\neg P1\neg Q, \\
N_7 &= \neg E\neg F\neg P\neg Q, & N_8 &= EG\neg F\neg P1\neg Q1, \\
N_9 &= EQ1\neg G\neg P\neg Q, & N_{10} &= P1Q1\neg E\neg F\neg G. \tag*{(38)}
\end{aligned} N 1 N 3 N 5 N 7 N 9 = G ¬ F ¬ P , = P 1¬ F ¬ P , = P 1¬ G ¬ P ¬ Q 1 , = ¬ E ¬ F ¬ P ¬ Q , = E Q 1¬ G ¬ P ¬ Q , N 2 N 4 N 6 N 8 N 10 = G ¬ F ¬ Q , = Q 1¬ F ¬ Q , = Q 1¬ G ¬ P 1¬ Q , = E G ¬ F ¬ P 1¬ Q 1 , = P 1 Q 1¬ E ¬ F ¬ G . (38) Conjunction signs are suppressed only in this display.
For N 1 , N 3 N_1,N_3 N 1 , N 3 σ p : C p × → Z \sigma_p:C_p^\times\to\mathbb{Z} σ p : C p × → Z P 1 P1 P 1 G G G F F F P P P p p p N 2 , N 4 N_2,N_4 N 2 , N 4 N 5 , N 6 N_5,N_6 N 5 , N 6 N 7 N_7 N 7 P 1 , Q 1 , G P1,Q1,G P 1 , Q 1 , G N 8 , N 9 N_8,N_9 N 8 , N 9 C 2 C_2 C 2 E E E N 10 N_{10} N 10 C 2 C_2 C 2 P 1 P1 P 1 C 2 C_2 C 2
One may choose the integer witness concretely: select a representative pair { a , − a } \{a,-a\} { a , − a } σ p ( a ) = 1 \sigma_p(a)=1 σ p ( a ) = 1 σ p ( − a ) = − 1 \sigma_p(-a)=-1 σ p ( − a ) = − 1 N 1 , N 3 N_1,N_3 N 1 , N 3 p p p Z \mathbb{Z} Z
For the C 2 C_2 C 2 N 8 , N 9 , N 10 N_8,N_9,N_{10} N 8 , N 9 , N 10
The integer-valued witnesses cannot be characters of a finite group, and the C 2 C_2 C 2 § 14 expands the coverage table, proving the reverse implication without a machine-generated Boolean premise.
Finally, degree two contains only inverse-pair rows inside each box. Table 2 shows that E E E § 14 includes an elementary count. Thus all 77 close first at three.
Adjoining the optional zero orbit doubles them, so 154 of the 255 supports have depth three.
The Quartic Skeleton
We next work modulo the local degree-four forms (6.2). Fourteen uniform zero-sum connector types, listed in Appendix A, attach the remaining local parameters. The specialization
P 1 [ a ] + G [ − a , b ] + G [ c , d ] + G [ − c , − b − d ] = 0 (39) P1[a] + G[-a,b] + G[c,d] + G[-c,-b-d] = 0 \tag*{(39)} P 1 [ a ] + G [ − a , b ] + G [ c , d ] + G [ − c , − b − d ] = 0 (39) and its p ↔ q p \leftrightarrow q p ↔ q
Table 8.0.
Some types necessarily occur in two rows. In particular, merging X10A and X10B merely because they have the same box counts would erase (39) . Appendix A supplies uniform nonvanishing parameter choices, including the p = 3 p = 3 p = 3 q = 5 q = 5 q = 5
We repeatedly use the following elementary isolation principle. If a connector image is r + b s r+bs r + b s s s s b b b s s s r r r 2 s 2s 2 s
Lemma 8.1 (Primary isolation). Let s s s r r r x + b 1 s = 0 x+b_1s = 0 x + b 1 s = 0 x + b 2 s = 0 x+b_2s = 0 x + b 2 s = 0 b 1 − b 2 b_1-b_2 b 1 − b 2 r r r s = x = 0 s = x = 0 s = x = 0 b 1 − b 2 = 2 b_1-b_2 = 2 b 1 − b 2 = 2
Proof. Subtraction gives ( b 1 − b 2 ) s = 0 (b_1-b_2)s = 0 ( b 1 − b 2 ) s = 0 r r r s = 0 s = 0 s = 0 x = 0 x = 0 x = 0 r r r
Theorem 8.2 (Quartic closure theorem). Among the masks not closed by degree three, the exact degree-four failure sets are
p , q ≥ 5 p,q \ge5 p , q ≥ 5
p = 3 p = 3 p = 3
Consequently 46 generic and 40 prime-three residual masks close first at degree four.
Generic branch
Suppose p , q ≥ 5 p,q \ge5 p , q ≥ 5 p p p q q q p p p P P P F F F G G G
P − P 1 : X 2 , P − F : X 6 , P − G : X 3 , P 1 − F : X 13 , P 1 − G : X 10 B , F − G : X 1. (40) P - P1 : X2,\qquad P - F : X6,\qquad P - G : X3,
\\
P1 - F : X13,\qquad P1 - G : X10B,\qquad F - G : X1. \tag*{(40)} P − P 1 : X 2 , P − F : X 6 , P − G : X 3 , P 1 − F : X 13 , P 1 − G : X 10 B , F − G : X 1. (40) This is a complete graph. The exchanged connectors give the complete graph on Q Q Q Q 1 Q1 Q 1 F F F G G G q q q
An edge in (40) means equality of the named local slope parameters. X1, for example, gives twice the F − G F-G F − G Lemma 8.1 removes the factor two. X10B gives the P 1 − G P1-G P 1 − G
The parity vertices are E E E P 1 P1 P 1 Q 1 Q1 Q 1 G G G E E E G G G P 1 − Q 1 P1-Q1 P 1 − Q 1 F F F P , Q P,Q P , Q P 1 P1 P 1 Q 1 Q1 Q 1 E E E G G G F F F P P P Q Q Q
These masks decode to P 1 ∪ Q 1 P1 \cup Q1 P 1 ∪ Q 1 P ∪ P 1 ∪ Q 1 P \cup P1 \cup Q1 P ∪ P 1 ∪ Q 1 Q ∪ P 1 ∪ Q 1 Q \cup P1 \cup Q1 Q ∪ P 1 ∪ Q 1
The prime-three branch
Let p = 3 p=3 p = 3 C q C_q C q
( e , p , u , c , f , g ) = ( E , P , P 1 , Q 1 , F , G ) . (e,p,u,c,f,g)=(E,P,P1,Q1,F,G). ( e , p , u , c , f , g ) = ( E , P , P 1 , Q 1 , F , G ) . with 2 e = 3 p = 2 c = 3 f = 0 2e=3p=2c=3f=0 2 e = 3 p = 2 c = 3 f = 0 u , g u,g u , g C 6 C_6 C 6 ( 3 , 4 , 1 , 3 , 4 , 1 ) (3,4,1,3,4,1) ( 3 , 4 , 1 , 3 , 4 , 1 )
F G : f + 2 g P P 1 : p + 2 u P G : p + 2 g P F : p + 2 f P 1 F : 2 u − 2 f P 1 G : u − g E G : e + 3 g E P 1 : e + 3 u E Q 1 : e + c Q 1 G : c + 3 g P 1 Q 1 F : u + c − f P P 1 Q 1 : p − u + c (41) \begin{aligned}
FG &: f+2g &\quad PP1 &: p+2u \\
PG &: p+2g &\quad PF &: p+2f \\
P1F &: 2u-2f &\quad P1G &: u-g \\
EG &: e+3g &\quad EP1 &: e+3u \\
EQ1 &: e+c &\quad Q1G &: c+3g \\
P1Q1F &: u+c-f &\quad PP1Q1 &: p-u+c \tag*{(41)}
\end{aligned} F G P G P 1 F E G E Q 1 P 1 Q 1 F : f + 2 g : p + 2 g : 2 u − 2 f : e + 3 g : e + c : u + c − f P P 1 P F P 1 G E P 1 Q 1 G P P 1 Q 1 : p + 2 u : p + 2 f : u − g : e + 3 u : c + 3 g : p − u + c (41) Every row maps to zero in C 6 C_6 C 6 C q C_q C q
The target map is
e , p , u , c , f , g ⟼ 3 , 4 , 1 , 3 , 4 , 1 in C 6 . (42) e,p,u,c,f,g \longmapsto3,4,1,3,4,1 \text{ in } C_6. \tag*{(42)} e , p , u , c , f , g ⟼ 3 , 4 , 1 , 3 , 4 , 1 in C 6 . (42) Each row of (41) visibly lies in its kernel. The free elements u , g u,g u , g C 2 C_2 C 2 C 3 C_3 C 3
If neither free generator occurs, ordinary connectors give the ambient target. If g g g u u u E E E P P P F F F Q 1 Q1 Q 1 (39) gives u = g u=g u = g u u u g g g E E E P P P F F F Q 1 Q1 Q 1 C 2 C_2 C 2
The division is exhaustive because u , g u,g u , g (42) . If exactly one is present, the listed anchor set contains every row in (41) with a nonzero coefficient on it. If both occur, X10B first reduces them to one common generator and the same anchor analysis applies. The exceptional Q 1 Q1 Q 1 C 2 C_2 C 2
Theorem 7.1 accounts for 77 masks. Removing those, E E E Theorem 8.2 .
Remark 8.3. The hexadecimal masks are only compact names. Appendix § 14 decodes the bit order and lists the routed families. The proof above is uniform in p , q p,q p , q
The finite atlases at n = 30 , 42 , 70 , 110 n=30,42,70,110 n = 30 , 42 , 70 , 110 Theorem 8.2 consists of the parameterized rows in Appendix § 13 , Lemma 8.1 , the complete primary graphs, and the fixed integral presentation (41) . None of those steps interpolates from a numerical prime sample.
Depth five, depth six, and the complete profiles
The failures in (8.2) split into two depth-five bridges and the critical depth-six bases. For each family we prove exactness in two steps. First we exhibit a bounded quotient with a nonzero restriction to the translation kernel, proving failure at the preceding degree. Then we write a zero-sum row representing a generator of that kernel. A late relation alone would give only an upper bound; a defect alone would give only a lower bound.
Proposition 9.1 (Generic critical pair). If p , q ≥ 5 p,q \ge5 p , q ≥ 5 P 1 ∪ Q 1 P1 \cup Q1 P 1 ∪ Q 1
K ( P 1 ∪ Q 1 ) / L ≤ 5 ( P 1 ∪ Q 1 ) ≅ C 2 (43) K(P1 \cup Q1)/\mathcal{L}_{\leq5}(P1 \cup Q1) \cong C_2 \tag*{(43)} K ( P 1 ∪ Q 1 ) / L ≤ 5 ( P 1 ∪ Q 1 ) ≅ C 2 (43) and closes at degree six.
