Introduction
A gravitational instanton is a complete oriented Ricci-flat Riemannian four-manifold with finite curvature energy. For an effective isometric two-torus action, the boundary intervals of the orbit space record the circles that collapse. These intervals are called rods; their endpoints are turning points, or nuts. A primitive integral vector specifies each collapsing circle. The ordered vectors encode topology, whereas their normalization determines the cone angles. The central question is which rod configurations admit smooth Ricci-flat realizations and how the resulting families of gravitational instantons are organized.
The discovery of the Chen–Teo instanton [8 ] showed that the regular asymptotically flat (AF) solutions extend beyond the Euclidean Kerr family, including its Schwarzschild limit. It disproved the proposed Riemannian analogue of black-hole uniqueness and opened the problem of describing a larger moduli space. Li and Sun subsequently proved the existence of an infinite sequence of AF toric instantons with arbitrarily many turning points [20 ] ; beyond the Kerr and Chen–Teo members, these metrics are non-Hermitian. Teo gave an explicit construction of the four-turning-point member, reducing the remaining balance condition to a quintic equation [26 ] . The present work seeks the structure beyond this sequence. We derive restrictions on the possible rod data, construct a certified catalogue of regular solutions, and use it to test two descriptions of the moduli space: one through root arrangements and polygon walks, the other through compositions of endpoint geometries.
The Belinski–Zakharov (BZ) method constructs metrics from finitely many spectral poles and charge parameters [5 , 6 ] . Twistor theory describes the same field equations through a rational matrix on the symmetry axis, called the patching matrix [27 , 13 ] . Harmonic-map methods instead start from prescribed rods and give a canonical metric, generally with conical singularities [19 , 20 ] . Two recent representation results make the finite algebraic approach particularly significant.
Dunajski, Mason and Tod give a complete twistor representation of toric gravitational instantons with asymptotically locally flat or asymptotically locally Euclidean ends. They construct the rational patching matrices recursively and recover the associated metrics by linear algebra [12 ] . Andersson, Dahl and Nilsson prove that simply connected toric instantons with AF or asymptotically locally flat (ALF) ends and a finite positive number of turning points admit a flat-seed BZ representation [4 ] . An ALF end has a circle of bounded length at infinity; precise conventions are given below. These results reduce geometric classification to finite data, but the cone-balance conditions still have to be solved.
Our first contribution is a factorization of the positive BZ construction by signed Pfaffian tau-functions. The algebraic starting point is the Cauchy–Pfaffian identity
det ( 1 − a i a j 1 − b i b j ) i , j = 1 n = τ n ( a , b ) τ n ( − a , b ) ∏ i = 1 n ( 1 − b i 2 ) . \det\left(\frac{1-a_i a_j}{1-b_i b_j}\right)_{i,j=1}^{n}
=
\frac{\tau_n(a,b)\,\tau_n(-a,b)}{\prod_{i=1}^{n}(1-b_i^2)}. det ( 1 − b i b j 1 − a i a j ) i , j = 1 n = ∏ i = 1 n ( 1 − b i 2 ) τ n ( a , b ) τ n ( − a , b ) . Here a = ( a i ) a=(a_i) a = ( a i ) b = ( b i ) b=(b_i) b = ( b i ) τ n \tau_n τ n b i b_i b i a i a_i a i [1 , 2 ] ; an explicit change of charge variables gives the Neugebauer–Kramer determinant representation [22 , 23 ] . The pure-spinor interpretation explains the Pfaffian relations; the positive moment representation makes bulk positivity and axis limits explicit. Polynomial factors of the patching matrix connect these formulas to the boundary description. The closed Hankel formulas of [4 ] give a complementary representation and a smooth-completion theorem; Section 4.5 supplies the dictionary needed to apply it here.
Our second contribution is a variational description of balance. For a marked rod word W W W ℓ j \ell_j ℓ j θ j \theta_j θ j M M M
b j = log 2 π θ j , Φ W ( ℓ ) = ∑ j ℓ j b j − 2 M . b_j = \log\frac{2\pi}{\theta_j}, \qquad\Phi_W(\ell) = \sum_j \ell_j b_j - 2M. b j = log θ j 2 π , Φ W ( ℓ ) = j ∑ ℓ j b j − 2 M . We prove that ∇ ℓ Φ W = b \nabla_\ell\Phi_W = b ∇ ℓ Φ W = b Φ W \Phi_W Φ W Φ W \Phi_W Φ W 2 M 2M 2 M § 6.5 .
The third contribution is a catalogue of regular metrics obtained from analytic restrictions, exhaustive enumeration and rigorous existence certificates. Positive moment identities imply integral determinant bounds and restrict the finite self-intersections to − 1 , 0 , 1 -1, 0, 1 − 1 , 0 , 1 [8 , 26 ] ; exact enumeration through twelve genuine turning points gives 359 AF candidates. For 356 of these rod-word classes we prove the existence of a smooth complete AF metric at a specified interior twist, and hence of a local family; three further candidates are excluded by a positive boundary gradient. The ALF census through eight turning points gives 52 certified isolated regular metrics on 49 words. The finite certificates establish exact solutions of the full normalized rod equations. The analytic positive representation and completion theorem establish positivity throughout the bulk, smooth rods and nuts, and completeness of the end. Thus the catalogue consists of proved existence results, rather than metrics accepted on the basis of small residuals or a sampled regularity grid. Full-interval AF existence retains the stated collision hypotheses, and the ALF census is restricted to the enumerated positive reduced sectors.
The catalogues also serve as experimental templates for a classification. The first approach retains the distinct rod directions and identifies them with generalized rank-two root systems, equivalently triangulated polygons [11 , 10 ] . The ordered walk on the directions is essential: the same polygon can support both obstructed and regular words. The second approach composes marked endpoint components using their boundary directions and twist residues. Nine components generate exactly the 356 retained AF words through twelve turning points. This is an exact finite combinatorial result, independent of solving the parent balance equations. It does not yet prove that every permitted assembly opens to a balanced family. The ALF catalogue supplies components for this construction, while the AF catalogue tests their compatibility and the predicted opposite endpoints.
Together with the representation result of [4 ] , these restrictions and certificates provide a route to a geometric classification: one must control all permitted sectors, prove the required collision matching, and identify equivalent metrics. The recursive patching-matrix construction of [12 ] addresses the representation stage; our cone potential, admissibility tests and catalogue address the regular locus within it. Neither recursive representation nor the observed endpoint rules alone establish the missing global existence statement.
Conventions and organization. The integer n n n N N N
d t 2 + d r 2 + r 2 ( d ϑ 2 + sin 2 ϑ d ϕ 2 ) , dt^2 + dr^2 + r^2(d\vartheta^2+\sin^2\vartheta\,d\phi^2), d t 2 + d r 2 + r 2 ( d ϑ 2 + sin 2 ϑ d ϕ 2 ) , with identifications ( t , ϕ ) ∼ ( t , ϕ + 2 π ) ∼ ( t + 2 π L , ϕ + 2 π β ) (t,\phi)\sim(t,\phi+2\pi)\sim(t+2\pi L,\phi+2\pi\beta) ( t , ϕ ) ∼ ( t , ϕ + 2 π ) ∼ ( t + 2 π L , ϕ + 2 π β ) L > 0 L>0 L > 0 − 1 < β < 0 -1<\beta<0 − 1 < β < 0
Sections 2–4 develop the algebraic representations; Sections 5–6 treat admissibility and balance. Sections 7 and 8 present the two classification approaches consecutively. Appendix A contains the AF equations, explicit low-order examples and catalogue. Appendix B develops the ALF marking and records its certified solutions and endpoint components; the final appendix supplies supplementary analytic arguments.
Three levels of the soliton construction
The inverse-scattering, determinant and tau descriptions encode the same solution. We fix their normalizations together, including the different flat seeds for even and odd dressing. Throughout this section, n n n N N N T T T I I I 1 \mathbf{1} 1 adj \operatorname{adj} adj i 2 = − 1 i^2=-1 i 2 = − 1
The original BZ method
The original BZ method. Let G G G 2 × 2 2 \times2 2 × 2 ν \nu ν
g = G a b d x a d x b + e 2 ν ( d ρ 2 + d z 2 ) , ( x 1 , x 2 ) = ( t , ϕ ) , det G = ρ 2 . (1) g = G_{ab} dx^a dx^b + e^{2\nu}(d\rho^2 + dz^2), \qquad(x^1,x^2) = (t,\phi), \qquad\det G = \rho^2. \tag*{(1)} g = G ab d x a d x b + e 2 ν ( d ρ 2 + d z 2 ) , ( x 1 , x 2 ) = ( t , ϕ ) , det G = ρ 2 . (1) On the open orbit space ρ > 0 \rho>0 ρ > 0
U = ρ G ρ G − 1 , V = ρ G z G − 1 . (2) U = \rho G_{\rho}G^{-1}, \qquad V = \rho G_{z}G^{-1}. \tag*{(2)} U = ρ G ρ G − 1 , V = ρ G z G − 1 . (2) The vacuum equations are
∂ ρ U + ∂ z V = 0 , (3) \partial_{\rho}U+\partial_{z}V=0, \tag*{(3)} ∂ ρ U + ∂ z V = 0 , (3) ν ρ = − 1 2 ρ + tr ( U 2 − V 2 ) 8 ρ , ν z = tr ( U V ) 4 ρ . (4) \nu_{\rho}=-\frac{1}{2\rho}+\frac{\operatorname{tr}(U^2-V^2)}{8\rho}, \qquad\nu_{z}=\frac{\operatorname{tr}(UV)}{4\rho}. \tag*{(4)} ν ρ = − 2 ρ 1 + 8 ρ tr ( U 2 − V 2 ) , ν z = 4 ρ tr ( U V ) . (4) For positive definite G G G Φ = ρ − 1 G \Phi=\rho^{-1}G Φ = ρ − 1 G ν \nu ν
The spectral problem. Let λ s p \lambda_{\mathrm{sp}} λ sp
D ρ Ψ = ρ U + λ s p V ρ 2 + λ s p 2 Ψ , D ρ = ∂ ρ + 2 ρ λ s p ρ 2 + λ s p 2 ∂ λ s p , \mathcal{D}_{\rho}\Psi=\frac{\rho U+\lambda_{\mathrm{sp}}V}{\rho^2+\lambda_{\mathrm{sp}}^2}\Psi, \qquad\mathcal{D}_{\rho}=\partial_{\rho}+\frac{2\rho\lambda_{\mathrm{sp}}}{\rho^2+\lambda_{\mathrm{sp}}^2}\partial_{\lambda_{\mathrm{sp}}}, D ρ Ψ = ρ 2 + λ sp 2 ρ U + λ sp V Ψ , D ρ = ∂ ρ + ρ 2 + λ sp 2 2 ρ λ sp ∂ λ sp , D z Ψ = ρ V − λ s p U ρ 2 + λ s p 2 Ψ , D z = ∂ z − 2 λ s p 2 ρ 2 + λ s p 2 ∂ λ s p . (5) \mathcal{D}_{z}\Psi=\frac{\rho V-\lambda_{\mathrm{sp}}U}{\rho^2+\lambda_{\mathrm{sp}}^2}\Psi, \qquad\mathcal{D}_{z}=\partial_{z}-\frac{2\lambda_{\mathrm{sp}}^2}{\rho^2+\lambda_{\mathrm{sp}}^2}\partial_{\lambda_{\mathrm{sp}}}. \tag*{(5)} D z Ψ = ρ 2 + λ sp 2 ρ V − λ sp U Ψ , D z = ∂ z − ρ 2 + λ sp 2 2 λ sp 2 ∂ λ sp . (5) Its compatibility gives (3) ; the generating matrix is normalized by Ψ ( 0 ) = G \Psi(0)=G Ψ ( 0 ) = G