Proof. The internal presentations are C 2 p C_{2p} C 2 p C 2 q C_{2q} C 2 q A 5 ( P 1 ∪ Q 1 ) = C 2 p ⊕ C 2 q A_5(P1\cup Q1)=C_{2p}\oplus C_{2q} A 5 ( P 1 ∪ Q 1 ) = C 2 p ⊕ C 2 q (43) .
If x , y x,y x , y C 2 p , C 2 q C_{2p},C_{2q} C 2 p , C 2 q C 2 C_2 C 2 p x − q y px-qy p x − q y
Choose a coordinate-zero triple ( 1 , 1 , − 2 ) (1,1,-2) ( 1 , 1 , − 2 )
This support is the sharpness family appearing in Casper–Orozco [3 ] ; the exact quotient and its placement in the full profile are the assertions used here.
Proposition 9.2 (Prime-three critical family). Let p = 3 p=3 p = 3
P 1 , Q ∪ P 1 , P 1 ∪ Q 1 , G , Q ∪ G , P 1 ∪ G , Q ∪ P 1 ∪ G (44) P_1,\quad Q\cup P_1,\quad P_1\cup Q_1,\quad G,\quad Q\cup G,\quad P_1\cup G,\quad Q\cup P_1\cup G \tag*{(44)} P 1 , Q ∪ P 1 , P 1 ∪ Q 1 , G , Q ∪ G , P 1 ∪ G , Q ∪ P 1 ∪ G (44) have fifth-order obstruction Z Z Z
Proof. Put A = P 1 = O 6 A=P_1=\mathcal{O}_6 A = P 1 = O 6 B = Q 1 = O 2 q B=Q_1=\mathcal{O}_{2q} B = Q 1 = O 2 q C = Q = O q C=Q=\mathcal{O}_q C = Q = O q D = G = O 6 q D=G=\mathcal{O}_{6q} D = G = O 6 q η \eta η C 3 C_3 C 3 A , D A,D A , D B , C B,C B , C
A , A ∪ C , A ∪ B , D , C ∪ D , A ∪ D , A ∪ C ∪ D , (45) A,\quad A\cup C,\quad A\cup B,\quad D,\quad C\cup D,\quad A\cup D,\quad A\cup C\cup D, \tag*{(45)} A , A ∪ C , A ∪ B , D , C ∪ D , A ∪ D , A ∪ C ∪ D , (45) every zero-sum row of length at most five has η = 0 \eta=0 η = 0 A ∪ B A\cup B A ∪ B A ∪ B A\cup B A ∪ B ± 3 \pm3 ± 3 A A A B B B B B B q q q
The local forms are
A : Z u , C : C q v , B : C 2 c ⊕ C q v , D : Z u ⊕ C q v . A:\mathbb{Z}u,\qquad C:C_qv,\qquad B:C_2c\oplus C_qv,\qquad D:\mathbb{Z}u\oplus C_qv. A : Z u , C : C q v , B : C 2 c ⊕ C q v , D : Z u ⊕ C q v . Available triples glue the q q q C ∪ D C\cup D C ∪ D A ∪ D A\cup D A ∪ D 6 u 6u 6 u A ∪ B A\cup B A ∪ B 3 u + c 3u+c 3 u + c η \eta η Z Z Z
The seven presentations may be summarized as
Table 9.3a.
The last column denotes the integer coefficient on the cyclic kernel; it is not division inside an abelian target. The visibility argument proves that this coefficient is zero on every row of length at most five.
Six positive A A A D D D 6 u 6u 6 u A ∪ B A\cup B A ∪ B A A A B B B q q q 3 u + c 3u+c 3 u + c
Proposition 9.3 (The two bridges). In both arithmetic regimes the bases
P ∪ P 1 ∪ Q 1 , Q ∪ P 1 ∪ Q 1 (46) P\cup P_1\cup Q_1,\qquad Q\cup P_1\cup Q_1 \tag*{(46)} P ∪ P 1 ∪ Q 1 , Q ∪ P 1 ∪ Q 1 (46) have exact depth five.
Proof. For the first base, the local forms are x a xa x a c p + x a p c_p+x a_p c p + x a p c q + y b c_q+yb c q + y b a p = a a_p=a a p = a c p + c q c_p+c_q c p + c q Q 1 Q_1 Q 1 q q q P 1 P_1 P 1 p p p P P P P 1 P_1 P 1 Q 1 Q_1 Q 1 q q q p , q p,q p , q
The lower bound is uniform in the primes. A representative of c p + c q c_p+c_q c p + c q 1 + 1 1+1 1 + 1 1 + 3 1+3 1 + 3
If p = 3 p=3 p = 3 P ∪ P 1 P\cup P_1 P ∪ P 1 α + 2 u = 0 \alpha+2u=0 α + 2 u = 0 3 α = 0 3\alpha=0 3 α = 0 C 6 C_6 C 6 Q 1 Q_1 Q 1 q q q Z ( 3 u + c ) \mathbb{Z}(3u+c) Z ( 3 u + c ) ± 3 \pm3 ± 3 P 1 P_1 P 1 Q 1 Q_1 Q 1 q q q
3 P 1 [ 1 ] + Q [ − y ] + Q 1 [ y ] = 0 3P_1[1]+Q[-y]+Q_1[y]=0 3 P 1 [ 1 ] + Q [ − y ] + Q 1 [ y ] = 0 represents 3 u + c 3u+c 3 u + c
Assembly of the 255 supports
The zero orbit splits off harmlessly. Indeed e 0 e_0 e 0 O 1 O_1 O 1 O 1 O_1 O 1
Indeed, for every d ≥ 1 d\geq1 d ≥ 1
Z S ∪ { 0 } = Z S ⊕ Z e 0 , L ≤ d ( S ∪ { 0 } ) = L ≤ d ( S ) ⊕ Z e 0 , (47) \mathbb{Z}^{S\cup\{0\}}=\mathbb{Z}^{S}\oplus\mathbb{Z}e_0,\qquad\mathcal{L}_{\leq d}(S\cup\{0\})=\mathcal{L}_{\leq d}(S)\oplus\mathbb{Z}e_0, \tag*{(47)} Z S ∪ { 0 } = Z S ⊕ Z e 0 , L ≤ d ( S ∪ { 0 }) = L ≤ d ( S ) ⊕ Z e 0 , (47) and the same direct sum holds for K K K
Proof of Theorem 1.1. The support E E E Theorem 7.1 gives 77 depth-three bases and hence 154 supports. If p , q ≥ 5 p,q\geq5 p , q ≥ 5 Theorem 8.2 gives 46 depth-four bases, Proposition 9.3 gives two depth-five bases, and Proposition 9.1 gives one depth-six base. Therefore
1 , 2 , 2 ( 77 ) = 154 , 2 ( 46 ) = 92 , 2 ( 2 ) = 4 , 2 ( 1 ) = 2 , (48) 1,\quad2,\quad2(77)=154,\quad2(46)=92,\quad2(2)=4,\quad2(1)=2, \tag*{(48)} 1 , 2 , 2 ( 77 ) = 154 , 2 ( 46 ) = 92 , 2 ( 2 ) = 4 , 2 ( 1 ) = 2 , (48) which sums to 255.
If p = 3 p=3 p = 3 Theorem 8.2 , Proposition 9.3 , and Proposition 9.2 . Hence
1 , 2 , 154 , 2 ( 40 ) = 80 , 4 , 2 ( 7 ) = 14 , (49) 1,\quad2,\quad154,\quad2(40)=80,\quad4,\quad2(7)=14, \tag*{(49)} 1 , 2 , 154 , 2 ( 40 ) = 80 , 4 , 2 ( 7 ) = 14 , (49) again summing to 255. Every family was accompanied both by closure and by a defect at the preceding degree, so these are first-closing depths rather than upper bounds. Proposition 2.1 identifies them with maximum separating-atom lengths.
As checks on the assembly,
1 + 2 + 154 + 92 + 4 + 2 = 255 , 1 + 2 + 154 + 80 + 4 + 14 = 255. (50) 1+2+154+92+4+2=255,\qquad1+2+154+80+4+14=255. \tag*{(50)} 1 + 2 + 154 + 92 + 4 + 2 = 255 , 1 + 2 + 154 + 80 + 4 + 14 = 255. (50) The cubic layer is identical in both regimes. The twelve-support decrease in the quartic entry when p = 3 p=3 p = 3
Corollary 6.2 now follows from both theorem lines: the single box G G G p = 3 p=3 p = 3 3 ∤ 2 p q 3\nmid2pq 3 ∤ 2 pq
Galois realization on every stable support
The preceding sections determine the lattice closing degree on every Aut ( C 2 p q ) \operatorname{Aut}(C_{2pq}) Aut ( C 2 pq )
We continue to write n = 2 p q n = 2pq n = 2 pq p < q p < q p < q σ u ( ζ n ) = ζ n u \sigma_u(\zeta_n) = \zeta_n^u σ u ( ζ n ) = ζ n u r 0 = 1 r_0 = 1 r 0 = 1
Non-torsion orbit phases and the low-depth layers
The phase families used at depths three and four come from an elementary norm-one construction.
Lemma 10.1 (Non-torsion cyclotomic phase). For every divisor m ∣ n m \mid n m ∣ n m > 2 m > 2 m > 2 z m ∈ Q ( ζ m ) × z_m \in\mathbb{Q}(\zeta_m)^\times z m ∈ Q ( ζ m ) × z m z m ‾ = 1 z_m\overline{z_m} = 1 z m z m = 1 z m z_m z m s m = n / m s_m = n/m s m = n / m
ψ m ( u s m ) = σ u ( z m ) ( u s m ∈ O m ) (51) \psi_m(us_m) = \sigma_u(z_m) \qquad(us_m \in\mathcal{O}_m) \tag*{(51)} ψ m ( u s m ) = σ u ( z m ) ( u s m ∈ O m ) (51) is well defined, Galois equivariant, and satisfies ψ m ( − s ) = ψ m ( s ) − 1 \psi_m(-s) = \psi_m(s)^{-1} ψ m ( − s ) = ψ m ( s ) − 1
Proof. For t ∈ Q t \in\mathbb{Q} t ∈ Q
z m ( t ) = t − ζ m t − ζ m − 1 . z_m(t) = \frac{t-\zeta_m}{t-\zeta_m^{-1}}. z m ( t ) = t − ζ m − 1 t − ζ m . Conjugation inverts this value. The map t ↦ z m ( t ) t \mapsto z_m(t) t ↦ z m ( t ) Q \mathbb{Q} Q t , u t,u t , u ( t − u ) ( ζ m − ζ m − 1 ) = 0 (t-u)(\zeta_m-\zeta_m^{-1}) = 0 ( t − u ) ( ζ m − ζ m − 1 ) = 0 t t t z m z_m z m
Two units carrying s m s_m s m Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) − 1 -1 − 1 z m z_m z m ψ m \psi_m ψ m
The following observation isolates the logical step used repeatedly below. It also makes clear which part comes from the existing lattice classifier and which part is new in this section.
Lemma 10.2 (Closing-degree witness). Let S S S L ≤ d − 1 ( S ) ⊊ L ≤ d ( S ) = K ( S ) \mathcal{L}_{\le d-1}(S) \subsetneq\mathcal{L}_{\le d}(S) = K(S) L ≤ d − 1 ( S ) ⊊ L ≤ d ( S ) = K ( S ) r = ( r s ) s ∈ S r = (r_s)_{s\in S} r = ( r s ) s ∈ S L ≤ d − 1 ( S ) \mathcal{L}_{\le d-1}(S) L ≤ d − 1 ( S ) d d d d − 1 d-1 d − 1 d d d
Proof. If r r r L ≤ d ( S ) = K ( S ) \mathcal{L}_{\le d}(S) = K(S) L ≤ d ( S ) = K ( S ) d d d Proposition 3.3 realizes r r r f ^ = 1 \hat{f}=1 f ^ = 1 g ^ = r \hat{g}=r g ^ = r S S S d d d
The zero orbit separates at every degree. Indeed, e 0 e_0 e 0 (47) ,
K ( S ∪ { 0 } ) = K ( S ) ⊕ Z e 0 , L ≤ d ( S ∪ { 0 } ) = L ≤ d ( S ) ⊕ Z e 0 . (52) K(S \cup\{0\}) = K(S) \oplus\mathbb{Z}e_0,\qquad\mathcal{L}_{\leq d}(S \cup\{0\}) = \mathcal{L}_{\leq d}(S) \oplus\mathbb{Z}e_0. \tag*{(52)} K ( S ∪ { 0 }) = K ( S ) ⊕ Z e 0 , L ≤ d ( S ∪ { 0 }) = L ≤ d ( S ) ⊕ Z e 0 . (52) Thus adjoining O 1 \mathcal{O}_1 O 1 O 1 \mathcal{O}_1 O 1
For the depth-two base E = O 2 E=\mathcal{O}_2 E = O 2 r E = 2 r_E=2 r E = 2 E E E ± 1 \pm1 ± 1 E + E = 0 E+E=0 E + E = 0 Proposition 3.3 proves exact rational and integer depth two, with or without O 1 \mathcal{O}_1 O 1
Each of the 77 depth-three bases other than E E E O m \mathcal{O}_m O m m > 2 m>2 m > 2 r = ψ m r=\psi_m r = ψ m r = 1 r=1 r = 1 Lemma 10.1 makes r r r O m \mathcal{O}_m O m
The nine depth-four families
For a point in a box with p p p a a a q q q b b b ψ p ( a ) \psi_p(a) ψ p ( a ) ψ q ( b ) \psi_q(b) ψ q ( b ) (51) . On boxes not named in the phase column, the ratio is one. Table 3 is a canonical first-match cover of every base that fails the cubic criterion and closes at degree four. The last column records the generic and prime-three first-match counts; those counts are checks on the routing, not premises of the cancellation proof.
Table 3. The nine canonical depth-four Galois lifts.
We first justify that the table is complete without invoking a finite search. The necessity half of Theorem 7.1 covers the complement of its ten positive clauses by N 1 , … , N 10 N1,\ldots,N10 N 1 , … , N 10
N 7 : ¬ E , ¬ F , ¬ P , ¬ Q . N7:\ \neg E,\neg F,\neg P,\neg Q. N 7 : ¬ E , ¬ F , ¬ P , ¬ Q . A nonempty N 7 N7 N 7 P 1 P1 P 1 Q 1 Q1 Q 1 G G G G G G N 1 N1 N 1 G G G P 1 P1 P 1 N 3 N3 N 3 Q 1 Q1 Q 1 N 4 N4 N 4
N 7 ⊂ N 1 ∪ N 3 ∪ N 4 , (53) N7 \subset N1\cup N3\cup N4, \tag*{(53)} N 7 ⊂ N 1 ∪ N 3 ∪ N 4 , (53) and the nine displayed families cover every cubic-negative mask.
Next we verify invisibility through degree three. Inverse pairs cancel by Lemma 10.1 . For N 1 N1 N 1 N 3 N3 N 3 Theorem 7.1 shows that only Q G G QGG QGG Q P 1 G QP1G QP 1 G p p p p p p a a a − a -a − a N 2 N2 N 2 N 4 N4 N 4 q q q P G G PGG P GG P Q 1 G PQ1G P Q 1 G N 5 N5 N 5 N 6 N6 N 6 N 8 N8 N 8 N 9 N9 N 9 E E E N 10 N10 N 10 P P 1 P 1 PP1P1 P P 1 P 1 Table 3 kills L ≤ 3 \mathcal{L}_{\leq3} L ≤ 3
The six cyclotomic rows are nontranslations by Lemma 10.1 . The three sign rows require separate arguments. In N 8 N8 N 8 1 ∈ G 1\in G 1 ∈ G − 1 -1 − 1 E E E N 9 N9 N 9 p ∈ Q 1 p\in Q1 p ∈ Q 1 − 1 -1 − 1 E = p q E=pq E = pq N 10 N10 N 10 q ∈ P 1 q\in P1 q ∈ P 1 − 1 -1 − 1 p ∈ Q 1 p\in Q1 p ∈ Q 1
The Quartic Skeleton says that every base counted in the last column has L ≤ 4 = K \mathcal{L}_{\leq4}=K L ≤ 4 = K Lemma 10.2 therefore applies to each displayed ratio: a degree-four row detects it, and the rational and integer realizations have exact depth four. There are 46 generic or 40 prime-three bases, and (10.2) doubles these to 92 or 80 supports. Thus the last column of Table 3 is used only after the symbolic cancellation and nontranslation arguments have been completed; it is never a substitute for either argument.
Depth five and six
The two depth-five bases are
S p = P ∪ P 1 ∪ Q 1 , S q = Q ∪ P 1 ∪ Q 1. S_p=P\cup P1\cup Q1,\qquad S_q=Q\cup P1\cup Q1. S p = P ∪ P 1 ∪ Q 1 , S q = Q ∪ P 1 ∪ Q 1. On either base put r = − 1 r=-1 r = − 1 P 1 P1 P 1 r = 1 r=1 r = 1 N 10 N10 N 10 − 1 -1 − 1 S p S_p S p Q 1 Q1 Q 1 q q q P P P P 1 P1 P 1 Q 1 Q1 Q 1 P 1 P1 P 1 P P P p p p S q S_q S q P 1 P1 P 1 Q 1 Q1 Q 1 p p p q q q
S p : P [ − a ] + P 1 [ a ] + Q 1 [ 1 ] + Q 1 [ 1 ] + Q 1 [ − 2 ] , S q : P 1 [ 1 ] + P 1 [ 1 ] + P 1 [ − 2 ] + Q [ − b ] + Q 1 [ b ] . (54) \begin{aligned}
S_p &: \quad P[-a]+P1[a]+Q1[1]+Q1[1]+Q1[-2],\\
S_q &: \quad P1[1]+P1[1]+P1[-2]+Q[-b]+Q1[b]. \tag*{(54)}
\end{aligned} S p S q : P [ − a ] + P 1 [ a ] + Q 1 [ 1 ] + Q 1 [ 1 ] + Q 1 [ − 2 ] , : P 1 [ 1 ] + P 1 [ 1 ] + P 1 [ − 2 ] + Q [ − b ] + Q 1 [ b ] . (54) All shown coordinates are nonzero. When p = 3 p=3 p = 3 1 , 1 , − 2 1,1,-2 1 , 1 , − 2 1 , 1 , 1 1,1,1 1 , 1 , 1
Suppose now that p , q ≥ 5 p,q \ge5 p , q ≥ 5 P 1 ∪ Q 1 P1 \cup Q1 P 1 ∪ Q 1 P 1 P1 P 1 1 + 1 1+1 1 + 1 3 + 1 3+1 3 + 1
P 1 [ 1 ] + P 1 [ 1 ] + P 1 [ − 2 ] + Q 1 [ 1 ] + Q 1 [ 1 ] + Q 1 [ − 2 ] (55) P1[1] + P1[1] + P1[-2] + Q1[1] + Q1[1] + Q1[-2] \tag*{(55)} P 1 [ 1 ] + P 1 [ 1 ] + P 1 [ − 2 ] + Q 1 [ 1 ] + Q 1 [ 1 ] + Q 1 [ − 2 ] (55) is zero-sum and contains three negative factors. It detects the ratio and proves exact depth six for the base and its zero-orbit copy.