w = z + λ s p 2 − ρ 2 2 λ s p (6) w=z+\frac{\lambda_{\mathrm{sp}}^2-\rho^2}{2\lambda_{\mathrm{sp}}} \tag*{(6)} w = z + 2 λ sp λ sp 2 − ρ 2 (6) is constant under both D ρ \mathcal{D}_{\rho} D ρ D z \mathcal{D}_{z} D z λ s p \lambda_{\mathrm{sp}} λ sp λ s p ↦ − ρ 2 / λ s p \lambda_{\mathrm{sp}}\mapsto-\rho^2/\lambda_{\mathrm{sp}} λ sp ↦ − ρ 2 / λ sp
The dimensionless coordinate used below is
ζ = i λ s p ρ , w = z − i ρ 2 ( ζ + ζ − 1 ) . (7) \zeta=\frac{i\lambda_{\mathrm{sp}}}{\rho}, \qquad w=z-\frac{i\rho}{2}(\zeta+\zeta^{-1}). \tag*{(7)} ζ = ρ i λ sp , w = z − 2 i ρ ( ζ + ζ − 1 ) . (7) The sheet involution becomes ζ ↦ ζ − 1 \zeta\mapsto\zeta^{-1} ζ ↦ ζ − 1
D ^ ρ Ψ = U − i ζ V ρ ( 1 − ζ 2 ) Ψ , D ^ ρ = ∂ ρ + ζ ( 1 + ζ 2 ) ρ ( 1 − ζ 2 ) ∂ ζ , D ^ z Ψ = V + i ζ U ρ ( 1 − ζ 2 ) Ψ , D ^ z = ∂ z + 2 i ζ 2 ρ ( 1 − ζ 2 ) ∂ ζ . (8) \begin{aligned}
\widehat{\mathcal{D}}_{\rho}\Psi&= \frac{U-i\zeta V}{\rho(1-\zeta^{2})}\Psi, &
\widehat{\mathcal{D}}_{\rho} &= \partial_{\rho}+\frac{\zeta(1+\zeta^{2})}{\rho(1-\zeta^{2})}\partial_{\zeta},\\
\widehat{\mathcal{D}}_{z}\Psi&= \frac{V+i\zeta U}{\rho(1-\zeta^{2})}\Psi, &
\widehat{\mathcal{D}}_{z} &= \partial_{z}+\frac{2i\zeta^{2}}{\rho(1-\zeta^{2})}\partial_{\zeta}. \tag*{(8)}
\end{aligned} D ρ Ψ D z Ψ = ρ ( 1 − ζ 2 ) U − i ζ V Ψ , = ρ ( 1 − ζ 2 ) V + i ζ U Ψ , D ρ D z = ∂ ρ + ρ ( 1 − ζ 2 ) ζ ( 1 + ζ 2 ) ∂ ζ , = ∂ z + ρ ( 1 − ζ 2 ) 2 i ζ 2 ∂ ζ . (8) These operators differentiate at fixed w w w ζ \zeta ζ
Dressing the seed. Given a seed generating matrix Ψ 0 \Psi_{0} Ψ 0 [5 , 6 ] sets
Ψ u n ( λ s p ) = X ( λ s p ) Ψ 0 ( λ s p ) , X ( λ s p ) = I + ∑ i = 1 n R i λ s p − μ i , R i = n i m i . (9) \Psi_{\mathrm{un}}(\lambda_{\mathrm{sp}})=\mathcal{X}(\lambda_{\mathrm{sp}})\Psi_{0}(\lambda_{\mathrm{sp}}),\qquad
\mathcal{X}(\lambda_{\mathrm{sp}})=I+\sum_{i=1}^{n}\frac{R_{i}}{\lambda_{\mathrm{sp}}-\mu_{i}},\qquad
R_{i}=\mathbf{n}_{i}\mathbf{m}_{i}. \tag*{(9)} Ψ un ( λ sp ) = X ( λ sp ) Ψ 0 ( λ sp ) , X ( λ sp ) = I + i = 1 ∑ n λ sp − μ i R i , R i = n i m i . (9) Here m i \mathbf{m}_{i} m i n i \mathbf{n}_{i} n i w = z i w=z_{i} w = z i
μ i = ρ 2 + ( z − z i ) 2 − ( z − z i ) , μ i 2 + 2 ( z − z i ) μ i − ρ 2 = 0. (10) \mu_{i}=\sqrt{\rho^{2}+(z-z_{i})^{2}}-(z-z_{i}),\qquad
\mu_{i}^{2}+2(z-z_{i})\mu_{i}-\rho^{2}=0. \tag*{(10)} μ i = ρ 2 + ( z − z i ) 2 − ( z − z i ) , μ i 2 + 2 ( z − z i ) μ i − ρ 2 = 0. (10) The second branch is − ρ 2 / μ i -\rho^{2}/\mu_{i} − ρ 2 / μ i z i z_{i} z i
Each pole also carries a constant, nonzero polarization row m 0 , i \mathbf{m}_{0,i} m 0 , i
m i = m 0 , i Ψ 0 ( μ i ) − 1 , N i = m i G 0 , Γ i j = m i G 0 m j T ρ 2 + μ i μ j . (11) \mathbf{m}_{i}=\mathbf{m}_{0,i}\Psi_{0}(\mu_{i})^{-1},\qquad
\mathbf{N}_{i}=\mathbf{m}_{i}G_{0},\qquad
\Gamma_{ij}=\frac{\mathbf{m}_{i}G_{0}\mathbf{m}_{j}^{T}}{\rho^{2}+\mu_{i}\mu_{j}}. \tag*{(11)} m i = m 0 , i Ψ 0 ( μ i ) − 1 , N i = m i G 0 , Γ ij = ρ 2 + μ i μ j m i G 0 m j T . (11) The projective polarization coordinate will be denoted by C i C_{i} C i C i C_{i} C i C i C_{i} C i
The symmetric-matrix reduction is
G u n = X ( λ s p ) G 0 X T ( − ρ 2 / λ s p ) . (12) G_{\mathrm{un}}=\mathcal{X}(\lambda_{\mathrm{sp}})G_{0}\mathcal{X}^{T}(-\rho^{2}/\lambda_{\mathrm{sp}}). \tag*{(12)} G un = X ( λ sp ) G 0 X T ( − ρ 2 / λ sp ) . (12) Cancellation of its residue at λ s p = μ i \lambda_{\mathrm{sp}}=\mu_{i} λ sp = μ i
∑ j Γ i j n j = N i T μ i , n i = ∑ j ( Γ − 1 ) i j N j T μ j . (13) \sum_{j}\Gamma_{ij}\mathbf{n}_{j}=\frac{\mathbf{N}_{i}^{T}}{\mu_{i}},\qquad
\mathbf{n}_{i}=\sum_{j}(\Gamma^{-1})_{ij}\frac{\mathbf{N}_{j}^{T}}{\mu_{j}}. \tag*{(13)} j ∑ Γ ij n j = μ i N i T , n i = j ∑ ( Γ − 1 ) ij μ j N j T . (13) Evaluating (9) at $\lambda_{\mathrm{sp}}=0 therefore gives
G u n = G 0 − ∑ i , j = 1 n ( Γ − 1 ) i j μ i μ j N i T N j . (14) G_{\mathrm{un}}=G_{0}-\sum_{i,j=1}^{n}\frac{(\Gamma^{-1})_{ij}}{\mu_{i}\mu_{j}}\mathbf{N}_{i}^{T}\mathbf{N}_{j}. \tag*{(14)} G un = G 0 − i , j = 1 ∑ n μ i μ j ( Γ − 1 ) ij N i T N j . (14) It solves the reduced matrix equation, but generally does not yet have determinant ρ 2 \rho^{2} ρ 2
det G u n = ( − 1 ) n ρ 2 n ∏ i μ i 2 det G 0 . (15) \det G_{\mathrm{un}}=(-1)^{n}\frac{\rho^{2n}}{\prod_{i}\mu_{i}^{2}}\det G_{0}. \tag*{(15)} det G un = ( − 1 ) n ∏ i μ i 2 ρ 2 n det G 0 . (15) Even and odd flat-seed conventions. Put Q ( λ s p ) = ρ 2 − 2 z λ s p − λ s p 2 Q(\lambda_{\mathrm{sp}})=\rho^{2}-2z\lambda_{\mathrm{sp}}-\lambda_{\mathrm{sp}}^{2} Q ( λ sp ) = ρ 2 − 2 z λ sp − λ sp 2
G 0 Ψ 0 ( λ s p ) m i N i n e v e n diag ( 1 , ρ 2 ) diag ( 1 , Q ( λ s p ) ) ( C i , μ i − 1 ) ( C i , ρ 2 μ i − 1 ) n o d d diag ( − 1 , 1 ) diag ( − 1 , 1 ) ( − C i , 1 ) ( C i , 1 ) (16) \begin{array}{c|c|c|c|c}
& G_{0} & \Psi_{0}(\lambda_{\mathrm{sp}}) & \mathbf{m}_{i} & \mathbf{N}_{i}\\
n\ \mathrm{even} & \operatorname{diag}(1,\rho^{2}) & \operatorname{diag}(1,Q(\lambda_{\mathrm{sp}})) & (C_{i},\mu_{i}^{-1}) & (C_{i},\rho^{2}\mu_{i}^{-1})\\
n\ \mathrm{odd} & \operatorname{diag}(-1,1) & \operatorname{diag}(-1,1) & (-C_{i},1) & (C_{i},1) \tag*{(16)}
\end{array} n even n odd G 0 diag ( 1 , ρ 2 ) diag ( − 1 , 1 ) Ψ 0 ( λ sp ) diag ( 1 , Q ( λ sp )) diag ( − 1 , 1 ) m i ( C i , μ i − 1 ) ( − C i , 1 ) N i ( C i , ρ 2 μ i − 1 ) ( C i , 1 ) (16) For even n n n m 0 , i = ( C i , − 2 z i ) \mathbf{m}_{0,i}=(C_{i},-2z_{i}) m 0 , i = ( C i , − 2 z i ) Q ( μ i ) = − 2 z i μ i Q(\mu_{i})=-2z_{i}\mu_{i} Q ( μ i ) = − 2 z i μ i z i = 0 z_{i}=0 z i = 0 n n n m 0 , i = ( C i , 1 ) \mathbf{m}_{0,i}=(C_{i},1) m 0 , i = ( C i , 1 ) − 1 -1 − 1 (1) . The odd dressing and determinant normalization produce the Riemannian section.
Define
λ i = i μ i ρ , Λ = ∏ i λ i , λ i + λ i − 1 = − 2 i ( z − z i ) ρ . (17) \lambda_i = \frac{i\mu_i}{\rho}, \qquad\Lambda= \prod_i \lambda_i, \qquad\lambda_i + \lambda_i^{-1} = -\frac{2i(z-z_i)}{\rho}. \tag*{(17)} λ i = ρ i μ i , Λ = i ∏ λ i , λ i + λ i − 1 = − ρ 2 i ( z − z i ) . (17) Thus λ i \lambda_i λ i ζ \zeta ζ i i i
G = h n G u n , h n = { Λ , n even , i ρ Λ , n odd . (18) G = h_n G_{\mathrm{un}}, \qquad h_n =
\begin{cases}
\Lambda, & n\ \text{even},\\
i\rho\Lambda, & n\ \text{odd}.
\end{cases} \tag*{(18)} G = h n G un , h n = { Λ , i ρ Λ , n even , n odd . (18) They give det G = ρ 2 \det G = \rho^2 det G = ρ 2 G G G h n h_n h n
The conformal factor obtained from (4) is
e 2 ν = k 0 det Γ ∏ i < j ( ρ 2 + μ i μ j ) 2 { ρ − n 2 / 2 ( ∏ i μ i ) 3 − n , n even , ρ − ( n − 1 ) 2 / 2 ( ∏ i μ i ) 2 − n , n odd . (19) e^{2\nu} = k_0 \det\Gamma\prod_{i<j}(\rho^2+\mu_i\mu_j)^2
\begin{cases}
\rho^{-n^2/2}\left(\prod_i\mu_i\right)^{3-n}, & n\ \text{even},\\
\rho^{-(n-1)^2/2}\left(\prod_i\mu_i\right)^{2-n}, & n\ \text{odd}.
\end{cases} \tag*{(19)} e 2 ν = k 0 det Γ i < j ∏ ( ρ 2 + μ i μ j ) 2 { ρ − n 2 /2 ( ∏ i μ i ) 3 − n , ρ − ( n − 1 ) 2 /2 ( ∏ i μ i ) 2 − n , n even , n odd . (19) The constant k 0 k_0 k 0 z i z_i z i
∂ ρ λ i = λ i ( 1 + λ i 2 ) ρ ( 1 − λ i 2 ) , ∂ z λ i = 2 i λ i 2 ρ ( 1 − λ i 2 ) . (20) \partial_\rho\lambda_i = \frac{\lambda_i(1+\lambda_i^2)}{\rho(1-\lambda_i^2)}, \qquad\partial_z\lambda_i = \frac{2i\lambda_i^2}{\rho(1-\lambda_i^2)}. \tag*{(20)} ∂ ρ λ i = ρ ( 1 − λ i 2 ) λ i ( 1 + λ i 2 ) , ∂ z λ i = ρ ( 1 − λ i 2 ) 2 i λ i 2 . (20) The determinant representation
For an integer α \alpha α
M i j ( α ) = 1 − C i C j λ i α λ j α 1 − λ i λ j , d α = det M ( α ) . (21) M_{ij}^{(\alpha)} = \frac{1-C_iC_j\lambda_i^\alpha\lambda_j^\alpha}{1-\lambda_i\lambda_j}, \qquad d_\alpha= \det M^{(\alpha)}. \tag*{(21)} M ij ( α ) = 1 − λ i λ j 1 − C i C j λ i α λ j α , d α = det M ( α ) . (21) Write a ( α ) = C λ α a^{(\alpha)} = C\lambda^\alpha a ( α ) = C λ α D λ = diag ( λ i ) D_\lambda= \operatorname{diag}(\lambda_i) D λ = diag ( λ i ) D μ = diag ( μ i ) D_\mu= \operatorname{diag}(\mu_i) D μ = diag ( μ i )
M ( α + 1 ) = M ( α ) + a ( α ) a ( α ) T = 11 T + D λ M ( α ) D λ (22) M^{(\alpha+1)} = M^{(\alpha)} + a^{(\alpha)}a^{(\alpha)T} = 11^T + D_\lambda M^{(\alpha)}D_\lambda \tag*{(22)} M ( α + 1 ) = M ( α ) + a ( α ) a ( α ) T = 1 1 T + D λ M ( α ) D λ (22) relate all neighboring determinants. The original BZ matrix is
Γ = { D μ − 1 M ( 1 ) D μ − 1 , n even , ρ − 2 M ( 0 ) , n odd . (23) \Gamma=
\begin{cases}
D_\mu^{-1}M^{(1)}D_\mu^{-1}, & n\ \text{even},\\
\rho^{-2}M^{(0)}, & n\ \text{odd}.
\end{cases} \tag*{(23)} Γ = { D μ − 1 M ( 1 ) D μ − 1 , ρ − 2 M ( 0 ) , n even , n odd . (23) Thus the determinant center is 1 for even dressing and 0 for odd dressing.
To include the off-diagonal entry, set
r α = 1 T ( M ( α ) ) − 1 a ( α ) = det ( M ( α ) + a ( α ) 1 T ) d α − 1. (24) r_\alpha= 1^T(M^{(\alpha)})^{-1}a^{(\alpha)} = \frac{\det(M^{(\alpha)}+a^{(\alpha)}1^T)}{d_\alpha}-1. \tag*{(24)} r α = 1 T ( M ( α ) ) − 1 a ( α ) = d α det ( M ( α ) + a ( α ) 1 T ) − 1. (24) Substitution into (14) , followed by the determinant lemma, gives
G = ( Λ d 0 / d 1 − i ρ r 1 / Λ − i ρ r 1 / Λ ρ 2 d 2 / ( Λ d 1 ) ) , n even . (25) G =
\begin{pmatrix}
\Lambda d_0/d_1 & -i\rho r_1/\Lambda\\
-i\rho r_1/\Lambda& \rho^2d_2/(\Lambda d_1)
\end{pmatrix},
\qquad n\ \text{even}. \tag*{(25)} G = ( Λ d 0 / d 1 − i ρ r 1 /Λ − i ρ r 1 /Λ ρ 2 d 2 / ( Λ d 1 ) ) , n even . (25) G = ( − i ρ Λ d − 1 / d 0 i ρ r 0 / Λ i ρ r 0 / Λ i ρ d 1 / ( Λ d 0 ) ) , n odd . (26) G =
\begin{pmatrix}
-i\rho\Lambda d_{-1}/d_0 & i\rho r_0/\Lambda\\
i\rho r_0/\Lambda& i\rho d_1/(\Lambda d_0)
\end{pmatrix},
\qquad n\ \text{odd}. \tag*{(26)} G = ( − i ρ Λ d − 1 / d 0 i ρ r 0 /Λ i ρ r 0 /Λ i ρ d 1 / ( Λ d 0 ) ) , n odd . (26) For comparison, the even off-diagonal entry can also be written as
G t ϕ = − i ρ Λ { det ( M ( 1 ) + C ( λ − 1 ) T ) d 1 − 1 } . (27) G_{t\phi} = -i\rho\Lambda\left\{\frac{\det(M^{(1)}+C(\lambda^{-1})^T)}{d_1}-1\right\}. \tag*{(27)} G tϕ = − i ρ Λ { d 1 det ( M ( 1 ) + C ( λ − 1 ) T ) − 1 } . (27) The equivalence follows from (22) . All expressions are understood on nonsingular charts, with removable singularities treated by taking limits of the complete quantities.
In these variables, the conformal factor is
e 2 ν = k 0 ∏ i < j ( 1 − λ i λ j ) 2 { ( − 1 ) n / 2 ρ ( n 2 − 2 n ) / 2 Λ 1 − n d 1 , n even , − i ρ ( n 2 − 2 n − 1 ) / 2 Λ 2 − n d 0 , n odd . (28) e^{2\nu}=k_0\prod_{i<j}(1-\lambda_i\lambda_j)^2
\begin{cases}
(-1)^{n/2}\rho^{(n^2-2n)/2}\Lambda^{1-n}d_1, & n\ \text{even},\\
-\mathrm{i}\rho^{(n^2-2n-1)/2}\Lambda^{2-n}d_0, & n\ \text{odd}.