It remains to realize the seven prime-three critical bases. Put
A = P 1 = O 6 , B = Q 1 = O 2 q , C = Q = O q , D = G = O 6 q . (56) A = P1 = O_6,\qquad B = Q1 = O_{2q},\qquad C = Q = O_q,\qquad D = G = O_{6q}. \tag*{(56)} A = P 1 = O 6 , B = Q 1 = O 2 q , C = Q = O q , D = G = O 6 q . (56) They are
A , A ∪ C , A ∪ B , D , C ∪ D , A ∪ D , A ∪ C ∪ D . (57) A,\ A \cup C,\ A \cup B,\ D,\ C \cup D,\ A \cup D,\ A \cup C \cup D. \tag*{(57)} A , A ∪ C , A ∪ B , D , C ∪ D , A ∪ D , A ∪ C ∪ D . (57) Choose a non-torsion norm-one z ∈ Q ( ζ 6 ) × z \in\mathbb{Q}(\zeta_6)^{\times} z ∈ Q ( ζ 6 ) × A , D A,D A , D ε ( s ) ∈ { 1 , − 1 } \varepsilon(s) \in\{1,-1\} ε ( s ) ∈ { 1 , − 1 } F 3 \mathbb{F}_3 F 3 r s = z ε ( s ) r_s = z^{\varepsilon(s)} r s = z ε ( s ) r = 1 r=1 r = 1 B , C B,C B , C 6 q 6q 6 q 1 1 1 − 1 -1 − 1 z z z ε \varepsilon ε
For a zero-sum multiset, let η \eta η F 3 \mathbb{F}_3 F 3 3 ∣ η 3 \mid\eta 3 ∣ η A ∪ B A \cup B A ∪ B η \eta η 6 ∣ η 6 \mid\eta 6 ∣ η ∣ η ∣ ≤ 5 |\eta| \le5 ∣ η ∣ ≤ 5 η = 0 \eta= 0 η = 0 A ∪ B A \cup B A ∪ B η = ± 3 \eta= \pm3 η = ± 3 A A A B B B B B B q q q
Degree six detects it explicitly. Six positive A A A A A A A ∪ B A \cup B A ∪ B D D D D [ 1 , b ] , D [ 1 , − b ] D[1,b],D[1,-b] D [ 1 , b ] , D [ 1 , − b ] A ∪ B A \cup B A ∪ B A A A B [ 1 ] , B [ 1 ] , B [ − 2 ] B[1],B[1],B[-2] B [ 1 ] , B [ 1 ] , B [ − 2 ] z 6 z^6 z 6 z 3 z^3 z 3 z z z
For reference, the detector assignment is displayed without suppressing any of the seven cases:
Table 10.
The coordinates in the middle column sum to zero in both CRT components. Because z z z L ≤ 6 = K \mathcal{L}_{\le6}=K L ≤ 6 = K
The realized pairs and the proof audit
For every ratio chosen above, the rational pair is completely explicit in Fourier coordinates. Namely,
f r ( x ) = 1 n ∑ s ∈ S ζ n s x , g r ( x ) = 1 n ∑ s ∈ S r s ζ n s x . f_r(x)=\frac{1}{n}\sum_{s\in S}\zeta_n^{sx},\qquad g_r(x)=\frac{1}{n}\sum_{s\in S}r_s\zeta_n^{sx}. f r ( x ) = n 1 s ∈ S ∑ ζ n s x , g r ( x ) = n 1 s ∈ S ∑ r s ζ n s x . Galois equivariance fixes both sums under every element of Gal ( Q ( ζ n ) / Q ) \operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) Gal ( Q ( ζ n ) / Q ) f r , g r ∈ Q C n f_r,g_r\in\mathbb{Q}^{C_n} f r , g r ∈ Q C n 1 1 1 r s r_s r s S S S S S S S S S D D D D f r , D g r Df_r,Dg_r D f r , D g r D D D k k k D k D^k D k
The proof obligations are summarized in the following table. Its middle columns cite the human argument that makes the ratio invisible below the claimed depth and the independent lattice statement that closes at that depth. This separation prevents the finite profile enumeration from being used as a surrogate for the new realization argument.
Table 5.
The depth-one row is direct, and the depth-two row uses Proposition 3.3 , whose hypotheses here are Galois equivariance, lower-degree cancellation, and nontranslation. Every row from depth three onward has supplied all four ingredients required by Lemma 10.2 : Galois equivariance, inverse symmetry, lower-degree cancellation, and nontranslation. The zero-orbit splitting in (10.2) then transfers each nonzero-base construction to its optional-zero copy without altering the applicable ingredients.
Assembly
We can now prove the signal-level theorem without changing the lattice classifier.
Proof of Theorem 1.2 . The inclusion of signal classes and Proposition 3.2 give
δ Z ( S ) ≤ δ Q ( S ) ≤ δ ( S ) . (58) \delta_{\mathbb{Z}}(S) \le\delta_{\mathbb{Q}}(S) \le\delta(S). \tag*{(58)} δ Z ( S ) ≤ δ Q ( S ) ≤ δ ( S ) . (58) The depth-one support was handled directly. For every support with δ ( S ) > 1 \delta(S) > 1 δ ( S ) > 1 L ≤ δ ( S ) − 1 ( S ) \mathcal{L}_{\leq\delta(S)-1}(S) L ≤ δ ( S ) − 1 ( S ) Proposition 3.3 therefore gives both reverse inequalities. The three depths are equal.
The underlying nonzero bases are partitioned as
d = 2 d = 3 d = 4 d = 5 d = 6 p = 3 1 77 40 2 7 p , q ≥ 5 1 77 46 2 1 (59) \begin{array}{c|ccccc}
& d=2 & d=3 & d=4 & d=5 & d=6 \\
p=3 & 1 & 77 & 40 & 2 & 7 \\
p,q\geq5 & 1 & 77 & 46 & 2 & 1 \tag*{(59)}
\end{array} p = 3 p , q ≥ 5 d = 2 1 1 d = 3 77 77 d = 4 40 46 d = 5 2 2 d = 6 7 1 (59) Restoring the optional zero orbit doubles every base and adds the sole depth-one support. This gives exactly the two profiles in Theorem 1.1 for both rational and integer signals.
The theorem establishes thresholds, not a complete moduli description. The exact unit-support torus is classified in § 5 ; on a stable support of depth greater than one, the construction above deliberately records only the Galois-equivariant lower-bound ratio needed for exactness.
Binary set reconstruction on C 2 p q C_{2pq} C 2 pq
For a subset A ⊂ C n A \subset C_n A ⊂ C n f A = 1 A f_A = \mathbf{1}_A f A = 1 A S A = supp f A ^ S_A = \operatorname{supp}\widehat{f_A} S A = supp f A k k k M k ( f A ) M_k(f_A) M k ( f A ) (12) ; order counts factors, and equivalence in this subsection means translation only. Let ρ T ( A ) \rho_T(A) ρ T ( A ) d d d d d d A A A [13 , 19 ] .
For a nonempty pattern T ⊂ C n T \subset C_n T ⊂ C n [ T ] [T] [ T ] μ A ( [ T ] ) \mu_A([T]) μ A ([ T ]) A A A Stab ( T ) = { h ∈ C n : T + h = T } \operatorname{Stab}(T) = \{h \in C_n : T + h = T\} Stab ( T ) = { h ∈ C n : T + h = T } d d d μ A ( [ T ] ) \mu_A([T]) μ A ([ T ]) 1 ≤ ∣ T ∣ ≤ d 1 \leq|T| \leq d 1 ≤ ∣ T ∣ ≤ d
Lemma 11.1 (Pebody/analytical deck equivalence). Let 1 ≤ d ≤ n 1 \leq d \leq n 1 ≤ d ≤ n A ⊂ C n A \subset C_n A ⊂ C n A = ∅ A = \varnothing A = ∅ ∣ A ∣ < d |A| < d ∣ A ∣ < d d d d d d d T = { 0 , t 1 , … , t j − 1 } T = \{0,t_1,\ldots,t_{j-1}\} T = { 0 , t 1 , … , t j − 1 } j j j
M j ( f A ; t 1 , … , t j − 1 ) = ∣ Stab ( T ) ∣ μ A ( [ T ] ) . M_j(f_A;t_1,\ldots,t_{j-1}) = |\operatorname{Stab}(T)|\,\mu_A([T]). M j ( f A ; t 1 , … , t j − 1 ) = ∣ Stab ( T ) ∣ μ A ([ T ]) . Pebody’s 2004 count-valued set deck records the left-hand side directly; the 2007 necklace convention records the right-hand multiplicity without the known stabilizer factor [17 , 18 ] .
Proof. The analytical value counts the translations x ∈ C n x \in C_n x ∈ C n x + T ⊂ A x+T \subset A x + T ⊂ A T T T A A A ∣ Stab ( T ) ∣ |\operatorname{Stab}(T)| ∣ Stab ( T ) ∣ d d d d d d 1 A m = 1 A \mathbf{1}_A^m=\mathbf{1}_A 1 A m = 1 A A = ∅ A=\varnothing A = ∅ ∣ A ∣ |A| ∣ A ∣
The exact deck becomes cumulative for indicators. If 1 ≤ k ≤ d 1 \le k \le d 1 ≤ k ≤ d
M k ( f A ; t 1 , … , t k − 1 ) = M d ( f A ; t 1 , … , t k − 1 , 0 , … , 0 ) , (60) M_k(f_A;t_1,\ldots,t_{k-1})=M_d(f_A;t_1,\ldots,t_{k-1},0,\ldots,0), \tag*{(60)} M k ( f A ; t 1 , … , t k − 1 ) = M d ( f A ; t 1 , … , t k − 1 , 0 , … , 0 ) , (60) because f A m = f A f_A^m=f_A f A m = f A
Proposition 11.2 (Supportwise binary crosswalk). Every nonempty A ⊂ C n A \subset C_n A ⊂ C n
ρ T ( A ) ≤ δ ( S A ) . (61) \rho_T(A)\le\delta(S_A). \tag*{(61)} ρ T ( A ) ≤ δ ( S A ) . (61) Equality need not hold.
Proof. Put d = δ ( S A ) d=\delta(S_A) d = δ ( S A ) d = 1 d=1 d = 1 S A = { 0 } S_A=\{0\} S A = { 0 } f A f_A f A
Assume d ≥ 2 d\ge2 d ≥ 2 B B B d d d (60) gives equality of the second decks, so Lemma 3.1 gives S B = S A = : S S_B=S_A=:S S B = S A =: S r s = f B ^ ( s ) / f A ^ ( s ) r_s=\widehat{f_B}(s)/\widehat{f_A}(s) r s = f B ( s ) / f A ( s ) (60) say that r r r L ≤ d ( S ) \mathcal{L}_{\le d}(S) L ≤ d ( S ) L ≤ d ( S ) = K ( S ) \mathcal{L}_{\le d}(S)=K(S) L ≤ d ( S ) = K ( S ) Proposition 3.2 makes r r r B B B A A A
For strictness, a singleton in C 30 C_{30} C 30 ρ T ( A ) = 1 < 3 = δ ( S A ) \rho_T(A)=1<3=\delta(S_A) ρ T ( A ) = 1 < 3 = δ ( S A ) (61) is used below.
The deep-support exclusion
Fix distinct odd primes p < q p<q p < q § 6 . The complete inverse-Fourier dictionary, before the constant orbit O 1 O_1 O 1
E O 2 ( − 1 ) ε e P O p U ( x ) Q O q T ( y ) P 1 O 2 p ( − 1 ) ε V ( x ) Q 1 O 2 q ( − 1 ) ε W ( y ) F O p q R ( x , y ) G O 2 p q ( − 1 ) ε Z ( x , y ) (62) \begin{array}{c|c|c|c|c|c}
E & O_2 & (-1)^\varepsilon e & P & O_p & U(x) \\
Q & O_q & T(y) & P1 & O_{2p} & (-1)^\varepsilon V(x) \\
Q1 & O_{2q} & (-1)^\varepsilon W(y) & F & O_{pq} & R(x,y) \\
G & O_{2pq} & (-1)^\varepsilon Z(x,y) & & & \tag*{(62)}
\end{array} E Q Q 1 G O 2 O q O 2 q O 2 pq ( − 1 ) ε e T ( y ) ( − 1 ) ε W ( y ) ( − 1 ) ε Z ( x , y ) P P 1 F O p O 2 p O pq U ( x ) ( − 1 ) ε V ( x ) R ( x , y ) (62) Here U , T , V , W U,T,V,W U , T , V , W R , Z R,Z R , Z O 1 O_1 O 1 f A ^ ( 0 ) = ∣ A ∣ > 0 \widehat{f_A}(0)=|A|>0 f A ( 0 ) = ∣ A ∣ > 0 (52) . Moreover, f A f_A f A (14) makes S A S_A S A
Theorem 1.1 leaves exactly the following bases above depth four; the mandatory O 1 \mathcal{O}_1 O 1
Table 11.4.
Thus a binary upper bound four is equivalent to excluding every row of (11.4) as an exact Fourier support.
There is also a useful elementary mask-polynomial obstruction for part of this list. General lower bounds for prescribed cyclotomic mask-polynomial divisors are studied in [KŁMS26], but no such general bound is needed here. For A ⊂ C 2 p q A \subset C_{2pq} A ⊂ C 2 pq A ( X ) = ∑ a ∈ A X a A(X) = \sum_{a \in A} X^a A ( X ) = ∑ a ∈ A X a s s s Φ s ( X ) ∣ A ( X ) \Phi_s(X) \mid A(X) Φ s ( X ) ∣ A ( X ) P 1 ∪ Q 1 P1 \cup Q1 P 1 ∪ Q 1 { 2 , p , q , p q , 2 p q } \{2,p,q,pq,2pq\} { 2 , p , q , pq , 2 pq } p = 3 p=3 p = 3 P 1 P1 P 1 G G G P 1 ∪ G P1 \cup G P 1 ∪ G { 2 , p , q , 2 q , p q , 2 p q } \{2,p,q,2q,pq,2pq\} { 2 , p , q , 2 q , pq , 2 pq } { 2 , p , q , 2 p , 2 q , p q } \{2,p,q,2p,2q,pq\} { 2 , p , q , 2 p , 2 q , pq } { 2 , p , q , 2 q , p q } \{2,p,q,2q,pq\} { 2 , p , q , 2 q , pq } 2 , p , q 2,p,q 2 , p , q Z [ X ] \mathbb{Z}[X] Z [ X ] Z [ X ] \mathbb{Z}[X] Z [ X ]
Φ 2 ( X ) Φ p ( X ) Φ q ( X ) ∣ A ( X ) . \Phi_2(X)\Phi_p(X)\Phi_q(X) \mid A(X). Φ 2 ( X ) Φ p ( X ) Φ q ( X ) ∣ A ( X ) . Evaluation at X = 1 X=1 X = 1
2 p q = Φ 2 ( 1 ) Φ p ( 1 ) Φ q ( 1 ) ∣ A ( 1 ) = ∣ A ∣ . 2pq = \Phi_2(1)\Phi_p(1)\Phi_q(1) \mid A(1) = |A|. 2 pq = Φ 2 ( 1 ) Φ p ( 1 ) Φ q ( 1 ) ∣ A ( 1 ) = ∣ A ∣. An indicator with any of the four exact supports is nonempty and proper: O 1 \mathcal{O}_1 O 1 0 < ∣ A ∣ < 2 p q 0 < |A| < 2pq 0 < ∣ A ∣ < 2 pq
Lemma 11.3 (Two finite-set rigidity facts). If a nonempty finite X ⊂ C X \subset\mathbb{C} X ⊂ C X + h = X X+h=X X + h = X h = 0 h=0 h = 0 X , Y ⊂ C X,Y \subset\mathbb{C} X , Y ⊂ C ∣ X + Y ∣ ≤ 2 |X+Y| \le2 ∣ X + Y ∣ ≤ 2 X , Y X,Y X , Y
Proof. For the first assertion, iteration of a nonzero h h h x + m h x+mh x + mh X X X Y Y Y Y Y Y
Theorem 11.4 (Binary exclusion of every deep support). No base in (11.4), after restoring O 1 \mathcal{O}_1 O 1 C 2 p q C_{2pq} C 2 pq
Proof. Assume first p , q ≥ 5 p,q \ge5 p , q ≥ 5 P 1 ∪ Q 1 P1 \cup Q1 P 1 ∪ Q 1
f ( ε , x , y ) = c + ( − 1 ) ε ( V ( x ) + W ( y ) ) . (63) f(\varepsilon,x,y)=c+(-1)^{\varepsilon}\bigl(V(x)+W(y)\bigr). \tag*{(63)} f ( ε , x , y ) = c + ( − 1 ) ε ( V ( x ) + W ( y ) ) . (63) Both summands are nonconstant. Adding the two parity slices shows that $2c \in {0,1,2}$. The extreme values force V + W = 0 V+W=0 V + W = 0 V ( C p ) + W ( C q ) ⊂ { − 1 / 2 , 1 / 2 } V(C_p)+W(C_q)\subset\{-1/2,1/2\} V ( C p ) + W ( C q ) ⊂ { − 1/2 , 1/2 }
On P ∪ P 1 ∪ Q 1 P\cup P_1\cup Q_1 P ∪ P 1 ∪ Q 1
f ( ε , x , y ) = c + U ( x ) + ( − 1 ) ε ( V ( x ) + W ( y ) ) . (64) f(\varepsilon,x,y)=c+U(x)+(-1)^{\varepsilon}(V(x)+W(y)). \tag*{(64)} f ( ε , x , y ) = c + U ( x ) + ( − 1 ) ε ( V ( x ) + W ( y )) . (64) For fixed x x x 2 ( c + U ( x ) ) 2(c+U(x)) 2 ( c + U ( x )) y y y W W W Q 1 Q_1 Q 1 x x x U U U U = 0 U=0 U = 0 P P P
The other bridge has f = c + T ( y ) + ( − 1 ) ε ( V ( x ) + W ( y ) ) f=c+T(y)+(-1)^{\varepsilon}(V(x)+W(y)) f = c + T ( y ) + ( − 1 ) ε ( V ( x ) + W ( y )) 2 ( c + T ( y ) ) 2(c+T(y)) 2 ( c + T ( y )) P 1 P_1 P 1 V V V T T T p = 3 p=3 p = 3
It remains to treat the seven p = 3 p=3 p = 3 C 2 × C 3 C_2\times C_3 C 2 × C 3 C 6 C_6 C 6 h : C 6 → { 0 , 1 } h:C_6\to\{0,1\} h : C 6 → { 0 , 1 } O 1 ∪ O 6 \mathcal{O}_1\cup\mathcal{O}_6 O 1 ∪ O 6
h 0 + h 3 = h 1 + h 4 = h 2 + h 5 = : s . (65) h_0+h_3=h_1+h_4=h_2+h_5=:s. \tag*{(65)} h 0 + h 3 = h 1 + h 4 = h 2 + h 5 =: s . (65) For s = 0 , 2 s=0,2 s = 0 , 2 s = 1 s=1 s = 1 h 0 + h 2 + h 4 = h 1 + h 3 + h 5 h_0+h_2+h_4=h_1+h_3+h_5 h 0 + h 2 + h 4 = h 1 + h 3 + h 5
Write A = P 1 A=P_1 A = P 1 B = Q 1 B=Q_1 B = Q 1 C = Q C=Q C = Q D = G D=G D = G y ∈ C q y\in C_q y ∈ C q O 1 ∪ O 6 \mathcal{O}_1\cup\mathcal{O}_6 O 1 ∪ O 6
Table 9.
In the last two rows, Fourier transformation in y y y y y y A ∪ B A\cup B A ∪ B
On that exceptional base,
f ( t , y ) = c + u ( t ) + ( − 1 ) t w ( y ) , (66) f(t,y)=c+u(t)+(-1)^t w(y), \tag*{(66)} f ( t , y ) = c + u ( t ) + ( − 1 ) t w ( y ) , (66) where u u u O 6 = { 1 , 5 } \mathcal{O}_6=\{1,5\} O 6 = { 1 , 5 } w w w t t t c + u ( t ) + w ( C q ) c+u(t)+w(C_q) c + u ( t ) + w ( C q ) { 0 , 1 } \{0,1\} { 0 , 1 } u u u u ( t + 2 ) = u ( t ) u(t+2)=u(t) u ( t + 2 ) = u ( t ) C 6 C_6 C 6 { 0 , 3 } = O 1 ∪ O 2 \{0,3\}=\mathcal{O}_1\cup\mathcal{O}_2 { 0 , 3 } = O 1 ∪ O 2 O 6 \mathcal{O}_6 O 6 u = 0 u=0 u = 0
Exact reconstruction and the reflection quotient
Proof of Theorem 1.3. Every nonempty binary exact support has depth at most four by Theorem 11.4 and the complete profile theorem. Proposition 11.2 therefore gives r s e t ( C 2 p q ) ≤ 4 r_{\mathrm{set}}(C_{2pq})\le4 r set ( C 2 pq ) ≤ 4 k = p q ≥ 15 k=pq\ge15 k = pq ≥ 15 C 2 k C_{2k} C 2 k
{ 0 } ∪ { 3 , … , k − 1 } ∪ { k + 1 , k + 2 } , { 0 , 1 } ∪ { 3 , … , k − 1 } ∪ { k + 2 } (67) \{0\}\cup\{3,\ldots,k-1\}\cup\{k+1,k+2\},\qquad\{0,1\}\cup\{3,\ldots,k-1\}\cup\{k+2\} \tag*{(67)} { 0 } ∪ { 3 , … , k − 1 } ∪ { k + 1 , k + 2 } , { 0 , 1 } ∪ { 3 , … , k − 1 } ∪ { k + 2 } (67) have the same analytical three-deck but are not translates [13 Theorem 2.22] . Hence order three is insufficient and the value is four.