\end{cases} \tag*{(28)} e 2 ν = k 0 i < j ∏ ( 1 − λ i λ j ) 2 { ( − 1 ) n /2 ρ ( n 2 − 2 n ) /2 Λ 1 − n d 1 , − i ρ ( n 2 − 2 n − 1 ) /2 Λ 2 − n d 0 , n even , n odd . (28) For real charges and imaginary λ i \lambda_i λ i − i -\mathrm{i} − i (19) and (17) ; it cannot be replaced independently by a real sign.
The determinant formulas now reduce to signed tau products. The diagonal entries use the common denominator and its neighboring shifts; the off-diagonal entry uses the rank-one correction in (24) .
The tau representation
Let a = ( a 1 , … , a n ) \boldsymbol{a}=(a_1,\ldots,a_n) a = ( a 1 , … , a n ) b = ( b 1 , … , b n ) \boldsymbol{b}=(b_1,\ldots,b_n) b = ( b 1 , … , b n ) [ n ] = { 1 , … , n } [n]=\{1,\ldots,n\} [ n ] = { 1 , … , n }
K ( b i , b j ) = b i − b j 1 − b i b j , Δ b ( S ) = ∏ i < j i , j ∈ S K ( b i , b j ) . (29) K(b_i,b_j)=\frac{b_i-b_j}{1-b_i b_j},\qquad\Delta_b(S)=\prod_{\substack{i<j\\i,j\in S}}K(b_i,b_j). \tag*{(29)} K ( b i , b j ) = 1 − b i b j b i − b j , Δ b ( S ) = i < j i , j ∈ S ∏ K ( b i , b j ) . (29) Empty products are one. Define the signed subset sum
τ n ( a , b ) = ∑ S ⊂ [ n ] ( − 1 ) ∑ i ∈ S a i ( ∏ i ∈ S a i ) Δ b ( S ) Δ b ( S c ) , τ ± ( α ) = τ n ( ∓ C λ α , λ ) . (30) \tau_n(a,b)=\sum_{S\subset[n]}(-1)^{\sum_{i\in S}a_i}\left(\prod_{i\in S}a_i\right)\Delta_b(S)\Delta_b(S^c),\qquad\tau_{\pm}^{(\alpha)}=\tau_n(\mp C\lambda^\alpha,\lambda). \tag*{(30)} τ n ( a , b ) = S ⊂ [ n ] ∑ ( − 1 ) ∑ i ∈ S a i ( i ∈ S ∏ a i ) Δ b ( S ) Δ b ( S c ) , τ ± ( α ) = τ n ( ∓ C λ α , λ ) . (30) The index sign is part of the definition. In particular, inserting or removing a slot changes the ordering signs of subsequent slots. Set
Π = ∏ i ( 1 − λ i 2 ) . (31) \Pi=\prod_i(1-\lambda_i^2). \tag*{(31)} Π = i ∏ ( 1 − λ i 2 ) . (31) Proposition 2.1 (Determinant factorization and symmetric shift). For either parity of n n n
d α = τ + ( α ) τ − ( α ) Π , (32) d_\alpha=\frac{\tau_+^{(\alpha)}\tau_-^{(\alpha)}}{\Pi}, \tag*{(32)} d α = Π τ + ( α ) τ − ( α ) , (32) τ + ( α + 1 ) τ − ( α − 1 ) + τ + ( α − 1 ) τ − ( α + 1 ) = 2 τ + ( α ) τ − ( α ) . (33) \tau_+^{(\alpha+1)}\tau_-^{(\alpha-1)}+\tau_+^{(\alpha-1)}\tau_-^{(\alpha+1)}=2\tau_+^{(\alpha)}\tau_-^{(\alpha)}. \tag*{(33)} τ + ( α + 1 ) τ − ( α − 1 ) + τ + ( α − 1 ) τ − ( α + 1 ) = 2 τ + ( α ) τ − ( α ) . (33) Proof. The Cauchy–Pfaffian factorization for ( ( 1 − a i a j ) / ( 1 − b i b j ) ) \left((1-a_i a_j)/(1-b_i b_j)\right) ( ( 1 − a i a j ) / ( 1 − b i b j ) ) (30) , divided by ∏ i ( 1 − b i 2 ) \prod_i(1-b_i^2) ∏ i ( 1 − b i 2 ) [17 ] . Its neighboring-weight identity gives (33) . The even and bordered odd Pfaffian representatives are described in § 3 . Equivalently, both identities follow by expanding in the a i a_i a i
For example, τ 1 ( a , b ) = 1 − a 1 \tau_1(a,b)=1-a_1 τ 1 ( a , b ) = 1 − a 1 τ 2 ( a , b ) = ( 1 − a 1 a 2 ) K ( b 1 , b 2 ) + a 2 − a 1 \tau_2(a,b)=(1-a_1a_2)K(b_1,b_2)+a_2-a_1 τ 2 ( a , b ) = ( 1 − a 1 a 2 ) K ( b 1 , b 2 ) + a 2 − a 1
Proposition 2.2 (Antisymmetric shift and parity). With the same conventions,
τ + ( α + 1 ) τ − ( α − 1 ) − τ + ( α − 1 ) τ − ( α + 1 ) = − 2 ( − 1 ) n Π Λ 2 1 T adj ( M ( α ) ) a ( α ) . (34) \tau_+^{(\alpha+1)}\tau_-^{(\alpha-1)}-\tau_+^{(\alpha-1)}\tau_-^{(\alpha+1)}
=-2(-1)^n\frac{\Pi}{\Lambda^2}\mathbf{1}^{T}\operatorname{adj}(M^{(\alpha)})a^{(\alpha)}. \tag*{(34)} τ + ( α + 1 ) τ − ( α − 1 ) − τ + ( α − 1 ) τ − ( α + 1 ) = − 2 ( − 1 ) n Λ 2 Π 1 T adj ( M ( α ) ) a ( α ) . (34) When d α ≠ 0 d_\alpha\ne0 d α = 0 right-hand side is − 2 ( − 1 ) n Π d α r α / Λ -2(-1)^n\Pi d_\alpha r_\alpha/\Lambda − 2 ( − 1 ) n Π d α r α /Λ
Proof. Work first where M = M ( α ) M=M^{(\alpha)} M = M ( α ) a = a ( α ) a=a^{(\alpha)} a = a ( α ) d = d α d=d_\alpha d = d α S = M − 1 S=M^{-1} S = M − 1
d α + 1 = d ( 1 + a T S a ) , d α − 1 = d Λ 2 ( 1 − 1 T S 1 ) . (35) d_{\alpha+1}=d(1+a^{T}Sa),\qquad d_{\alpha-1}=\frac{d}{\Lambda^2}(1-\mathbf{1}^{T}S\mathbf{1}). \tag*{(35)} d α + 1 = d ( 1 + a T S a ) , d α − 1 = Λ 2 d ( 1 − 1 T S 1 ) . (35) Since M + a a T − 11 T = D Λ M D Λ M + aa^{T} - 11^{T} = D_{\Lambda}MD_{\Lambda} M + a a T − 1 1 T = D Λ M D Λ
( 1 + a T S a ) ( 1 − 1 T S 1 ) + ( 1 T S a ) 2 = Λ 2 . (36) (1 + a^{T}Sa)(1 - 1^{T}S1) + (1^{T}Sa)^2 = \Lambda^2. \tag*{(36)} ( 1 + a T S a ) ( 1 − 1 T S 1 ) + ( 1 T S a ) 2 = Λ 2 . (36) Consequently,
d α 2 − d α + 1 d α − 1 = d α 2 Λ 2 ( 1 T S a ) 2 . (37) d_{\alpha}^{2} - d_{\alpha+1}d_{\alpha-1} = \frac{d_{\alpha}^{2}}{\Lambda^2}(1^{T}Sa)^2. \tag*{(37)} d α 2 − d α + 1 d α − 1 = Λ 2 d α 2 ( 1 T S a ) 2 . (37) By Proposition 2.1 , the square of the left-hand side of (34) is therefore the square of its proposed right-hand side. The sign is fixed by setting a 1 = ⋯ = a n − 1 = 0 a_1 = \cdots= a_{n-1} = 0 a 1 = ⋯ = a n − 1 = 0 b = λ b = \lambda b = λ D ∗ = Δ b ( [ n ] ) Δ b ( [ n − 1 ] ) D_* = \Delta_b([n])\Delta_b([n-1]) D ∗ = Δ b ([ n ]) Δ b ([ n − 1 ]) 2 ( − 1 ) n a n ( b n − b n − 1 ) D ∗ 2(-1)^n a_n(b_n - b_n^{-1})D_* 2 ( − 1 ) n a n ( b n − b n − 1 ) D ∗
1 T adj ( M ) a = Λ ∏ n a n ( b n − 1 − b n ) D ∗ . (38) 1^{T}\operatorname{adj}(M)a = \frac{\Lambda}{\prod_n}a_n(b_n^{-1} - b_n)D_*. \tag*{(38)} 1 T adj ( M ) a = ∏ n Λ a n ( b n − 1 − b n ) D ∗ . (38) This proves the stated sign. Rational continuation gives the adjugate identity without the invertibility assumption. □
Together with the symmetric shift, the proposition also gives the polarized determinant identity
τ + ( α + 1 ) τ − ( α − 1 ) = ∏ det ( M ( α ) − ( − 1 ) n Λ a ( α ) 1 T ) . (39) \tau_{+}^{(\alpha+1)}\tau_{-}^{(\alpha-1)} = \prod\det\left(M^{(\alpha)} - \frac{(-1)^n}{\Lambda}a^{(\alpha)}1^{T}\right). \tag*{(39)} τ + ( α + 1 ) τ − ( α − 1 ) = ∏ det ( M ( α ) − Λ ( − 1 ) n a ( α ) 1 T ) . (39) Reversing all charges reverses the rank-one term and exchanges the two shifted products.
Remark 2.3 (Alekseev’s determinant reconstruction). Alekseev’s determinant form [1 ] is the BZ residue formula with each inverse-matrix contraction replaced by a rank-one determinant quotient. To make the comparison explicit in our conventions, put ξ a = ( N i a / μ i ) i = 1 n \xi_a = (N_{ia}/\mu_i)_{i=1}^n ξ a = ( N ia / μ i ) i = 1 n a a a c a c_a c a (14) as
G a b = h n { ( G 0 ) a b + c a c b [ 1 − det ( Γ + ξ a ξ b T / ( c a c b ) ) det Γ ] } . (40) G_{ab} = h_n\left\{(G_0)_{ab} + c_ac_b\left[1 - \frac{\det\left(\Gamma+ \xi_a\xi_b^{T}/(c_ac_b)\right)}{\det\Gamma}\right]\right\}. \tag*{(40)} G ab = h n { ( G 0 ) ab + c a c b [ 1 − det Γ det ( Γ + ξ a ξ b T / ( c a c b ) ) ] } . (40) with no summation over a , b a,b a , b (23) then give (25) –(26) , including the off-diagonal correction (24) . Thus the determinant numerator and denominator reconstruct the same metric, with the normalization h n h_n h n [2 ] .
In tau variables, the rank-one correction has a more specific meaning. The denominator is the product τ + ( α ) τ − ( α ) / Π \tau_{+}^{(\alpha)}\tau_{-}^{(\alpha)}/\Pi τ + ( α ) τ − ( α ) /Π − ( − 1 ) n / Λ -(-1)^n/\Lambda − ( − 1 ) n /Λ a ( α ) 1 T a^{(\alpha)}1^{T} a ( α ) 1 T τ + ( α + 1 ) τ − ( α − 1 ) / Π \tau_{+}^{(\alpha+1)}\tau_{-}^{(\alpha-1)}/\Pi τ + ( α + 1 ) τ − ( α − 1 ) /Π (39) ; reversing the update gives the opposite pair of shifts. Their difference reconstructs G t ϕ G_{t\phi} G tϕ
The symmetric and antisymmetric shifts recover respectively the diagonal and off-diagonal metric data. The complete Gram matrices are
G = 1 τ + ( 1 ) τ − ( 1 ) ( Λ τ + ( 0 ) τ − ( 0 ) i ρ 2 ( τ + ( 2 ) τ − ( 0 ) − τ + ( 0 ) τ − ( 2 ) ) i ρ 2 ( τ + ( 2 ) τ − ( 0 ) − τ + ( 0 ) τ − ( 2 ) ) ρ 2 Λ τ + ( 2 ) τ − ( 2 ) ) , n even , (41) G = \frac{1}{\tau_{+}^{(1)}\tau_{-}^{(1)}}\begin{pmatrix}
\Lambda\tau_{+}^{(0)}\tau_{-}^{(0)} & \frac{i\rho}{2}\left(\tau_{+}^{(2)}\tau_{-}^{(0)}-\tau_{+}^{(0)}\tau_{-}^{(2)}\right) \\
\frac{i\rho}{2}\left(\tau_{+}^{(2)}\tau_{-}^{(0)}-\tau_{+}^{(0)}\tau_{-}^{(2)}\right) & \frac{\rho^2}{\Lambda}\tau_{+}^{(2)}\tau_{-}^{(2)}
\end{pmatrix}, \qquad n\ \text{even}, \tag*{(41)} G = τ + ( 1 ) τ − ( 1 ) 1 Λ τ + ( 0 ) τ − ( 0 ) 2 i ρ ( τ + ( 2 ) τ − ( 0 ) − τ + ( 0 ) τ − ( 2 ) ) 2 i ρ ( τ + ( 2 ) τ − ( 0 ) − τ + ( 0 ) τ − ( 2 ) ) Λ ρ 2 τ + ( 2 ) τ − ( 2 ) , n even , (41) and
G = ρ τ + ( 0 ) τ − ( 0 ) ( − i Λ τ + ( − 1 ) τ − ( − 1 ) i 2 ( τ + ( 1 ) τ − ( − 1 ) − τ + ( − 1 ) τ − ( 1 ) ) i 2 ( τ + ( 1 ) τ − ( − 1 ) − τ + ( − 1 ) τ − ( 1 ) ) i Λ τ + ( 1 ) τ − ( 1 ) ) , n odd . (42) G = \frac{\rho}{\tau_{+}^{(0)}\tau_{-}^{(0)}}\begin{pmatrix}
-i\Lambda\tau_{+}^{(-1)}\tau_{-}^{(-1)} & \frac{i}{2}\left(\tau_{+}^{(1)}\tau_{-}^{(-1)}-\tau_{+}^{(-1)}\tau_{-}^{(1)}\right) \\
\frac{i}{2}\left(\tau_{+}^{(1)}\tau_{-}^{(-1)}-\tau_{+}^{(-1)}\tau_{-}^{(1)}\right) & \frac{i}{\Lambda}\tau_{+}^{(1)}\tau_{-}^{(1)}
\end{pmatrix}, \qquad n\ \text{odd}. \tag*{(42)} G = τ + ( 0 ) τ − ( 0 ) ρ − i Λ τ + ( − 1 ) τ − ( − 1 ) 2 i ( τ + ( 1 ) τ − ( − 1 ) − τ + ( − 1 ) τ − ( 1 ) ) 2 i ( τ + ( 1 ) τ − ( − 1 ) − τ + ( − 1 ) τ − ( 1 ) ) Λ i τ + ( 1 ) τ − ( 1 ) , n odd . (42) In both cases, (33) gives det G = ρ 2 \det G=\rho^2 det G = ρ 2 d 1 d_1 d 1 d 0 d_0 d 0 (28) by (32) expresses the conformal factor through the same tau data. Thus the signed tau functions, the pole data and k 0 k_0 k 0
Ernst potentials from tau ratios.