Keep the deck oriented, but now call two outputs equivalent when one is a translation or a reflected translation of the other; denote the resulting number by r s e t T / I , o r r_{\mathrm{set}}^{T/I,\mathrm{or}} r set T / I , or
Corollary 11.5 (Oriented deck modulo translations and reflections). For distinct odd primes p < q p<q p < q
r s e t T / I , o r ( C 2 p q ) = 4. (68) r_{\mathrm{set}}^{T/I,\mathrm{or}}(C_{2pq})=4. \tag*{(68)} r set T / I , or ( C 2 pq ) = 4. (68) Proof. Theorem 1.3 gives the upper bound. For the lower bound set m = q m=q m = q n ˉ = 2 p \bar n=2p n ˉ = 2 p C 2 p q C_{2pq} C 2 pq q x + i qx+i q x + i x ∈ C 2 p x\in C_{2p} x ∈ C 2 p i ∈ C q i\in C_q i ∈ C q C 2 p C_{2p} C 2 p
H A = { 0 , 2 , … , 2 p − 2 } , H B = { 0 , 1 , … , p − 1 } . H_A=\{0,2,\ldots,2p-2\},\qquad H_B=\{0,1,\ldots,p-1\}. H A = { 0 , 2 , … , 2 p − 2 } , H B = { 0 , 1 , … , p − 1 } . In C q C_q C q J B = { 0 , 1 , − 1 } J_B=\{0,1,-1\} J B = { 0 , 1 , − 1 } J A = C q ∖ J B J_A=C_q\setminus J_B J A = C q ∖ J B ℓ − 1 = 1 \ell_{-1}=1 ℓ − 1 = 1 ℓ i = 0 \ell_i=0 ℓ i = 0 D i = H A + ℓ i D_i=H_A+\ell_i D i = H A + ℓ i i ∈ J A i\in J_A i ∈ J A D i = H B + ℓ i D_i=H_B+\ell_i D i = H B + ℓ i i ∈ J B i\in J_B i ∈ J B
A = { q x + i : x ∈ D i , i ∈ C q } , A ′ = { q x + i : x ∈ D i + 21 { i = 0 } , i ∈ C q } . (69) A=\{qx+i:x\in D_i,\ i\in C_q\},\qquad A'=\{qx+i:x\in D_i+2\mathbf{1}_{\{i=0\}},\ i\in C_q\}. \tag*{(69)} A = { q x + i : x ∈ D i , i ∈ C q } , A ′ = { q x + i : x ∈ D i + 2 1 { i = 0 } , i ∈ C q } . (69) This is Simon’s Proposition 3.11 with i 0 = 0 i_0=0 i 0 = 0 δ = 2 \delta=2 δ = 2 j = 1 j=1 j = 1 ℓ − 1 + ℓ 1 = 1 = 2 ℓ 0 + 2 − 1 \ell_{-1}+\ell_1=1=2\ell_0+2-1 ℓ − 1 + ℓ 1 = 1 = 2 ℓ 0 + 2 − 1 C 2 p C_{2p} C 2 p p q pq pq q q q ∣ J B ∣ = 3 \lvert J_B\rvert=3 ∣ J B ∣ = 3 J B J_B J B j = 2 j=2 j = 2 ℓ − 2 + ℓ 2 = 0 \ell_{-2}+\ell_2=0 ℓ − 2 + ℓ 2 = 0 [19 Corollary 1.41 and Propositions 3.11, 3.13] .
Load-bearing controls and theorem boundary
The binary hypothesis cannot be replaced even by a three-letter alphabet. The function ( 0 , 0 , 1 , 2 , 2 , 1 ) (0,0,1,2,2,1) ( 0 , 0 , 1 , 2 , 2 , 1 ) C 6 C_6 C 6 O 1 ∪ O 6 O_1\cup O_6 O 1 ∪ O 6 − 3 ± i 3 -3\pm i\sqrt{3} − 3 ± i 3 (65) is genuinely an idempotent two-value phenomenon.
Nor does Theorem 11.4 collapse the weighted signal problem. The integer functions
( 14 , 7 , − 7 , − 14 , − 7 , 7 ) , ( 2 , − 11 , − 13 , − 2 , 11 , 13 ) (70) (14,7,-7,-14,-7,7),\qquad(2,-11,-13,-2,11,13) \tag*{(70)} ( 14 , 7 , − 7 , − 14 , − 7 , 7 ) , ( 2 , − 11 , − 13 , − 2 , 11 , 13 ) (70) on C 6 C_6 C 6 O 6 O_6 O 6
Finally, the two Keleti–Kolountzakis sets in (67) satisfy F = ( k + 2 ) − E F=(k+2)-E F = ( k + 2 ) − E (69) , including the failed reflection test, is therefore logically necessary rather than a second presentation of the same witness.
Scope boundary. The result is restricted to binary indicators on C 2 p q C_{2pq} C 2 pq p < q p<q p < q
Interpretation and prior-art boundary
The common invariant of the paper is the filtration
L ≤ d ( S ) ⊂ K ( S ) . \mathcal{L}_{\le d}(S)\subset K(S). L ≤ d ( S ) ⊂ K ( S ) . It has three readings that should not be conflated. Its quotient-group completion is the supportwise separating-atom invariant of Proposition 2.1; its lattice points are the first dilations of the zero-sum face (11) ; and its complex character group is the phase ambiguity modulo translations in Proposition 3.2. All three readings now give the same closing threshold on every stable C 2 p q C_{2pq} C 2 pq
The two algebraic readings may be compressed as
L ≤ d ( S ) = K ( S ) ⟺ all complex invisible ratios are translations, ⟺ all separating atoms over S have length ≤ d . (71) \mathcal{L}_{\le d}(S)=K(S)\ \Longleftrightarrow\ \text{all complex invisible ratios are translations,}
\\
\Longleftrightarrow\ \text{all separating atoms over }S\text{ have length }\le d. \tag*{(71)} L ≤ d ( S ) = K ( S ) ⟺ all complex invisible ratios are translations, ⟺ all separating atoms over S have length ≤ d . (71) For rational signals, equality gives sufficiency immediately, but failure initially produces only a complex character obstruction. Realizing it by two rational Fourier vectors requires Galois covariance. Section 10 constructs such a lower-bound ratio on every stable C 2 p q C_{2pq} C 2 pq U n U_n U n
The historical higher-autocorrelation literature established translation invariance, support-generation and ratio-character uniqueness mechanisms, and global finite-cyclic bounds [2 , 16 , 4 , 12 , 8 ] . Konyagin–Lev supply the universal Freiman ambient group used in Section 4 [14 ] ; the modern separating-invariant works supply the zero-sum generation criterion, non-closed-field Galois-stable criterion, quotient-group and degree language [5 , 6 , 7 SZZ26, 10 ] ; and the recent autocorrelation papers supply the exact Fourier and rationality criteria and the support-specific questions [1 , 3 ] . The zero-sum face reformulation connects the result to the polyhedral counting framework of Han–Wang–Zhang–Zhang [11 ] .
For subsets, Pebody supplies the finite-abelian reconstruction framework, the cyclic upper bound six, the multiset classification and the even-order value-four conjecture [17 , 18 ] . Together with Keleti–Kolountzakis’s even-order three-deck obstruction, that classification already gives the exact infinite families C 2 a C_{2^a} C 2 a a ≥ 4 a\ge4 a ≥ 4 C 2 ℓ k C_{2\ell^k} C 2 ℓ k ℓ > 3 \ell>3 ℓ > 3 2 k > 10 2^k>10 2 k > 10 [13 ] . The family C 2 p q C_{2pq} C 2 pq [19 ] . The binary increment here is the uniform exclusion of all deep 2 p q 2pq 2 pq
The overlap with Domokos’s 2026 criterion is architectural and direct: Theorem 5.4 there decides separation by testing integral generation on every Galois-stable character subset of a regular representation. Here a support S S S K ( S ) K(S) K ( S ) ( 1 , 2 , 154 , 80 , 4 , 14 ) (1, 2, 154, 80, 4, 14) ( 1 , 2 , 154 , 80 , 4 , 14 ) ( 1 , 2 , 154 , 92 , 4 , 2 ) (1, 2, 154, 92, 4, 2) ( 1 , 2 , 154 , 92 , 4 , 2 )
Within that boundary, the contribution assembled here is one theorem chain. For unit supports of all even cyclic groups, the fourth-order quotient has a sharp dichotomy, the exceptional branch closes at six, and its entire rational phase family is the norm-one torus over Q ( ζ 6 ) \mathbb{Q}(\zeta_6) Q ( ζ 6 ) C 2 p q C_{2pq} C 2 pq Q \mathbb{Q} Q Z \mathbb{Z} Z
The regime change has a local explanation. If 3 ∤ 2 p q 3 \nmid2pq 3 ∤ 2 pq P 1 ∪ Q 1 P_1 \cup Q_1 P 1 ∪ Q 1 p = 3 p = 3 p = 3 P 1 P_1 P 1 G G G
Table 4. The theorem boundary of the unified paper.
Several stronger-sounding statements are deliberately absent. We do not claim absolute priority, a complete ambiguity-torus parametrization on every atlas row, a classification for arbitrary non-stable supports, or a filtration through all cyclotomic subfields. We also do not claim the full even cyclic set-reconstruction conjecture. Nor do the exact-arithmetic replays in Appendix C replace any uniform proof. They audit the theorem architecture and its finite routing only.
The computations do have a reproducibility role. They check Boolean exhaustiveness, connector images, endpoint branches, and independent Smith presentations. Mutation controls reject missing X10B sign glue, altered failure sets, and corrupted atlases. The direction of dependence remains unchanged: the written lemmas imply the theorem, while the receipts test their implementation and transcription.
The maximum orbit-stable local separating degree on C 2 p q C_{2pq} C 2 pq
Several separate questions remain natural: a complete rational ambiguity parametrization on every atlas row, analogous Skeletons for higher-rank groups, and robust reconstruction from approximate data. They require, respectively, a finer cocycle/descent analysis, a larger orbit geometry, and quantitative conditioning estimates. The exact cyclic result here is a base case for those directions, not a claim to solve them.
Local reductions and uniform connector schemata
In CRT coordinates write
P [ a ] = ( 0 , a , 0 ) , Q [ b ] = ( 0 , 0 , b ) , P 1 [ a ] = ( 1 , a , 0 ) , Q 1 [ b ] = ( 1 , 0 , b ) , F [ a , b ] = ( 0 , a , b ) , G [ a , b ] = ( 1 , a , b ) , E = ( 1 , 0 , 0 ) . \begin{aligned}
P[a] &= (0,a,0), \qquad Q[b] = (0,0,b), \\
P1[a] &= (1,a,0), \qquad Q1[b] = (1,0,b), \\
F[a,b] &= (0,a,b), \qquad G[a,b] = (1,a,b), \qquad E = (1,0,0).
\end{aligned} P [ a ] P 1 [ a ] F [ a , b ] = ( 0 , a , 0 ) , Q [ b ] = ( 0 , 0 , b ) , = ( 1 , a , 0 ) , Q 1 [ b ] = ( 1 , 0 , b ) , = ( 0 , a , b ) , G [ a , b ] = ( 1 , a , b ) , E = ( 1 , 0 , 0 ) . Expanded local reductions
We record the endpoint details suppressed from Proposition 6.1. Let a target-valued function kill the appropriate rows.
On a parity-zero punctured line, pairs give f ( − x ) = − f ( x ) f(-x)=-f(x) f ( − x ) = − f ( x )
f ( x + y ) = f ( x ) + f ( y ) ( x , y , x + y ≠ 0 ) . (72) f(x+y)=f(x)+f(y) \qquad(x,y,x+y\ne0). \tag*{(72)} f ( x + y ) = f ( x ) + f ( y ) ( x , y , x + y = 0 ) . (72) Fixing a generator of the prime cycle and walking through its nonzero multiples proves f ( x ) = x u f(x)=xu f ( x ) = xu r u = 0 ru=0 r u = 0 r ≥ 5 r\ge5 r ≥ 5 ( x , 1 , 1 , − x − 2 ) (x,1,1,-x-2) ( x , 1 , 1 , − x − 2 )
f ( x + 2 ) − f ( x ) = 2 f ( 1 ) . (73) f(x+2)-f(x)=2f(1). \tag*{(73)} f ( x + 2 ) − f ( x ) = 2 f ( 1 ) . (73) The translation-by-two cycle on C r C_r C r f ( x ) = c + x u f(x)=c+xu f ( x ) = c + xu 2 c = 0 2c=0 2 c = 0 r ≥ 7 r\ge7 r ≥ 7 ( 2 , 2 , 2 , r − 6 ) (2,2,2,r-6) ( 2 , 2 , 2 , r − 6 ) 4 c + r u = 0 4c+ru=0 4 c + r u = 0 r u = 0 ru=0 r u = 0 r = 5 r=5 r = 5 ( 1 , 1 , 1 , 2 ) (1,1,1,2) ( 1 , 1 , 1 , 2 ) 4 c + 5 u = 0 4c+5u=0 4 c + 5 u = 0 5 u = 0 5u=0 5 u = 0 c + x u c+xu c + xu r = 3 r=3 r = 3 Z \mathbb{Z} Z
For the parity-zero rectangle F F F p , q ≥ 5 p,q \ge5 p , q ≥ 5 s ∈ C p s \in C_p s ∈ C p
Δ s ( x , y ) = f ( x + s , y ) − f ( x , y ) \Delta_s(x,y)=f(x+s,y)-f(x,y) Δ s ( x , y ) = f ( x + s , y ) − f ( x , y ) when both endpoints are in the punctured rectangle. Given two base points, choose c c c
{ 0 , − s , − x − s , − x ′ − s } \{0,-s,-x-s,-x'-s\} { 0 , − s , − x − s , − x ′ − s } and d d d { 0 , − y , − y ′ } \{0,-y,-y'\} { 0 , − y , − y ′ } p ≥ 5 p \ge5 p ≥ 5 q ≥ 7 q \ge7 q ≥ 7 p = 5 p=5 p = 5
( x + s , y ) + ( c , d ) = ( x , y ) + ( c + s , d ) (74) (x+s,y)+(c,d)=(x,y)+(c+s,d) \tag*{(74)} ( x + s , y ) + ( c , d ) = ( x , y ) + ( c + s , d ) (74) and to the primed version makes Δ s \Delta_s Δ s Δ s + t = Δ s + Δ t \Delta_{s+t}=\Delta_s+\Delta_t Δ s + t = Δ s + Δ t f ( a , b ) = c + a u + b v f(a,b)=c+au+bv f ( a , b ) = c + a u + b v p u = q v = 0 pu=qv=0 p u = q v = 0 2 c = 0 2c=0 2 c = 0 ( 1 , 1 ) , ( 1 , 1 ) , ( − 2 , − 2 ) (1,1),(1,1),(-2,-2) ( 1 , 1 ) , ( 1 , 1 ) , ( − 2 , − 2 ) 3 c = 0 3c=0 3 c = 0 c = 0 c=0 c = 0
When p = 3 p=3 p = 3 h ( b ) = f ( 1 , b ) h(b)=f(1,b) h ( b ) = f ( 1 , b ) C 3 C_3 C 3 h h h q q q q = 5 q=5 q = 5 h ( b ) = u + b v h(b)=u+bv h ( b ) = u + b v 3 u = q v = 0 3u=qv=0 3 u = q v = 0 f ( a , b ) = a u + b v f(a,b)=au+bv f ( a , b ) = a u + b v
For the parity-one rectangle G G G
( 1 , x + s , y ) , ( 1 , − x , − y ) , ( 1 , x ′ , y ′ ) , ( 1 , − x ′ − s , − y ′ ) (75) (1,x+s,y),\ (1,-x,-y),\ (1,x',y'),\ (1,-x'-s,-y') \tag*{(75)} ( 1 , x + s , y ) , ( 1 , − x , − y ) , ( 1 , x ′ , y ′ ) , ( 1 , − x ′ − s , − y ′ ) (75) sum to zero whenever their coordinates are nonzero. Pair reduction of (A.4) makes horizontal differences base-point independent; the vertical argument is identical. This gives c + a u + b v c+au+bv c + a u + b v 2 c = p u = q v = 0 2c=pu=qv=0 2 c = p u = q v = 0 p = 3 p=3 p = 3 C q × C_q^\times C q × h ( b ) = u + b v h(b)=u+bv h ( b ) = u + b v u u u q v = 0 qv=0 q v = 0 Table 2 , including the smallest generic prime.
Table 2. Cumulative local box presentations.
The fourteen structural types
Every displayed parameter is nonzero, and every derived parameter is required to be nonzero. The following table gives the fourteen structural types; an exchanged orientation means p ↔ q p \leftrightarrow q p ↔ q P ↔ Q P \leftrightarrow Q P ↔ Q P 1 ↔ Q 1 P1 \leftrightarrow Q1 P 1 ↔ Q 1
Table 2 (PDF p. 39).
caption:
:::
A count string uses the order ( E , P , Q , P 1 , Q 1 , F , G ) (E, P, Q, P1, Q1, F, G) ( E , P , Q , P 1 , Q 1 , F , G ) − 1 -1 − 1 − 2 -2 − 2 a = c = 1 a = c = 1 a = c = 1 a + c = 2 ≠ 0 a + c = 2 \ne0 a + c = 2 = 0 a = c = b = d = 1 a = c = b = d = 1 a = c = b = d = 1 − 1 -1 − 1 − 2 -2 − 2 p = 3 p = 3 p = 3 q = 5 q = 5 q = 5
The A and B choices of X10 share a box-count type but have different images in the local presentation. X10B is needed for the P 1 − G P1 - G P 1 − G (40) and for the integral relation u = g u = g u = g (41) . This parameter distinction is why a count-pattern catalogue alone is insufficient for the Quartic Skeleton.
For completeness, the connector functions are partitioned as follows. X2, X3, and X6 attach prime-axis slopes to odd lines or rectangles. X1, X4, X7, X9, X10B, X13, and X14 compare rectangle slopes and cross-box parity. X5, X8, X10A/B, X11, and X12 anchor parity constants. Since two is invertible in both odd-primary layers, a doubled slope difference imposes the same equality as an undoubled one. In the p = 3 p = 3 p = 3 (41) .
Connector images in the prime-three layer
For p = 3 p = 3 p = 3 C q C_q C q ( e , p , u , c , f , g ) (e, p, u, c, f, g) ( e , p , u , c , f , g ) § 8 . The required images are
Table 11.
Substitution of the local forms into each parameterized zero-sum row gives this table. Every image vanishes under ( e , p , u , c , f , g ) ↦ ( 3 , 4 , 1 , 3 , 4 , 1 ) (e, p, u, c, f, g) \mapsto(3, 4, 1, 3, 4, 1) ( e , p , u , c , f , g ) ↦ ( 3 , 4 , 1 , 3 , 4 , 1 ) C 6 C_6 C 6 C q C_q C q Lemma 8.1 . Thus (A.5) is an integral presentation, not the result of evaluating at one value of q q q
In the generic branch, primary projection turns the same substitutions into graph edges. The p p p P P P P 1 P1 P 1 F F F G G G q q q Q Q Q Q 1 Q1 Q 1 F F F G G G E E E P 1 P1 P 1 Q 1 Q1 Q 1 G G G G G G P 1 − Q 1 P1-Q1 P 1 − Q 1 § 8 . These images account for every use of the connector table in the proof.
Two exact closing rows
The generic critical pair P 1 ∪ Q 1 P1 \cup Q1 P 1 ∪ Q 1 ( 1 , 1 , − 2 ) (1,1,-2) ( 1 , 1 , − 2 ) P 1 P1 P 1 G G G 6 u 6u 6 u P 1 ∪ Q 1 P1 \cup Q1 P 1 ∪ Q 1 P 1 P1 P 1 Q 1 Q1 Q 1 q q q 3 u + c 3u+c 3 u + c
Boolean routing and exact mask lists
The hexadecimal encoding assigns bit values
E = 0 x 01 , P = 0 x 02 , Q = 0 x 04 , P 1 = 0 x 08 , Q 1 = 0 x 10 , F = 0 x 20 , G = 0 x 40. (76) E = 0x01,\quad P = 0x02,\quad Q = 0x04,\quad P1 = 0x08,\quad Q1 = 0x10,\quad F = 0x20,\quad G = 0x40. \tag*{(76)} E = 0 x 01 , P = 0 x 02 , Q = 0 x 04 , P 1 = 0 x 08 , Q 1 = 0 x 10 , F = 0 x 20 , G = 0 x 40. (76) For example, 0 x 1 a = P ∪ P 1 ∪ Q 1 0x1a = P \cup P1 \cup Q1 0 x 1 a = P ∪ P 1 ∪ Q 1
The 77 cubic masks
Evaluating the ten clauses of (35) by the eight assignments of ( E , P , Q ) (E,P,Q) ( E , P , Q )
( E , P , Q ) 000 100 010 110 001 101 011 111 total # 6 5 9 9 9 9 15 15 77 (77) \begin{array}{c|cccccccc|c}
(E,P,Q) & 000 & 100 & 010 & 110 & 001 & 101 & 011 & 111 & \text{total}\\
\# & 6 & 5 & 9 & 9 & 9 & 9 & 15 & 15 & 77 \tag*{(77)}
\end{array} ( E , P , Q ) # 000 6 100 5 010 9 110 9 001 9 101 9 011 15 111 15 total 77 (77) Here each three-character label lists the truth values in the displayed order ( E , P , Q ) (E,P,Q) ( E , P , Q ) (35) and enumerating the sixteen assignments of ( P 1 , Q 1 , F , G ) (P1,Q1,F,G) ( P 1 , Q 1 , F , G )
The following matrix expands every positive clause. “Optional” means that the indicated box may be either selected or absent under that clause; each line is used only when it is selected. The symbols e e e x x x y y y z z z E E E P 1 P1 P 1 Q 1 Q1 Q 1 G G G
In T 6 T_6 T 6 T 7 T_7 T 7 T 8 T_8 T 8 E E E (34) .