Metric and Killing descriptions. Write
G = ( A B B D ) , A D − B 2 = ρ 2 . (43) G=\begin{pmatrix} A & B \\ B & D \end{pmatrix},\qquad AD-B^2=\rho^2. \tag*{(43)} G = ( A B B D ) , A D − B 2 = ρ 2 . (43) The metric Ernst pair is defined algebraically by
E M , ± = ρ ∓ i B A . (44) \mathcal{E}_{M,\pm}=\frac{\rho\mp iB}{A}. \tag*{(44)} E M , ± = A ρ ∓ i B . (44) For a real Riemannian metric the two components are complex conjugates. Conversely, setting E ± = E M , ± E_\pm=\mathcal{E}_{M,\pm} E ± = E M , ±
A = 2 ρ E + + E − , B = i ρ ( E + − E − ) E + + E − , D = 2 ρ E + E − E + + E − . (45) A=\frac{2\rho}{E_++E_-},\qquad B=\frac{i\rho(E_+-E_-)}{E_++E_-},\qquad D=\frac{2\rho E_+E_-}{E_++E_-}. \tag*{(45)} A = E + + E − 2 ρ , B = E + + E − i ρ ( E + − E − ) , D = E + + E − 2 ρ E + E − . (45) For a chosen Killing field ξ \xi ξ
F ξ = g ( ξ , ξ ) , d χ ξ = ∗ g ( ξ ♭ ∧ d ξ ♭ ) , E ξ , ± = F ξ ± χ ξ . (46) F_\xi=g(\xi,\xi),\qquad d\chi_\xi={*}_{g}(\xi^\flat\wedge d\xi^\flat),\qquad E_{\xi,\pm}=F_\xi\pm\chi_\xi. \tag*{(46)} F ξ = g ( ξ , ξ ) , d χ ξ = ∗ g ( ξ ♭ ∧ d ξ ♭ ) , E ξ , ± = F ξ ± χ ξ . (46) The twist one-form is closed in vacuum, so its primitive exists locally; its additive constant must be specified. Fix the orientation d t ∧ d ϕ ∧ d ρ ∧ d z dt\wedge d\phi\wedge d\rho\wedge dz d t ∧ d ϕ ∧ d ρ ∧ d z
χ t , ρ = A B z − B A z ρ , χ t , z = − A B ρ − B A ρ ρ . (47) \chi_{t,\rho}=\frac{AB_z-BA_z}{\rho},\qquad\chi_{t,z}=-\frac{AB_\rho-BA_\rho}{\rho}. \tag*{(47)} χ t , ρ = ρ A B z − B A z , χ t , z = − ρ A B ρ − B A ρ . (47) χ ϕ , ρ = − D B z − B D z ρ , χ ϕ , z = D B ρ − B D ρ ρ . (48) \chi_{\phi,\rho}=-\frac{DB_z-BD_z}{\rho},\qquad\chi_{\phi,z}=\frac{DB_\rho-BD_\rho}{\rho}. \tag*{(48)} χ ϕ , ρ = − ρ D B z − B D z , χ ϕ , z = ρ D B ρ − B D ρ . (48) The base conformal factor cancels from these equations. Both components of E ξ , ± E_{\xi,\pm} E ξ , ±
Both descriptions satisfy the paired vacuum Ernst equation
O ± ( E + , E − ) : = ( E + + E − ) Δ E ± − 2 ∇ E ± ⋅ ∇ E ± = 0 , (49) \mathcal{O}_{\pm}(E_+,E_-):=(E_++E_-)\Delta E_\pm-2\nabla E_\pm\cdot\nabla E_\pm=0, \tag*{(49)} O ± ( E + , E − ) := ( E + + E − ) Δ E ± − 2∇ E ± ⋅ ∇ E ± = 0 , (49) where
Δ = ∂ ρ 2 + ρ − 1 ∂ ρ + ∂ z 2 , ∇ f ⋅ ∇ h = f ρ h ρ + f z h z . (50) \Delta=\partial_\rho^2+\rho^{-1}\partial_\rho+\partial_z^2,\qquad\nabla f\cdot\nabla h=f_\rho h_\rho+f_z h_z. \tag*{(50)} Δ = ∂ ρ 2 + ρ − 1 ∂ ρ + ∂ z 2 , ∇ f ⋅ ∇ h = f ρ h ρ + f z h z . (50) The dot product is bilinear, without complex conjugation. For the metric pair, (49) becomes ( ℜ E M ) Δ E M = ∇ E M ⋅ ∇ E M (\Re\mathcal{E}_M)\Delta\mathcal{E}_M=\nabla\mathcal{E}_M\cdot\nabla\mathcal{E}_M ( ℜ E M ) Δ E M = ∇ E M ⋅ ∇ E M
F ξ Δ F ξ = ∇ F ξ ⋅ ∇ F ξ + ∇ χ ξ ⋅ ∇ χ ξ , F ξ Δ χ ξ = 2 ∇ F ξ ⋅ ∇ χ ξ . (51) F_\xi\Delta F_\xi=\nabla F_\xi\cdot\nabla F_\xi+\nabla\chi_\xi\cdot\nabla\chi_\xi,\qquad F_\xi\Delta\chi_\xi=2\nabla F_\xi\cdot\nabla\chi_\xi. \tag*{(51)} F ξ Δ F ξ = ∇ F ξ ⋅ ∇ F ξ + ∇ χ ξ ⋅ ∇ χ ξ , F ξ Δ χ ξ = 2∇ F ξ ⋅ ∇ χ ξ . (51) These are different real forms of the same paired equation. The conversion from the metric description to the Killing description is the norm–twist Bäcklund transformation, expressed differentially by (47) –(48) ; see also [27 ] .
The twist from spectral infinity. The first coefficient of the dressing matrix already supplies the required primitives. At fixed ( ρ , z ) (\rho,z) ( ρ , z )
X ( λ s p ) = I + W λ s p + O ( λ s p − 2 ) , W = ∑ i R i = ∑ i , j ( Γ − 1 ) i j N j T m i μ j . (52) \mathcal{X}(\lambda_{\rm sp})=I+\frac{\mathcal{W}}{\lambda_{\rm sp}}+O(\lambda_{\rm sp}^{-2}),\qquad\mathcal{W}=\sum_i R_i=\sum_{i,j}(\Gamma^{-1})_{ij}\frac{N_j^T m_i}{\mu_j}. \tag*{(52)} X ( λ sp ) = I + λ sp W + O ( λ sp − 2 ) , W = i ∑ R i = i , j ∑ ( Γ − 1 ) ij μ j N j T m i . (52) This is spectral infinity, not an asymptotic limit in spacetime. Let ( U 0 , V 0 ) (U_0,V_0) ( U 0 , V 0 ) ( U u n , V u n ) (U_{\rm un},V_{\rm un}) ( U un , V un )
W ρ = V u n − V 0 , W z = − U u n + U 0 . (53) \mathcal{W}_{\rho}=V_{\rm un}-V_0,\qquad\mathcal{W}_{z}=-U_{\rm un}+U_0. \tag*{(53)} W ρ = V un − V 0 , W z = − U un + U 0 . (53) Indeed, the two Lax coefficients start as V / λ s p V/\lambda_{\rm sp} V / λ sp − U / λ s p -U/\lambda_{\rm sp} − U / λ sp λ s p − 1 \lambda_{\rm sp}^{-1} λ sp − 1 (47) –(48) therefore identify the dressing-normalized primitives as
χ t = W 12 , χ ϕ = − W 21 . (54) \chi_t=\mathcal{W}_{12},\qquad\chi_\phi=-\mathcal{W}_{21}. \tag*{(54)} χ t = W 12 , χ ϕ = − W 21 . (54) There is no remaining integration in the evaluation of these entries. For example, when n n n S = ( M ( 1 ) ) − 1 S=(M^{(1)})^{-1} S = ( M ( 1 ) ) − 1
W 12 = ∑ i , j ( μ i S i j μ j ) C j μ j 1 μ i = 1 T S C . (55) \mathcal{W}_{12}=\sum_{i,j}(\mu_iS_{ij}\mu_j)\frac{C_j}{\mu_j}\frac{1}{\mu_i}=\mathbf{1}^TSC. \tag*{(55)} W 12 = i , j ∑ ( μ i S ij μ j ) μ j C j μ i 1 = 1 T S C . (55) Using (16) and (23) for both parities gives
Table 56.
Proposition 2.4 (Three Ernst descriptions of one BZ metric). In the Killing frame (1) , with the orientation and twist constants fixed above, the three Ernst pairs of the same flat-seed metric are
Table 57.
Each row satisfies (49) . The metric pair is complex conjugate on the real slice; the two Killing pairs are real.
Proof. The first row follows from (44) , (41) –(42) and the symmetric shift. For the other rows, the symmetric shift identifies the half-sums with A A A D D D (22) transfer the inverse matrices to the required center.
For example, in the even case,
d 0 d 1 1 T ( M ( 0 ) ) − 1 C = 1 T ( M ( 1 ) ) − 1 C . (56) \frac{d_0}{d_1}\mathbf{1}^T(M^{(0)})^{-1}C=\mathbf{1}^T(M^{(1)})^{-1}C. \tag*{(56)} d 1 d 0 1 T ( M ( 0 ) ) − 1 C = 1 T ( M ( 1 ) ) − 1 C . (56) Using (34) at α = 0 \alpha= 0 α = 0
χ t = Λ 2 ( τ + ( − 1 ) τ + ( 1 ) − τ − ( − 1 ) τ − ( 1 ) ) (57) \chi_t = \frac{\Lambda}{2}\left(\frac{\tau_{+}^{(-1)}}{\tau_{+}^{(1)}}-\frac{\tau_{-}^{(-1)}}{\tau_{-}^{(1)}}\right) \tag*{(57)} χ t = 2 Λ ( τ + ( 1 ) τ + ( − 1 ) − τ − ( 1 ) τ − ( − 1 ) ) (57) while the corresponding half-sum is Λ d 0 / d 1 = A \Lambda d_0/d_1 = A Λ d 0 / d 1 = A ϕ \phi ϕ M ( 2 ) = 11 T + D λ M ( 1 ) D λ M^{(2)} = 11^{T} + D_{\lambda}M^{(1)}D_{\lambda} M ( 2 ) = 1 1 T + D λ M ( 1 ) D λ − 1 -1 − 1 (34) . The identification with the geometric twists follows from (53) –(54) , rather than from the Ernst equation alone.
The shifted family. Introduce
F α , ± = ρ α Λ τ ± ( α + 1 ) τ ± ( α − 1 ) , F ^ α , ± = s n F α , ± , s n = { 1 , n even , i , n odd . (61) \mathcal{F}_{\alpha,\pm}=\frac{\rho^\alpha}{\Lambda}\frac{\tau_{\pm}^{(\alpha+1)}}{\tau_{\pm}^{(\alpha-1)}},\qquad
\widehat{\mathcal{F}}_{\alpha,\pm}=s_n\mathcal{F}_{\alpha,\pm},\qquad
s_n=
\begin{cases}
1,& n\ \text{even},\\
i,& n\ \text{odd}.
\end{cases} \tag*{(61)} F α , ± = Λ ρ α τ ± ( α − 1 ) τ ± ( α + 1 ) , F α , ± = s n F α , ± , s n = { 1 , i , n even , n odd . (61) If k n = 1 k_n=1 k n = 1 n n n k n = 0 k_n=0 k n = 0 n n n
E M , ± = F ^ k n , ± , E ∂ t , ± = F ^ k n − 1 , ± − 1 , E ∂ ϕ , ± = F ^ k n + 1 , ± . (62) \mathcal{E}_{M,\pm}=\widehat{\mathcal{F}}_{k_n,\pm},\qquad
\mathcal{E}_{\partial_t,\pm}=\widehat{\mathcal{F}}_{k_n-1,\pm}^{-1},\qquad
\mathcal{E}_{\partial_\phi,\pm}=\widehat{\mathcal{F}}_{k_n+1,\pm}. \tag*{(62)} E M , ± = F k n , ± , E ∂ t , ± = F k n − 1 , ± − 1 , E ∂ ϕ , ± = F k n + 1 , ± . (62) The neighboring-shift relation is not restricted to these two centers.
Proposition 2.6 (Shifted Ernst family). For either parity of n n n F ^ α , ± \widehat{F}_{\alpha,\pm} F α , ± (49) . This holds for integer α \alpha α α \alpha α E M , ± [ α ] = F ^ α , ± \mathcal{E}^{[\alpha]}_{M,\pm}=\widehat{F}_{\alpha,\pm} E M , ± [ α ] = F α , ±
E ∂ t , ± [ α ] = F ^ α − 1 , ± − 1 , E ∂ ϕ , ± [ α ] = F ^ α + 1 , ± . (63) \mathcal{E}^{[\alpha]}_{\partial t,\pm}=\widehat{F}^{-1}_{\alpha-1,\pm},\qquad\mathcal{E}^{[\alpha]}_{\partial\phi,\pm}=\widehat{F}_{\alpha+1,\pm}. \tag*{(63)} E ∂ t , ± [ α ] = F α − 1 , ± − 1 , E ∂ ϕ , ± [ α ] = F α + 1 , ± . (63) This is a statement about the complexified local construction, before imposing a real slice and global regularity.
Proof. For even n n n
G 0 [ α ] = diag ( ρ 1 − α , ρ 1 + α ) , Ψ 0 [ α ] = diag ( Q ( 1 − α ) / 2 , Q ( 1 + α ) / 2 ) . (64) G^{[\alpha]}_0=\operatorname{diag}\left(\rho^{1-\alpha},\rho^{1+\alpha}\right),\qquad\Psi^{[\alpha]}_0=\operatorname{diag}\left(Q^{(1-\alpha)/2},Q^{(1+\alpha)/2}\right). \tag*{(64)} G 0 [ α ] = diag ( ρ 1 − α , ρ 1 + α ) , Ψ 0 [ α ] = diag ( Q ( 1 − α ) /2 , Q ( 1 + α ) /2 ) . (64) Its currents are U 0 = diag ( 1 − α , 1 + α ) U_0=\operatorname{diag}(1-\alpha,1+\alpha) U 0 = diag ( 1 − α , 1 + α ) V 0 = 0 V_0=0 V 0 = 0 w = z i w=z_i w = z i
m i = ( η i μ i ( α − 1 ) / 2 , μ i − ( α + 1 ) / 2 ) , C i = i 1 − α η i . (65) m_i=\left(\eta_i\mu_i^{(\alpha-1)/2},\mu_i^{-(\alpha+1)/2}\right),\qquad C_i=\mathrm{i}^{1-\alpha}\eta_i. \tag*{(65)} m i = ( η i μ i ( α − 1 ) /2 , μ i − ( α + 1 ) /2 ) , C i = i 1 − α η i . (65) For odd n n n
G 0 [ α ] = diag ( − ρ − α , ρ α ) , Ψ 0 [ α ] = diag ( − Q − α / 2 , Q α / 2 ) , (66) G^{[\alpha]}_0=\operatorname{diag}\left(-\rho^{-\alpha},\rho^\alpha\right),\qquad\Psi^{[\alpha]}_0=\operatorname{diag}\left(-Q^{-\alpha/2},Q^{\alpha/2}\right), \tag*{(66)} G 0 [ α ] = diag ( − ρ − α , ρ α ) , Ψ 0 [ α ] = diag ( − Q − α /2 , Q α /2 ) , (66) m i = ( − η i μ i α / 2 , μ i − α / 2 ) , C i = i − α η i . m_i=\left(-\eta_i\mu_i^{\alpha/2},\mu_i^{-\alpha/2}\right),\qquad C_i=\mathrm{i}^{-\alpha}\eta_i. m i = ( − η i μ i α /2 , μ i − α /2 ) , C i = i − α η i . In both cases Γ \Gamma Γ M ( α ) M^{(\alpha)} M ( α ) (18) , the determinant lemma yields
G t t [ α ] = s n − 1 ρ 1 − α Λ d α − 1 d α , G ϕ ϕ [ α ] = s n ρ 1 + α Λ d α + 1 d α , G^{[\alpha]}_{tt}=s_n^{-1}\rho^{1-\alpha}\Lambda\frac{d_{\alpha-1}}{d_\alpha},\qquad G^{[\alpha]}_{\phi\phi}=\frac{s_n\rho^{1+\alpha}}{\Lambda}\frac{d_{\alpha+1}}{d_\alpha}, G tt [ α ] = s n − 1 ρ 1 − α Λ d α d α − 1 , G ϕϕ [ α ] = Λ s n ρ 1 + α d α d α + 1 , G t ϕ [ α ] = i ρ 2 τ + ( α + 1 ) τ − ( α − 1 ) − τ + ( α − 1 ) τ − ( α + 1 ) τ + ( α ) τ − ( α ) . (67) G^{[\alpha]}_{t\phi}=\frac{\mathrm{i}\rho}{2}\frac{\tau_+^{(\alpha+1)}\tau_-^{(\alpha-1)}-\tau_+^{(\alpha-1)}\tau_-^{(\alpha+1)}}{\tau_+^{(\alpha)}\tau_-^{(\alpha)}}. \tag*{(67)} G tϕ [ α ] = 2 i ρ τ + ( α ) τ − ( α ) τ + ( α + 1 ) τ − ( α − 1 ) − τ + ( α − 1 ) τ − ( α + 1 ) . (67) Thus its metric Ernst pair is F ^ α , ± \widehat{F}_{\alpha,\pm} F α , ± (52) for these rows gives
χ t [ α ] = s n ρ 1 − α 1 T ( M ( α ) ) − 1 ( C λ α − 1 ) , \chi_t^{[\alpha]}=s_n\rho^{1-\alpha}\mathbf{1}^{T}(M^{(\alpha)})^{-1}(C\lambda^{\alpha-1}), χ t [ α ] = s n ρ 1 − α 1 T ( M ( α ) ) − 1 ( C λ α − 1 ) , χ ϕ [ α ] = − s n − 1 ρ 1 + α ( λ − 1 ) T ( M ( α ) ) − 1 ( C λ α ) . (68) \chi_\phi^{[\alpha]}=-s_n^{-1}\rho^{1+\alpha}(\lambda^{-1})^{T}(M^{(\alpha)})^{-1}(C\lambda^\alpha). \tag*{(68)} χ ϕ [ α ] = − s n − 1 ρ 1 + α ( λ − 1 ) T ( M ( α ) ) − 1 ( C λ α ) . (68) The two shift identities now give (63) . Finally, simultaneous inversion of the two components preserves (49) . □
For real C i C_i C i α \alpha α
{ F ^ α , − = F ^ ‾ α , + , n + α odd , F ^ α , + , F ^ α , − ∈ R , n + α even . (69) \begin{cases}
\widehat{F}_{\alpha,-}=\overline{\widehat{F}}_{\alpha,+}, & n+\alpha\ \text{odd},\\
\widehat{F}_{\alpha,+},\widehat{F}_{\alpha,-}\in\mathbb{R}, & n+\alpha\ \text{even}. \tag*{(69)}
\end{cases} { F α , − = F α , + , F α , + , F α , − ∈ R , n + α odd , n + α even . (69) The metric center and its neighbors therefore have precisely the different reality structures required in (62) . For noninteger shifts a real form requires a corresponding choice of charge phases. Changing α \alpha α ρ α \rho^\alpha ρ α (61) .
Choice of Killing field. The oriented exchange ( ∂ t , ∂ ϕ ) ↦ ( ∂ ϕ , − ∂ t ) (\partial_t,\partial_\phi) \mapsto(\partial_\phi,-\partial_t) ( ∂ t , ∂ ϕ ) ↦ ( ∂ ϕ , − ∂ t )
ρ ± i B D = 1 E M , ± . (70) \frac{\rho\pm iB}{D}=\frac{1}{\mathcal{E}_{M,\pm}}. \tag*{(70)} D ρ ± i B = E M , ± 1 . (70) The two Killing Ernst pairs of a fixed metric, however, are the distinct neighboring expressions in (62) , not generally reciprocals of one another. For ξ = a ∂ t + b ∂ ϕ \xi=a\partial_t+b\partial_\phi ξ = a ∂ t + b ∂ ϕ a , b a,b a , b
F ξ = a 2 A + 2 a b B + b 2 D , χ ξ = a 2 χ t + a b χ m i x + b 2 χ ϕ . (71) F_\xi=a^2A+2abB+b^2D,\qquad\chi_\xi=a^2\chi_t+ab\chi_{\mathrm{mix}}+b^2\chi_\phi. \tag*{(71)} F ξ = a 2 A + 2 ab B + b 2 D , χ ξ = a 2 χ t + ab χ mix + b 2 χ ϕ . (71) The same dressing coefficient supplies the mixed primitive:
χ m i x = W 22 − W 11 + { − 2 z , n even , 0 , n odd . (72) \chi_{\mathrm{mix}}=W_{22}-W_{11}+\begin{cases}-2z,&n\ \text{even},\\0,&n\ \text{odd}.\end{cases} \tag*{(72)} χ mix = W 22 − W 11 + { − 2 z , 0 , n even , n odd . (72) The seed term follows from U 0 , 11 − U 0 , 22 = − 2 U_{0,11}-U_{0,22}=-2 U 0 , 11 − U 0 , 22 = − 2 0 0 0 c c c c 2 c^2 c 2
Geometry of the tau-function
The subset sum (30) has two complementary uses. Its Pfaffian form explains the bilinear identities and makes insertion of additional solitons transparent. Its subset form makes degenerations and axis limits easy to read. We retain both descriptions, using the same variables a , b a,b a , b § 2 .
Pfaffian identities
Define the skew matrix
H i j = ( 1 − a i a j ) K ( b i , b j ) + a j − a i . (73) H_{ij}=(1-a_i a_j)K(b_i,b_j)+a_j-a_i. \tag*{(73)} H ij = ( 1 − a i a j ) K ( b i , b j ) + a j − a i . (73) Then
τ n ( a , b ) = { Pf H , n even , Pf ( 0 ( 1 − a ) T a − 1 H ) , n odd . (74) \tau_n(a,b)=\begin{cases}
\operatorname{Pf}H,&n\ \text{even},\\
\operatorname{Pf}\begin{pmatrix}0&(1-a)^T\\a-1&H\end{pmatrix},&n\ \text{odd}.
\end{cases} \tag*{(74)} τ n ( a , b ) = ⎩ ⎨ ⎧ Pf H , Pf ( 0 a − 1 ( 1 − a ) T H ) , n even , n odd . (74) The bordering convention fixes the sign for odd n n n n = 1 n=1 n = 1 1 − a 1 1-a_1 1 − a 1 a i a_i a i (30) [17 ] .
Several determinant squares now have a common explanation. For even n n n det H = τ n 2 \det H=\tau_n^2 det H = τ n 2 n n n d α = τ + ( α ) τ − ( α ) / Π d_\alpha=\tau_+^{(\alpha)}\tau_-^{(\alpha)}/\Pi d α = τ + ( α ) τ − ( α ) /Π
d α 2 − d α + 1 d α − 1 = ( 1 T adj ( M ( α ) ) a ( α ) Λ ) 2 . (75) d_\alpha^2-d_{\alpha+1}d_{\alpha-1}=\left(\frac{\mathbf{1}^T\operatorname{adj}(M^{(\alpha)})a^{(\alpha)}}{\Lambda}\right)^2. \tag*{(75)} d α 2 − d α + 1 d α − 1 = ( Λ 1 T adj ( M ( α ) ) a ( α ) ) 2 . (75) The square does not determine a sign. The signed identity (34) fixes that sign before the metric is reconstructed.
The spectral shifts are small operations on the skew matrix. If a = C b α a=C b^\alpha a = C b α x α = C b α − 1 x_\alpha=C b^\alpha-1 x α = C b α − 1
H α + 1 − H α = x α + 1 x α T − x α x α + 1 T . (76) H_{\alpha+1}-H_\alpha=x_{\alpha+1}x_\alpha^T-x_\alpha x_{\alpha+1}^T. \tag*{(76)} H α + 1 − H α = x α + 1 x α T − x α x α + 1 T . (76) Likewise charge reversal changes H H H
Pf ( H + x y T − y x T ) = Pf ( H ) ( 1 + y T H − 1 x ) . (77) \operatorname{Pf}(H+xy^T-yx^T)=\operatorname{Pf}(H)(1+y^T H^{-1}x). \tag*{(77)} Pf ( H + x y T − y x T ) = Pf ( H ) ( 1 + y T H − 1 x ) . (77) After clearing denominators this extends across singular charts. It explains why neighbouring tau-functions, rather than unrelated determinants, suffice for the metric and Ernst fields.
The insertion recurrence
Adjoin a slot with charge weight A A A ζ \zeta ζ k i = K ( b i , ζ ) k_i=K(b_i,\zeta) k i = K ( b i , ζ ) L k = ∏ i k i L_k=\prod_i k_i L k = ∏ i k i
τ n + 1 ( a , A ; b , ζ ) = L k τ n ( a / k , b ) + ( − 1 ) n + 1 A τ n ( a k , b ) . (78) \tau_{n+1}(a,A;b,\zeta)=L_k\tau_n(a/k,b)+(-1)^{n+1}A\tau_n(ak,b). \tag*{(78)} τ n + 1 ( a , A ; b , ζ ) = L k τ n ( a / k , b ) + ( − 1 ) n + 1 A τ n ( ak , b ) . (78) Products and quotients of weight vectors are componentwise. An empty new slot couples to every empty old slot; an occupied new slot couples to every occupied old slot. The two terms in (78) express precisely this alternative.
For a second new slot ( B , η ) (B,\eta) ( B , η ) h i = K ( b i , η ) h_i=K(b_i,\eta) h i = K ( b i , η ) L h = ∏ i h i L_h=\prod_i h_i L h = ∏ i h i c = K ( ζ , η ) c=K(\zeta,\eta) c = K ( ζ , η )
τ n + 2 ( a , A , B ; b , ζ , η ) = c L k L h τ n ( a / ( k h ) , b ) + ( − 1 ) n + 1 A L h τ n ( a k / h , b ) + ( − 1 ) n + 2 B L k τ n ( a h / k , b ) − A B c τ n ( a k h , b ) . (79) \begin{aligned}
\tau_{n+2}(a,A,B;b,\zeta,\eta)={}&cL_kL_h\tau_n(a/(kh),b)\\
&+(-1)^{n+1}AL_h\tau_n(ak/h,b)\\
&+(-1)^{n+2}BL_k\tau_n(ah/k,b)-ABc\tau_n(akh,b). \tag*{(79)}
\end{aligned} τ n + 2 ( a , A , B ; b , ζ , η ) = c L k L h τ n ( a / ( k h ) , b ) + ( − 1 ) n + 1 A L h τ n ( ak / h , b ) + ( − 1 ) n + 2 B L k τ n ( ah / k , b ) − A B c τ n ( ak h , b ) . (79) The four terms correspond to the four occupations of the new pair. For n = 0 n=0 n = 0 ( 1 − A B ) c + B − A (1-AB)c+B-A ( 1 − A B ) c + B − A
The same idea appears in block elimination. For an invertible even skew block B 0 B_0 B 0
Pf ( A 0 E − E T B 0 ) = Pf ( B 0 ) Pf ( A 0 + E B 0 − 1 E T ) . (80) \operatorname{Pf}\begin{pmatrix} A_0 & E \\ -E^{T} & B_0 \end{pmatrix}
=\operatorname{Pf}(B_0)\operatorname{Pf}(A_0+EB_0^{-1}E^{T}). \tag*{(80)} Pf ( A 0 − E T E B 0 ) = Pf ( B 0 ) Pf ( A 0 + E B 0 − 1 E T ) . (80) A distant group of solitons can consequently change the effective local skew matrix even when the group itself has left the observation region. This is the algebraic reason to retain the cross terms in a cluster limit. A smaller number of visible poles does not, by itself, give the balance equations of an isolated lower-order solution.
The pure-spinor viewpoint
There is a geometric structure behind these formulas. Let E E E e 1 , … , e n e^1,\ldots,e^n e 1 , … , e n E ⊕ E ∗ E\oplus E^* E ⊕ E ∗ H H H
s H = exp ( 1 2 ∑ i , j H i j e i ∧ e j ) . (81) s_H=\exp\left(\frac{1}{2}\sum_{i,j}H_{ij}e^i\wedge e^j\right). \tag*{(81)} s H = exp ( 2 1 i , j ∑ H ij e i ∧ e j ) . (81) Its coordinates are the principal Pfaffians of H H H
Thus the tau-function is one coordinate of a structured object, rather than an isolated polynomial. The principal Pfaffians obey quadratic relations because the corresponding isotropic planes lie on an orthogonal Grassmannian. The bilinear identities encountered in the metric are concrete specializations of these relations. Rank-two updates change the skew form by a decomposable two-form, while a change of Pfaffian chart changes the coordinate description of the same geometric construction. This viewpoint explains why the algebra closes on a small collection of neighbouring quantities.
Related pure-spinor and Pfaffian coordinates occur in the B B B [28 ] . No Grassmannian classification or hierarchy of additional times is required below. In particular, identifying this finite Pfaffian geometry does not by itself specify a BKP evolution or a new field equation. Its role here is organizational: it explains the recurrences, the determinant squares and the compatibility of the descendants already used in the BZ metric. The two-component metric and boundary spinors introduced below are descendants of this data; they are not the full Cartan pure spinor (83).