The complete positive list is
Table 6. Complete cubic sufficiency routing, clauses T 1 T_1 T 1 T 5 T_5 T 5
Table 7. Complete cubic sufficiency routing, clauses T 6 T_6 T 6 T 10 T_{10} T 10
02 03 04 05 06 07 0a 0b 0e 0f 14 15 16 17 1f 20 21 22 23 24 25 26 27 2a 2b 2e 2f 34 35 36 37 38 3a 3b 3c 3d 3e 3f 46 4e 4f 56 57 5e 5f 60 61 62 63 64 65 66 67 68 69 6a 6b 6c 6d 6e 6f 70 71 72 73 74 75 76 77 78 79 7a 7b 7c 7d 7e 7f.
Each entry is hexadecimal and carries the prefix 0x understood. The necessity witnesses in § 7 may be summarized by
Table B.3.
Formula (7.4) is verified by the same fixed truth table: every assignment failing (7.1) satisfies at least one N i N_i N i N i N_i N i
The Boolean verification can also be organized without hexadecimal masks. Split first on whether G G G G G G F F F p p p q q q N 1 N_1 N 1 N 2 N_2 N 2 F F F F , G F,G F , G G G G P 1 P1 P 1 P P P F F F N 3 N_3 N 3 N 4 N_4 N 4 N 5 , N 6 N_5,N_6 N 5 , N 6 N 7 N_7 N 7 N 8 , N 9 , N 10 N_8,N_9,N_{10} N 8 , N 9 , N 10
For the integer defects, choose an odd function supported on one opposite pair. A triple that can see it either contains the opposite argument and cancels, or requires one of the boxes excluded by its N i N_i N i Z \mathbb{Z} Z C 2 C_2 C 2
Quartic routing lists
After removing E E E
08 09 0c 0d 10 11 12 13 19 1b 1d 1e 28 29 2c 2d 30 31 32 33 39 40 41 42 43 44 45 47 48 49 4a 4b 4c 4d 50 51 52 53 54 55 58 59 5a 5b 5c 5d.
The three omitted failures are 18, 1a, 1c. When p = 3 p = 3 p = 3
09 0d 10 11 12 13 19 1b 1d 1e 28 29 2c 2d 30 31 32 33 39 41 42 43 45 47 49 4a 4b 4d 50 51 52 53 54 55 58 59 5a 5b 5c 5d.
The nine omitted failures are 08, 0c, 18, 1a, 1c, 40, 44, 48, 4c.
The bridge masks are 1 a = P ∪ P 1 ∪ Q 1 1a = P \cup P1 \cup Q1 1 a = P ∪ P 1 ∪ Q 1 1 c = Q ∪ P 1 ∪ Q 1 1c = Q \cup P1 \cup Q1 1 c = Q ∪ P 1 ∪ Q 1 18 = P 1 ∪ Q 1 18 = P1 \cup Q1 18 = P 1 ∪ Q 1
08 , 0 c , 18 , 40 , 44 , 48 , 4 c , 08,\ 0c,\ 18,\ 40,\ 44,\ 48,\ 4c, 08 , 0 c , 18 , 40 , 44 , 48 , 4 c , which decode exactly to the seven supports in (44) . This recovers the base counts 77 + 46 + 2 + 1 + 1 = 127 77 + 46 + 2 + 1 + 1 = 127 77 + 46 + 2 + 1 + 1 = 127 77 + 40 + 2 + 7 + 1 = 127 77 + 40 + 2 + 7 + 1 = 127 77 + 40 + 2 + 7 + 1 = 127 E E E
The list is a routing index, not a second proof. Membership in the cubic list routes to Theorem 7.1 ; membership in a quartic list routes to the generic graph or prime-three integral case of Theorem 8.2 ; the remaining nine masks route to Propositions 9.1 , 9.2 , and 9.3 . Each route is uniform in the primes and carries a defect at the preceding order.
Reproducibility and assurance boundary
All theorem proofs are contained in the preceding sections. The companion repository supplies exact-arithmetic, symbolic, manuscript and package audits. They detect transcription, routing, endpoint, convention and software errors; no finite replay is used to infer a statement for all primes.
Table 8. Assurance layers and their roles.
From the repository root, the reviewer first selects independently controlled absolute Python and Git executables. In a POSIX shell the standard routes are
PYTHON_ABSOLUTE=/absolute/path/to/python
GIT_ABSOLUTE=/absolute/path/to/git
trusted() {
target=$1
shift
"$PYTHON_ABSOLUTE" -I -S -B scripts/runtime_bootstrap.py \
--git-executable "$GIT_ABSOLUTE" --target "$target" -- "$@"
}trusted scripts/verify.py --profile manifest
trusted scripts/verify.py --profile core
trusted scripts/verify.py --profile full
"$PYTHON_ABSOLUTE" -I -S -B -O scripts/runtime_bootstrap.py \
--git-executable "$GIT_ABSOLUTE" --target scripts/verify.py -- \
--profile full
trusted scripts/verify_manifest_only.py
trusted scripts/verify_git_checkout.py --profile fullThe equivalent PowerShell invocation and the exact runtime trust boundary are recorded in REPRODUCE.md. The displayed commands are the actual bootstrap route: direct execution of a verifier is intentionally rejected. The core route runs the unit, UAG, CRT, Skeleton and bounded binary checks. Individual scripts write runtime JSON under results/ and generated presentations under certificates/; the stable aggregate is results/public_package_verification.json. The source manifest and release attestation bind the distributed inputs and title-named PDF.
Build and visual verification
The title-named distributed PDF is produced from a clean source-only copy by the repository’s locked builder:
TOOL_CACHE_PARENT=/absolute/path/to/TectonicProject
TECTONIC_CACHE_ROOT="$TOOL_CACHE_PARENT/Tectonic/bundles"
trusted scripts/build_locked_pdf.py \
--tectonic-executable /absolute/path/to/tectonic \
--tectonic-release-archive /absolute/path/to/release.zip \
--tectonic-cache-root "$TECTONIC_CACHE_ROOT" \
--source-dir /absolute/path/to/clean/source \
--output-dir /absolute/path/to/empty/outputThe resolved cache-root argument must end literally in TectonicProject/Tectonic/bundles; the builder rejects any other layout before compilation. The tracked pdf_toolchain.lock.json binds the Tectonic release asset, executable and resource-bundle identity. The tracked tectonic_cache_payloads.lock.json enumerates a canonical closed cache of 535 payloads by relative path, byte count and SHA-256; it contains every resource read by a clean build, plus a conservatively retained locked superset. The wrapper rejects missing or extra payloads, symlinks, junctions and other reparse points; verifies the closed payload tree both before and after compilation; and then passes the locked bundle explicitly together with --only-cached. An empty cache can first be materialized with scripts/materialize_locked_tectonic_cache.py; in the current freeze the clean build fetches 393 resources and the materializer explicitly adds the other 142 locked entries. That one online bootstrap must be followed by the complete payload check, while accepted release builds are offline. A different executable, payload tree, bundle or command fails closed. A conventional pdfLaTeX/BibTeX build remains the independent cross-engine lane:
job=Autocorrelation_Phase_Lattices_on_Cyclic_Groups
pdflatex -halt-on-error -jobname="$job" main.texbibtex "$job"
pdflatex -halt-on-error -jobname="$job" main.tex
pdflatex -halt-on-error -jobname="$job" main.texTwo clean locked Tectonic builds are compared byte for byte. The final PDF is rendered independently and every page is inspected for clipping, overflow, overlap, broken tables and unintended blank pages. Extracted text is scanned for unresolved references. Every cited key must occur in the repository-local refs.bib, every bibliography entry must be cited, and the four reconstruction sources must retain their checked identifiers.
Package and claim boundary
The repository contains only manuscript source, verification code, generated certificates and review documentation. It redistributes no third-party paper PDFs. The bounded binary audit does not replace the all-prime exclusion proof. The optional translation/reflection corollary keeps the analytical deck oriented and changes only the output quotient. Reproduction establishes author-operated package coherence; it does not establish priority, independent external reproduction, publication, journal acceptance or a DOI.
Data and code availability
The theorem proofs are contained in this manuscript. The companion repository is available at
https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups.
It contains the exact-arithmetic checks, symbolic certificates, source-boundary records and deterministic replay described in Appendix § 15 . Those artifacts are assurance layers rather than premises of the proofs. Third-party full texts are not redistributed.
Acknowledgment of prior frameworks
The Fourier monomial and Galois criteria used here are taken from Casper–Orozco; the classical translation-invariant and global sixth-order lines include Adler–Konheim and Grünbaum–Moore. The universal ambient group belongs to the Konyagin–Lev framework. Domokos supplies the zero-sum generation criterion, including its non-closed-field, Galois-stable-support form; the quotient-group/separating-atom language is drawn from the separating-invariant literature. Precise uses are identified at the relevant statements. The supportwise realization step here consists of explicit Galois-equivariant lower-bound ratios and does not re-claim those general criteria.
AI assistance and verification status
This work involved substantial AI assistance using GPT-5.6 Sol via OpenAI Codex, as reported by the author. AI systems contributed to mathematical exploration, candidate proof arguments, verification code, and manuscript drafting and revision. The author directed the research and is responsible for the claims and source attributions. The computational checks and AI reviews are assurance evidence, not independent human peer review or an end-to-end formal proof. At the time of this Hexagon submission, the author reports that no part of the contribution has been checked and understood by a human.