Surviving sectors in the rod limit
For the remainder of this section take N = n = 2 m N=n=2m N = n = 2 m I j = ( z j , z j + 1 ) I_j=(z_j,z_{j+1}) I j = ( z j , z j + 1 ) I 0 = ( − ∞ , z 1 ) I_0=(-\infty,z_1) I 0 = ( − ∞ , z 1 ) I N = ( z N , ∞ ) I_N=(z_N,\infty) I N = ( z N , ∞ ) [10 ] , the coordinates λ i \lambda_i λ i ρ \rho ρ i ≤ j i\le j i ≤ j ρ − 1 \rho^{-1} ρ − 1 i > j i>j i > j
In that chart the term with ∣ S ∣ = k \lvert S\rvert=k ∣ S ∣ = k
ρ ( k 2 ) + ( 2 m − k 2 ) = ρ m ( m − 1 ) + ( k − m ) 2 . (82) \rho^{\binom{k}{2}+\binom{2m-k}{2}}=\rho^{m(m-1)+(k-m)^2}. \tag*{(82)} ρ ( 2 k ) + ( 2 2 m − k ) = ρ m ( m − 1 ) + ( k − m ) 2 . (82) The middle sector k = m k=m k = m b i α b_i^\alpha b i α α k \alpha k α k α = 2 \alpha=2 α = 2 k = m − 1 k=m-1 k = m − 1 k = m + 1 k=m+1 k = m + 1
Define the resulting Vandermonde sectors by
W k ( C , z ) = ∑ ∣ S ∣ = k ( − 1 ) ∑ i ∈ S i C S Δ z ( S ) Δ z ( S c ) , Δ z ( S ) = ∏ i < j i , j ∈ S ( z i − z j ) . (83) W_k(C,z)=\sum_{\lvert S\rvert=k}(-1)^{\sum_{i\in S}i}C_S\Delta_z(S)\Delta_z(S^c),\qquad\Delta_z(S)=\prod_{\substack{i<j\\i,j\in S}}(z_i-z_j). \tag*{(83)} W k ( C , z ) = ∣ S ∣ = k ∑ ( − 1 ) ∑ i ∈ S i C S Δ z ( S ) Δ z ( S c ) , Δ z ( S ) = i < j i , j ∈ S ∏ ( z i − z j ) . (83) Here C S = ∏ i ∈ S C i C_S=\prod_{i\in S}C_i C S = ∏ i ∈ S C i i i i V k V_k V k
( C i , C i z i , … , C i z i k − 1 , 1 , z i , … , z i N − k − 1 ) , \left(C_i,C_i z_i,\ldots,C_i z_i^{k-1},1,z_i,\ldots,z_i^{N-k-1}\right), ( C i , C i z i , … , C i z i k − 1 , 1 , z i , … , z i N − k − 1 ) , then Laplace expansion gives
W k = ( − 1 ) k 2 + ( N − k ) ( N − k − 1 ) / 2 det V k . (84) W_k=(-1)^{k^2+(N-k)(N-k-1)/2}\det V_k. \tag*{(84)} W k = ( − 1 ) k 2 + ( N − k ) ( N − k − 1 ) /2 det V k . (84) This form avoids an exponential subset expansion when evaluating the axis data.
A cone-normalized rod vector r j \mathbf r_j r j G G G I j I_j I j
lim ρ → 0 G ( r j , r j ) ρ 2 e 2 ν = 1. \lim_{\rho\to0}\frac{G(\mathbf r_j,\mathbf r_j)}{\rho^2e^{2\nu}}=1. ρ → 0 lim ρ 2 e 2 ν G ( r j , r j ) = 1. Let C [ j ] C^{[j]} C [ j ] j j j k 0 k_0 k 0 [19 ] , a consistent choice is
r j = k 0 γ m ( ∏ i < j C i ) ( − 2 W m + 1 ( C [ j ] , z ) W m ( C [ j ] , z ) ) , γ m = ( − 1 ) m ( m + 1 ) / 2 2 m ( m − 1 ) . (85) \mathbf r_j=\sqrt{k_0}\,\gamma_m\left(\prod_{i<j}C_i\right)\left(\frac{-2W_{m+1}(C^{[j]},z)}{W_m(C^{[j]},z)}\right),\qquad\gamma_m=(-1)^{m(m+1)/2}2^{m(m-1)}. \tag*{(85)} r j = k 0 γ m ( i < j ∏ C i ) ( W m ( C [ j ] , z ) − 2 W m + 1 ( C [ j ] , z ) ) , γ m = ( − 1 ) m ( m + 1 ) /2 2 m ( m − 1 ) . (85) An individual rod may be reversed in sign. Its magnitude cannot be discarded: it is fixed by the cone normalization. In homogeneous coordinates C i = P i / Q i C_i=P_i/Q_i C i = P i / Q i C S C_S C S P S Q S c P_SQ_{S^c} P S Q S c j j j P i ↔ Q i P_i\leftrightarrow Q_i P i ↔ Q i C i = 0 C_i=0 C i = 0 C i = ∞ C_i=\infty C i = ∞
Finally, the metric itself has a two-column factorization. In the even convention set
ψ ± = 1 τ ± ( 1 ) ( Λ τ ± ( 0 ) ± i ρ Λ − 1 / 2 τ ± ( 2 ) ) , G = 1 2 ( ψ + ψ + T + ψ − ψ − T ) . (86) \psi_\pm=\frac{1}{\tau_\pm^{(1)}}\begin{pmatrix}\sqrt{\Lambda}\,\tau_\pm^{(0)}\\ \pm i\rho\Lambda^{-1/2}\tau_\pm^{(2)}\end{pmatrix},\qquad G=\frac{1}{2}\left(\psi_+\psi_+^{T}+\psi_-\psi_-^{T}\right). \tag*{(86)} ψ ± = τ ± ( 1 ) 1 ( Λ τ ± ( 0 ) ± i ρ Λ − 1/2 τ ± ( 2 ) ) , G = 2 1 ( ψ + ψ + T + ψ − ψ − T ) . (86) This is a factorization of the bilinear formula of § 2 . Taking its rod limit together with the conformal factor gives (87). Keeping the kinematic prefactors and the common conformal normalization is essential: the projective null direction alone contains no cone angle.
Spinor factorization of the patching matrix
The bulk tau-function determines the metric, while its axis restriction has a smaller rational representative. This section develops that boundary representation for genuine even data n = N = 2 m n = N = 2m n = N = 2 m
Positive boundary data
Put
D ( w ) = ∏ i = 1 N ( w − z i ) , Z = diag ( z 1 , … , z N ) . D(w) = \prod_{i=1}^{N}(w-z_i), \qquad Z = \operatorname{diag}(z_1,\ldots,z_N). D ( w ) = i = 1 ∏ N ( w − z i ) , Z = diag ( z 1 , … , z N ) . The positive real sector is specified by the weights
ϖ i = C i − 1 ( C i + 1 ) D ′ ( z i ) > 0 , [ P i : Q i ] = [ 1 + ϖ i D ′ ( z i ) : 1 − ϖ i D ′ ( z i ) ] . (87) \varpi_i = \frac{C_i-1}{(C_i+1)D'(z_i)} > 0, \qquad[P_i:Q_i] = [1+\varpi_iD'(z_i):1-\varpi_iD'(z_i)]. \tag*{(87)} ϖ i = ( C i + 1 ) D ′ ( z i ) C i − 1 > 0 , [ P i : Q i ] = [ 1 + ϖ i D ′ ( z i ) : 1 − ϖ i D ′ ( z i )] . (87) Here [ P i : Q i ] [P_i:Q_i] [ P i : Q i ] C i = P i / Q i C_i = P_i/Q_i C i = P i / Q i Q i ≠ 0 Q_i \ne0 Q i = 0 C i = ∞ C_i = \infty C i = ∞ § 5 .
Let F = g ( ∂ t , ∂ t ) F = g(\partial_t,\partial_t) F = g ( ∂ t , ∂ t ) χ = χ t \chi= \chi_t χ = χ t § 2 , with χ \chi χ w w w patching matrix
P ( w ) = 1 F ( 1 − χ − χ χ 2 − F 2 ) exterior axis . P(w) = \frac{1}{F}
\begin{pmatrix}
1 & -\chi\\
-\chi& \chi^2-F^2
\end{pmatrix}_{\text{exterior axis}}. P ( w ) = F 1 ( 1 − χ − χ χ 2 − F 2 ) exterior axis . In the positive BZ sector it has the representation
P ( w ) = J 0 + ∑ i = 1 N s i s i T w − z i , J 0 = diag ( 1 , − 1 ) , det P = − 1. (88) P(w) = J_0+\sum_{i=1}^{N}\frac{s_i s_i^{T}}{w-z_i}, \qquad J_0 = \operatorname{diag}(1,-1), \qquad\det P=-1. \tag*{(88)} P ( w ) = J 0 + i = 1 ∑ N w − z i s i s i T , J 0 = diag ( 1 , − 1 ) , det P = − 1. (88) The real columns s i = ( u i , v i ) T s_i=(u_i,v_i)^T s i = ( u i , v i ) T u = ( u 1 , … , u N ) T u=(u_1,\ldots,u_N)^T u = ( u 1 , … , u N ) T v = ( v 1 , … , v N ) T v=(v_1,\ldots,v_N)^T v = ( v 1 , … , v N ) T v j v_j v j
The representation follows by restricting the BZ Ernst fields to an exterior axis. The converse boundary construction starts from rod-adapted solutions of the axis Lax system: matching at each nut gives a rational transition factor, and the ordered product is constrained by symmetry and its value at infinity [21 , 27 , 12 ] . This requires the local rod and nut models and the normalized end. Rationality alone is not a global regularity criterion. The explicit Chen–Teo ALF family [9 ] and its twistor factorization [13 ] illustrate the distinction between algebraic reconstruction and the regular subfamily.
Positivity in (88) concerns the residues; P P P − 1 -1 − 1
P ( η ) − P ( w ) w − η = ∑ i s i s i T ( w − z i ) ( η − z i ) . (89) \frac{P(\eta)-P(w)}{w-\eta}=\sum_i\frac{s_i s_i^T}{(w-z_i)(\eta-z_i)}. \tag*{(89)} w − η P ( η ) − P ( w ) = i ∑ ( w − z i ) ( η − z i ) s i s i T . (89) For real sample points away from the poles its block matrix is a positive semidefinite Gram matrix. In particular − P ( w ) -P(w) − P ( w )
We use the mass M M M J J J
G t t = 1 − 2 M r + O ( r − 2 ) , G t ϕ = − 2 J sin 2 ϑ r + O ( r − 2 ) , ρ = r sin ϑ , z = r cos ϑ . G_{tt}=1-\frac{2M}{r}+O(r^{-2}), \qquad G_{t\phi}=-\frac{2J\sin^2\vartheta}{r}+O(r^{-2}), \qquad\rho=r\sin\vartheta,\quad z=r\cos\vartheta. G tt = 1 − r 2 M + O ( r − 2 ) , G tϕ = − r 2 J sin 2 ϑ + O ( r − 2 ) , ρ = r sin ϑ , z = r cos ϑ . In the AF frame, expansion of (88) gives
∥ u ∥ 2 = ∥ v ∥ 2 = 2 M , u T v = 0 , J = − 1 2 u T Z v . (90) \lVert u\rVert^2=\lVert v\rVert^2=2M, \qquad u^Tv=0, \qquad J=-\frac{1}{2}u^TZv. \tag*{(90)} ∥ u ∥ 2 = ∥ v ∥ 2 = 2 M , u T v = 0 , J = − 2 1 u T Z v . (90) The vanishing mixed zeroth moment is the AF alignment condition. Mass and angular momentum are moments of the same data that determine the normalized rods.
Polynomial spinors
On a nonsingular interpolation chart, let e e e m m m deg o ≤ m − 1 \deg o \le m-1 deg o ≤ m − 1
e ( z i ) + C i o ( z i ) = 0 , i = 1 , … , 2 m . (91) e(z_i) + C_i o(z_i) = 0,\qquad i=1,\ldots,2m. \tag*{(91)} e ( z i ) + C i o ( z i ) = 0 , i = 1 , … , 2 m . (91) In homogeneous coordinates the equation is Q i e ( z i ) + P i o ( z i ) = 0 Q_i e(z_i) + P_i o(z_i) = 0 Q i e ( z i ) + P i o ( z i ) = 0 e , o e,o e , o
f + = e + o , f − = e − o . f_+ = e + o,\qquad f_- = e-o. f + = e + o , f − = e − o . The general boundary representation retains two independent complementary polynomials g + , g − g_+,g_- g + , g − m m m
f + g + + f − g − = 2 D . (92) f_+g_+ + f_-g_- = 2D. \tag*{(92)} f + g + + f − g − = 2 D . (92) Set
A = f + f − , Q = D − f + g + , C = − g + g − . (93) \mathcal{A}=f_+f_-,\qquad Q=D-f_+g_+,\qquad C=-g_+g_-. \tag*{(93)} A = f + f − , Q = D − f + g + , C = − g + g − . (93) Then
P ( w ) = 1 D ( w ) ( A ( w ) Q ( w ) Q ( w ) C ( w ) ) , A C − Q 2 = − D 2 . (94) P(w)=\frac{1}{D(w)}
\begin{pmatrix}
\mathcal{A}(w) & Q(w)\\
Q(w) & C(w)
\end{pmatrix},
\qquad\mathcal{A}C-Q^2=-D^2. \tag*{(94)} P ( w ) = D ( w ) 1 ( A ( w ) Q ( w ) Q ( w ) C ( w ) ) , A C − Q 2 = − D 2 . (94) Indeed, writing X = f + g + X=f_+g_+ X = f + g + Y = f − g − Y=f_-g_- Y = f − g − − X Y − ( D − X ) 2 = − D 2 -XY-(D-X)^2=-D^2 − X Y − ( D − X ) 2 = − D 2 X + Y = 2 D X+Y=2D X + Y = 2 D
D − Q = f + g + , D + Q = f − g − . (95) D-Q=f_+g_+,\qquad D+Q=f_-g_-. \tag*{(95)} D − Q = f + g + , D + Q = f − g − . (95) These relations are useful in both directions. Given the charges, (91) is a linear system for the coefficients of e , o e,o e , o [ P i : Q i ] = [ − e ( z i ) : o ( z i ) ] [P_i:Q_i]=[-e(z_i):o(z_i)] [ P i : Q i ] = [ − e ( z i ) : o ( z i )] (92) alone does not select a metric. A simultaneous zero of e , o e,o e , o
The determinant lemma gives a spectral interpretation:
A ( w ) = det ( w I − Z + u u T ) , − C ( w ) = det ( w I − Z − v v T ) . (96) \mathcal{A}(w)=\det(wI-Z+uu^{T}),\qquad-C(w)=\det(wI-Z-vv^{T}). \tag*{(96)} A ( w ) = det ( w I − Z + u u T ) , − C ( w ) = det ( w I − Z − v v T ) . (96) Thus the combined roots of f + , f − f_+,f_- f + , f − Z − u u T Z-uu^{T} Z − u u T g + , g − g_+,g_- g + , g − Z + v v T Z+vv^{T} Z + v v T e e e o o o
A phantom is treated at the level of the complete matrix. A factor w − z p w-z_p w − z p D D D z p z_p z p
The positive real matrix reconstruction
Let V i k = z i k − 1 V_{ik}=z_i^{k-1} V ik = z i k − 1 1 ≤ k ≤ m 1\le k\le m 1 ≤ k ≤ m e m e_m e m R m \mathbb{R}^m R m
W + = diag ( ϖ i ) V , W − = diag ( 1 ϖ i D ′ ( z i ) ) V , W_+=\operatorname{diag}(\sqrt{\varpi_i})V,\qquad
W_-=\operatorname{diag}\left(\frac{1}{\sqrt{\varpi_i}D'(z_i)}\right)V, W + = diag ( ϖ i ) V , W − = diag ( ϖ i D ′ ( z i ) 1 ) V , a = W + ( W + T W + ) − 1 e m , c = W − ( W − T W − ) − 1 e m , a=W_+(W_+^{T}W_+)^{-1}e_m,\qquad
c=W_-(W_-^{T}W_-)^{-1}e_m, a = W + ( W + T W + ) − 1 e m , c = W − ( W − T W − ) − 1 e m , T = 2 W + ( W + T W + ) − 1 W + T − I , u = a + c , v = a − c . (97) T=2W_+(W_+^{T}W_+)^{-1}W_+^{T}-I,\qquad
u=a+c,\quad v=a-c. \tag*{(97)} T = 2 W + ( W + T W + ) − 1 W + T − I , u = a + c , v = a − c . (97) Both Gram matrices are positive definite. Lagrange interpolation makes the two weighted polynomial spaces orthogonal complements and gives
T = T T , T 2 = I , T v = u , [ Z , T ] = u w T − v u T , [ Z + v v T , T ] = 0 , (98) T=T^{T},\qquad T^{2}=I,\qquad Tv=u,\qquad[Z,T]=uw^{T}-vu^{T},\qquad[Z+vv^{T},T]=0, \tag*{(98)} T = T T , T 2 = I , T v = u , [ Z , T ] = u w T − v u T , [ Z + v v T , T ] = 0 , (98) where [ X , Y ] = X Y − Y X [X,Y]=XY-YX [ X , Y ] = X Y − Y X
Let T j T_j T j j × j j\times j j × j v [ j ] = ( v 1 , … , v j ) T v_{[j]}=(v_1,\ldots,v_j)^T v [ j ] = ( v 1 , … , v j ) T
r j = ( − 1 ) j ( 2 v [ j ] T adj ( T j ) v [ j ] , det T j ) , r 0 = ( 0 , 1 ) . (99) r_j=(-1)^j\left(2v_{[j]}^{T}\operatorname{adj}(T_j)v_{[j]},\det T_j\right),\qquad r_0=(0,1). \tag*{(99)} r j = ( − 1 ) j ( 2 v [ j ] T adj ( T j ) v [ j ] , det T j ) , r 0 = ( 0 , 1 ) . (99) This adjugate expression continues through det T j = 0 \det T_j=0 det T j = 0
For bulk reconstruction put
R i = ρ 2 + ( z − z i ) 2 , M ± = diag ( R i ± ( z − z i ) ) , B = M − + T M + T . R_i=\sqrt{\rho^2+(z-z_i)^2},\qquad M_{\pm}=\operatorname{diag}(R_i\pm(z-z_i)),\qquad\mathcal{B}=M_{-}+TM_{+}T. R i = ρ 2 + ( z − z i ) 2 , M ± = diag ( R i ± ( z − z i )) , B = M − + T M + T . For ρ > 0 \rho>0 ρ > 0 M ± M_{\pm} M ± B > 0 \mathcal{B}>0 B > 0
g = f ( d t − ω d ϕ ) 2 + ρ 2 f − 1 d ϕ 2 + e 2 k f − 1 ( d ρ 2 + d z 2 ) , e 2 ν = e 2 k f . g=f(dt-\omega\,d\phi)^2+\rho^2f^{-1}d\phi^2+e^{2k}f^{-1}(d\rho^2+dz^2),\qquad e^{2\nu}=\frac{e^{2k}}{f}. g = f ( d t − ω d ϕ ) 2 + ρ 2 f − 1 d ϕ 2 + e 2 k f − 1 ( d ρ 2 + d z 2 ) , e 2 ν = f e 2 k . The BZ reconstruction is
f − 1 = 1 + 2 v T B − 1 v , ω = − 2 v T B − 1 M − u , e 2 k = det B 2 N ∏ i R i . (100) f^{-1}=1+2v^{T}\mathcal{B}^{-1}v,\qquad\omega=-2v^{T}\mathcal{B}^{-1}M_{-}u,\qquad e^{2k}=\frac{\det\mathcal{B}}{2^{N}\prod_iR_i}. \tag*{(100)} f − 1 = 1 + 2 v T B − 1 v , ω = − 2 v T B − 1 M − u , e 2 k = 2 N ∏ i R i det B . (100) It proves positivity and absence of off-axis poles in this chamber. Axis completion still requires the conditions in the next section.
For axis limits it is preferable to reconstruct the Gram matrix directly. Put R = diag ( R i ) \mathcal{R}=\operatorname{diag}(R_i) R = diag ( R i )
B ^ = B + 2 v v T = R + T R T + u u T + v v T , w = M − u . (101) \widehat{\mathcal{B}}=\mathcal{B}+2vv^{T}=\mathcal{R}+T\mathcal{R}T+uu^{T}+vv^{T},\qquad w=M_{-}u. \tag*{(101)} B = B + 2 v v T = R + T R T + u u T + v v T , w = M − u . (101) Then the same metric is given by
G t t = 1 − 2 v T B ^ − 1 v , G t ϕ = 2 v T B ^ − 1 w , G_{tt}=1-2v^{T}\widehat{\mathcal{B}}^{-1}v,\qquad G_{t\phi}=2v^{T}\widehat{\mathcal{B}}^{-1}w, G tt = 1 − 2 v T B − 1 v , G tϕ = 2 v T B − 1 w , G ϕ ϕ = ρ 2 + 2 u T M − u − 2 w T B ^ − 1 w , e 2 ν = det B ^ 2 N ∏ i R i . (102) G_{\phi\phi}=\rho^2+2u^{T}M_{-}u-2w^{T}\widehat{\mathcal{B}}^{-1}w,\qquad e^{2\nu}=\frac{\det\widehat{\mathcal{B}}}{2^{N}\prod_iR_i}. \tag*{(102)} G ϕϕ = ρ 2 + 2 u T M − u − 2 w T B − 1 w , e 2 ν = 2 N ∏ i R i det B . (102) The first two entries follow from the rank-one inverse formula. For the third, use M − M + = ρ 2 I M_{-}M_{+}=\rho^2I M − M + = ρ 2 I T 2 = I T^2=I T 2 = I
M − − M − B − 1 M − = ρ 2 T B − 1 T , M_{-}-M_{-}\mathcal{B}^{-1}M_{-}=\rho^2T\mathcal{B}^{-1}T, M − − M − B − 1 M − = ρ 2 T B − 1 T , and then apply the same rank-one formula. This proves (102) directly from (100) . Unlike the separate norm and rotation coordinates, these expressions do not divide by a leading principal minor. Their positivity on the closed axis is used in Proposition 6.3.
Role in computation. Polynomial interpolation and factorization recover charges and asymptotic moments without large subset expansions. The real matrix form supplies stable bulk evaluation and the variation identities used in § 6 . Both retain the cone normalization. The original homogeneous subset rods remain an independent check of the numerical balance equations.
Positive reduced moment ranks
The matrix identities admit a formulation that does not require equal positive and negative ranks. Let 1 ≤ k ≤ N / 2 1\le k\le N/2 1 ≤ k ≤ N /2 ω i \omega_i ω i
W i a = ω i z i a − 1 ( 1 ≤ a ≤ k ) , P = W ( W T W ) − 1 W T . \mathcal{W}_{ia}=\sqrt{\omega_i}z_i^{a-1}\quad(1\le a\le k),\qquad\mathcal{P}=\mathcal{W}(\mathcal{W}^{T}\mathcal{W})^{-1}\mathcal{W}^{T}. W ia = ω i z i a − 1 ( 1 ≤ a ≤ k ) , P = W ( W T W ) − 1 W T . With e k e_k e k R k \mathbb{R}^k R k
a = W ( W T W ) − 1 e k , c = ( I − P ) ( ω i z i k ) i = 1 N , a=\mathcal{W}(\mathcal{W}^{T}\mathcal{W})^{-1}e_k,\qquad c=(I-\mathcal{P})(\sqrt{\omega_i}z_i^k)_{i=1}^{N}, a = W ( W T W ) − 1 e k , c = ( I − P ) ( ω i z i k ) i = 1 N , T = 2 P − I , u = a + c , v = a − c . (103) T=2\mathcal{P}-I,\qquad u=a+c,\qquad v=a-c. \tag*{(103)} T = 2 P − I , u = a + c , v = a − c . (103) Polynomial orthogonality gives all identities in (98) . Common weight scaling sends a a a s − 1 / 2 a s^{-1/2}a s − 1/2 a c c c s 1 / 2 c s^{1/2}c s 1/2 c u T v = 0 u^{T}v=0 u T v = 0 (99) . Thus the positive bulk reconstruction and residue moments retain their form. In these coordinates the reduced slot number is 2 ( N − k ) 2(N-k) 2 ( N − k ) N − 2 k N-2k N − 2 k
The essential positivity property is irreducibility of the weighted polynomial space. If a diagonal matrix preserves it, its diagonal entries are the values of a polynomial of degree below k k k 2 k − 2 < N 2k-2<N 2 k − 2 < N
Comparison with the positive Hankel family
The closed family of Andersson–Dahl–Nilsson [4 Sections 7–8] uses the same positive discrete moment data. The following dictionary also fixes the cone normalization. For signed weights η i \eta_i η i
h a ( η ) = ∑ i η i z i a , H r ( η ) = det ( h a + b ( η ) ) a , b = 0 r − 1 , H 0 = 1. h_a(\eta)=\sum_i\eta_i z_i^a,\qquad H_r(\eta)=\det\left(h_{a+b}(\eta)\right)_{a,b=0}^{r-1},\qquad H_0=1. h a ( η ) = i ∑ η i z i a , H r ( η ) = det ( h a + b ( η ) ) a , b = 0 r − 1 , H 0 = 1. Let H = ( h a + b ) a , b = 0 k − 1 H=\left(h_{a+b}\right)_{a,b=0}^{k-1} H = ( h a + b ) a , b = 0 k − 1 c = ( h k , … , h 2 k − 1 ) T c=\left(h_k,\ldots,h_{2k-1}\right)^T c = ( h k , … , h 2 k − 1 ) T e = e k e=e_k e = e k Σ k = c T adj ( H ) e \Sigma_k=c^T\operatorname{adj}(H)e Σ k = c T adj ( H ) e s z = ∑ i z i s_z=\sum_i z_i s z = ∑ i z i
det ( H c + e ( c + e ) T h 2 k + s z ) = H k + 1 − 2 Σ k − H k − 1 + s z H k . (104) \det\begin{pmatrix} H & c+e \\ (c+e)^T & h_{2k}+s_z \end{pmatrix}=H_{k+1}-2\Sigma_k-H_{k-1}+s_zH_k. \tag*{(104)} det ( H ( c + e ) T c + e h 2 k + s z ) = H k + 1 − 2 Σ k − H k − 1 + s z H k . (104) It follows first where H H H H k − 1 H_{k-1} H k − 1
For the signed axis weights η i = ( − 1 ) 1 i ≤ j ω i \eta_i=(-1)^{\mathbf{1}_{i\leq j}}\omega_i η i = ( − 1 ) 1 i ≤ j ω i
t j = ( H k + 1 − 2 Σ k − H k − 1 + s z H k , H k ) . t_j=\left(H_{k+1}-2\Sigma_k-H_{k-1}+s_zH_k,\ H_k\right). t j = ( H k + 1 − 2 Σ k − H k − 1 + s z H k , H k ) . In the notation of [4 Equation (5.22)] , the shifted determinant is H k + 1 ( a ∞ ) = H k + 1 − 2 a ∞ Σ k − a ∞ 2 H k − 1 H_{k+1}^{(a_\infty)}=H_{k+1}-2a_\infty\Sigma_k-a_\infty^2H_{k-1} H k + 1 ( a ∞ ) = H k + 1 − 2 a ∞ Σ k − a ∞ 2 H k − 1 ( H k , H k + 1 ( 1 ) ) \left(H_k,H_{k+1}^{(1)}\right) ( H k , H k + 1 ( 1 ) ) a ∞ = 1 a_\infty=1 a ∞ = 1 [4 Theorem 8.5] identifies the corresponding metrics, including the reduced phantom sectors. A common positive factor is fixed by the conformal normalization.
This dictionary permits direct use of the analytic axis and end results of [4 Propositions 7.5, 8.7–8.9 and 11.4] . Positive separated data whose complete normalized rods form one primitive unimodular lattice have smooth complete AF or ALF completion, with the end type determined by the exterior rods. Accordingly, an exact root of the full vector equations in the appendices proves regularity; a direction-only root does not. This completion result does not establish a root’s existence or the collision comparison needed for an entire AF parameter interval.
Regularity of the rod structure
Rod regularity means smooth collapse of primitive circles in one period lattice, including at their intersections [7 , 24 ] . The positive-family completion result in § 4.5 makes these normalized rod conditions sufficient for the AF and ALF ends considered here.
Fix the marked period lattice
Λ p e r = Z A + Z B , A = ( 0 , 1 ) , B = ( L , β ) , L > 0 , (105) \Lambda_{\mathrm{per}}=\mathbb{Z}A+\mathbb{Z}B,\qquad A=(0,1),\qquad B=(L,\beta),\qquad L>0, \tag*{(105)} Λ per = Z A + Z B , A = ( 0 , 1 ) , B = ( L , β ) , L > 0 , (105) where a lattice vector generates a flow of period 2 π 2\pi 2 π ( t , ϕ ) (t,\phi) ( t , ϕ ) ( A , − B ) (A,-B) ( A , − B ) L L L
v j = p j A − q j B = ( − q j L , p j − q j β ) . v_j=p_jA-q_jB=(-q_jL,p_j-q_j\beta). v j = p j A − q j B = ( − q j L , p j − q j β ) . A vector is primitive when gcd ( p j , q j ) = 1 \gcd(p_j,q_j)=1 g cd( p j , q j ) = 1 det Z ( v j − 1 , v j ) = 1 \det_{\mathbb{Z}}(v_{j-1},v_j)=1 det Z ( v j − 1 , v j ) = 1 v 0 = A v_0=A v 0 = A v 1 = − B v_1=-B v 1 = − B Z \mathbb{Z} Z ( A , − B ) (A,-B) ( A , − B ) L L L
The balance equations
A regular gravitational instanton in this paper is a smooth, complete, oriented, positive definite Ricci-flat four-manifold with ∫ M ∣ R m ∣ 2 d v o l g < ∞ \int_M |\mathrm{Rm}|^2\,\mathrm{dvol}_g<\infty ∫ M ∣ Rm ∣ 2 dvol g < ∞ R m \mathrm{Rm} Rm
a positive nonsingular metric for ρ > 0 \rho>0 ρ > 0
primitive vanishing vectors in one period lattice, with transverse angle 2 π 2\pi 2 π
adjacent primitive vectors forming a lattice basis at every nut;
the smooth rod and nut normal forms and the prescribed complete AF end with finite curvature energy.
The positive reconstruction (100) gives the bulk requirement. The finite lattice and cone conditions, together with the local BZ expansions established in Proposition 6.3, give smooth axis completion by Corollary 6.4. The end estimates supply completeness and curvature decay [19 , 20 ] .
For a primitive rod generator define its cone coefficient and angle by
κ j 2 = lim ρ → 0 G ( v j , v j ) ρ 2 e 2 ν , θ j = 2 π κ j , b j = log 2 π θ j . (106) \kappa_j^2=\lim_{\rho\to0}\frac{G(v_j,v_j)}{\rho^2 e^{2\nu}},\qquad\theta_j=2\pi\kappa_j,\qquad b_j=\log\frac{2\pi}{\theta_j}. \tag*{(106)} κ j 2 = ρ → 0 lim ρ 2 e 2 ν G ( v j , v j ) , θ j = 2 π κ j , b j = log θ j 2 π . (106) The cone-normalized vector is
r j = κ j − 1 v j = e b j v j . r_j=\kappa_j^{-1}v_j=e^{b_j}v_j. r j = κ j − 1 v j = e b j v j . It is the vector obtained in (85) and (99) . Regularity therefore requires the full vector equations
with consistent signs, rather than only equality of their directions. The signed rod sequence is determined by its self-intersection word W = ( d 1 , … , d N − 1 ) W=(d_1,\ldots,d_{N-1}) W = ( d 1 , … , d N − 1 )
v j + 1 = − v j − 1 − d j v j , ( p 0 , q 0 ) = ( 1 , 0 ) , ( p 1 , q 1 ) = ( 0 , 1 ) . (107) v_{j+1}=-v_{j-1}-d_jv_j,\qquad(p_0,q_0)=(1,0),\qquad(p_1,q_1)=(0,1). \tag*{(107)} v j + 1 = − v j − 1 − d j v j , ( p 0 , q 0 ) = ( 1 , 0 ) , ( p 1 , q 1 ) = ( 0 , 1 ) . (107) Here d j d_j d j j j j
ℓ j = z j + 1 − z j , Area ( S j ) = 2 π L ℓ j , x j = ℓ j / L . (108) \ell_j=z_{j+1}-z_j,\qquad\operatorname{Area}(S_j)=2\pi L\ell_j,\qquad x_j=\ell_j/L. \tag*{(108)} ℓ j = z j + 1 − z j , Area ( S j ) = 2 π L ℓ j , x j = ℓ j / L . (108) Thus x j x_j x j 2 π L 2 2\pi L^2 2 π L 2
The positive charge chamber
For N = 2 m N=2m N = 2 m D ′ ( z i ) = ( − 1 ) i D'(z_i)=(-1)^i D ′ ( z i ) = ( − 1 ) i (87) is equivalent to
∣ C i ∣ < 1 ( i odd ) , ∣ C i ∣ > 1 ( i even ) . (109) |C_i|<1\quad(i\ \text{odd}),\qquad|C_i|>1\quad(i\ \text{even}). \tag*{(109)} ∣ C i ∣ < 1 ( i odd ) , ∣ C i ∣ > 1 ( i even ) . (109) The adjacent-rod determinant has a square factor. For example, where T i − 1 T_{i-1} T i − 1
T i = ( T i − 1 t i t i T T i i ) , η i = v i − t i T T i − 1 − 1 v [ i − 1 ] . T_i=\begin{pmatrix}T_{i-1}&t_i\\ t_i^{T}&T_{ii}\end{pmatrix},\qquad\eta_i=v_i-t_i^{T}T_{i-1}^{-1}v_{[i-1]}. T i = ( T i − 1 t i T t i T ii ) , η i = v i − t i T T i − 1 − 1 v [ i − 1 ] . The rod formula gives
det ( r i − 1 , r i ) = 2 ( det T i − 1 ) 2 η i 2 > 0. (110) \det(r_{i-1},r_i)=2(\det T_{i-1})^2\eta_i^2>0. \tag*{(110)} det ( r i − 1 , r i ) = 2 ( det T i − 1 ) 2 η i 2 > 0. (110) For i = 1 i=1 i = 1 det T 0 = 1 \det T_0=1 det T 0 = 1 η 1 = v 1 \eta_1=v_1 η 1 = v 1
The charge values C i = ± 1 C_i = \pm1 C i = ± 1 [6 ] . An infinite affine charge is not such a wall. Membership in the open chamber is a statement about an individual metric, not a uniform separation estimate. At fixed distinct directions, separated nodes give charge compactness in § 6 . Colliding nodes can invalidate that argument; simultaneous charge and length degeneration must be controlled separately.
The block determinant restriction.
Theorem 5.1 (Block determinant bound). For a balanced metric in the genuine positive even BZ chamber,
∣ det Z ( v a , v b ) ∣ < b − a ( b − a ≥ 2 ) . (111) \left|\det_{\mathbb{Z}}(v_a,v_b)\right| < b-a \qquad(b-a \ge2). \tag*{(111)} Z det ( v a , v b ) < b − a ( b − a ≥ 2 ) . (111) Equivalently, the integral determinant is at most b − a − 1 b-a-1 b − a − 1
Proof mechanism. Let ρ i L \rho_i^{\mathrm{L}} ρ i L ρ i R \rho_i^{\mathrm{R}} ρ i R i i i a , b a,b a , b
det ( r a , r b ) = ∑ i = a + 1 b det ( r i − 1 , r i ) ρ i L ρ i R . \det(r_a,r_b)=\sum_{i=a+1}^{b}\det(r_{i-1},r_i)\rho_i^{\mathrm{L}}\rho_i^{\mathrm{R}}. det ( r a , r b ) = i = a + 1 ∑ b det ( r i − 1 , r i ) ρ i L ρ i R . The unflipped, sign-adjusted minor is a sum of positive occupation weights. A nonempty partial flip produces a signed sum in which both signs occur. Its absolute value is therefore strictly smaller than the unflipped sum. Consequently each ratio has absolute value at most one, and in every summand of a block containing at least two nuts at least one ratio has absolute value strictly below one. At balance, all adjacent determinants equal L L L ∣ det ( r a , r b ) ∣ < ( b − a ) L \left|\det(r_a,r_b)\right| < (b-a)L ∣ det ( r a , r b ) ∣ < ( b − a ) L
For two nuts this yields
d j ∈ { − 1 , 0 , 1 } . (112) d_j \in\{-1,0,1\}. \tag*{(112)} d j ∈ { − 1 , 0 , 1 } . (112) Longer blocks give additional restrictions. For example, the four successive entries ( − 1 , 1 , − 1 , 1 ) (-1,1,-1,1) ( − 1 , 1 , − 1 , 1 )
The admissible twist interval.
Every singular value of a principal block of the orthogonal matrix T T T ∣ det T j ∣ = 1 \left|\det T_j\right|=1 ∣ det T j ∣ = 1 (97) is irreducible, so
∣ r j ϕ ∣ < 1 , θ j > 2 π ∣ v j ϕ ∣ ( 0 < j < N ) . (113) |r_j^\phi| < 1,\qquad\theta_j > 2\pi|v_j^\phi| \qquad(0<j<N). \tag*{(113)} ∣ r j ϕ ∣ < 1 , θ j > 2 π ∣ v j ϕ ∣ ( 0 < j < N ) . (113) Thus balance requires
∣ p j − q j β ∣ < 1 ( 0 < j < N ) . (114) |p_j-q_j\beta|<1 \qquad(0<j<N). \tag*{(114)} ∣ p j − q j β ∣ < 1 ( 0 < j < N ) . (114) An interior rod parallel to the exterior rod is impossible: primitiveness would give ( p j , q j ) = ( ± 1 , 0 ) (p_j,q_j)=(\pm1,0) ( p j , q j ) = ( ± 1 , 0 )
For a word W W W
I W = { β ∈ ( − 1 , 0 ) : ∣ p j − q j β ∣ < 1 for 1 ≤ j < N } . I_W=\{\beta\in(-1,0): |p_j-q_j\beta|<1\text{ for }1\le j<N\}. I W = { β ∈ ( − 1 , 0 ) : ∣ p j − q j β ∣ < 1 for 1 ≤ j < N } . It is the intersection of explicit open intervals with rational endpoints. For N ≥ 3 N\ge3 N ≥ 3 d 1 = − 1 d_1=-1 d 1 = − 1 − 1 < β < 0 -1<\beta<0 − 1 < β < 0 v N = σ v 0 v_N=\sigma v_0 v N = σ v 0 σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 }
∣ q j ∣ ≤ min ( j − 1 , N − j − 1 ) ( 2 ≤ j ≤ N − 2 ) , ∣ q 1 ∣ = ∣ q N − 1 ∣ = 1. |q_j|\le\min(j-1,N-j-1)\qquad(2\le j\le N-2),\qquad|q_1|=|q_{N-1}|=1. ∣ q j ∣ ≤ min ( j − 1 , N − j − 1 ) ( 2 ≤ j ≤ N − 2 ) , ∣ q 1 ∣ = ∣ q N − 1 ∣ = 1. In particular, for even N ≥ 4 N \ge4 N ≥ 4 N / 2 − 1 N/2-1 N /2 − 1 ( − 1 , 0 ) (-1,0) ( − 1 , 0 )
Winding determines the reduced sector
The number of dressing slots is not determined by the number of nuts alone. Consider a positive even representation reduced by removing redundant opposite-sign phantom pairs. Let h h h
n = N + h , k = N − h 2 . n=N+h,\qquad k=\frac{N-h}{2}. n = N + h , k = 2 N − h . The following extension uses the positive reduced moment representation; it is not an assertion about arbitrary rational patching matrices.
For ordered genuine nodes and positive weights ω i \omega_i ω i
μ q [ j ] = ∑ i = 1 N ϵ i [ j ] ω i z i q , ϵ i [ j ] = { − 1 i ≤ j , 1 i > j . \mu_q^{[j]}=\sum_{i=1}^{N}\epsilon_i^{[j]}\omega_i z_i^q,\qquad
\epsilon_i^{[j]}=
\begin{cases}
-1 & i\le j,\\
1 & i>j.
\end{cases} μ q [ j ] = i = 1 ∑ N ϵ i [ j ] ω i z i q , ϵ i [ j ] = { − 1 1 i ≤ j , i > j . For k ≥ 1 k\ge1 k ≥ 1
M j = ( μ a + b [ j ] ) a , b = 0 k − 1 , c j = ( μ k [ j ] , … , μ 2 k − 1 [ j ] ) T , e k = ( 0 , … , 0 , 1 ) T . M_j=\left(\mu_{a+b}^{[j]}\right)_{a,b=0}^{k-1},\qquad
c_j=\left(\mu_k^{[j]},\ldots,\mu_{2k-1}^{[j]}\right)^T,\qquad
e_k=(0,\ldots,0,1)^T. M j = ( μ a + b [ j ] ) a , b = 0 k − 1 , c j = ( μ k [ j ] , … , μ 2 k − 1 [ j ] ) T , e k = ( 0 , … , 0 , 1 ) T . After normalizing the positive parameter at infinity to one, the two rod components, before a common frame change and common scale, are
t j = ( det ( M j c j + e k ( c j + e k ) T μ 2 k [ j ] + ∑ i z i ) , det M j ) . (115) t_j=\left(\det\begin{pmatrix}
M_j & c_j+e_k\\
(c_j+e_k)^T & \mu_{2k}^{[j]}+\sum_i z_i
\end{pmatrix},\det M_j\right). \tag*{(115)} t j = ( det ( M j ( c j + e k ) T c j + e k μ 2 k [ j ] + ∑ i z i ) , det M j ) . (115) Here t j t_j t j t i t_i t i (110) . These formulas follow by taking the same-sign phantom limit of the homogeneous axis sectors before removing the common phantom factors.
The polarized identity in these coordinates is
det ( t a , t b ) = 2 ∑ i = a + 1 b ω i Ψ a ( z i ) Ψ b ( z i ) , Ψ j ( x ) = det ( M j c j + e k ( 1 , x , … , x k − 1 ) x k ) . \det(t_a,t_b)=2\sum_{i=a+1}^{b}\omega_i\Psi_a(z_i)\Psi_b(z_i),\qquad
\Psi_j(x)=\det\begin{pmatrix}
M_j & c_j+e_k\\
(1,x,\ldots,x^{k-1}) & x^k
\end{pmatrix}. det ( t a , t b ) = 2 i = a + 1 ∑ b ω i Ψ a ( z i ) Ψ b ( z i ) , Ψ j ( x ) = det ( M j ( 1 , x , … , x k − 1 ) c j + e k x k ) . At nut i i i Ψ \Psi Ψ k − 1 k-1 k − 1 k k k k ≥ 1 k\ge1 k ≥ 1
det M j = ∑ ∣ S ∣ = k ( − 1 ) ∣ S ∩ { 1 , … , j } ∣ ∏ i ∈ S ω i ∏ a < b a , b ∈ S ( z a − z b ) 2 . \det M_j=\sum_{\lvert S\rvert=k}(-1)^{\lvert S\cap\{1,\ldots,j\}\rvert}\prod_{i\in S}\omega_i\prod_{\substack{a<b\\a,b\in S}}(z_a-z_b)^2. det M j = ∣ S ∣ = k ∑ ( − 1 ) ∣ S ∩ { 1 , … , j }∣ i ∈ S ∏ ω i a < b a , b ∈ S ∏ ( z a − z b ) 2 . Both signs occur for a proper prefix, so ∣ det M j ∣ < det M 0 \lvert\det M_j\rvert<\det M_0 ∣ det M j ∣ < det M 0
The maximal case k = 0 k=0 k = 0 t j = ( μ 0 [ j ] + ∑ i z i , 1 ) t_j=(\mu_0^{[j]}+\sum_i z_i,1) t j = ( μ 0 [ j ] + ∑ i z i , 1 )
det ( t 0 , t N ) = 2 ∑ i ω i > 0. \det(t_0,t_N)=2\sum_i\omega_i>0. det ( t 0 , t N ) = 2 i ∑ ω i > 0. Their exterior lines do not agree, so this case is excluded by AF closure. It must not be excluded from the ALF problem by an AF argument.
Proposition 5.2 (AF phantom restriction). In the positive reduced representation above, a regular AF metric with N ≥ 3 N\ge3 N ≥ 3
h ≡ N ( m o d 2 ) , 3 h ≤ N . h\equiv N\pmod2,\qquad3h\le N. h ≡ N ( mod 2 ) , 3 h ≤ N . Equality occurs only for N = 3 N=3 N = 3 h = 1 h=1 h = 1 ( − 1 , − 1 ) (-1,-1) ( − 1 , − 1 ) h < N / 3 h<N/3 h < N /3 N > 3 N>3 N > 3
Winding argument. Let w \mathbf{w} w ( 0 , π ) (0,\pi) ( 0 , π ) π w \pi\mathbf{w} π w w \mathbf{w} w f i = ( 1 , z i , … , z i k − 1 ) T f_i=(1,z_i,\ldots,z_i^{k-1})^T f i = ( 1 , z i , … , z i k − 1 ) T
M j = M 0 − 2 ∑ i ≤ j ω i f i f i T , M 0 > 0 , M N = − M 0 . M_j=M_0-2\sum_{i\leq j}\omega_i f_i f_i^T,\qquad M_0>0,\qquad M_N=-M_0. M j = M 0 − 2 i ≤ j ∑ ω i f i f i T , M 0 > 0 , M N = − M 0 . Continuous prefix flips give exactly k k k (115) are affine during each rank-one flip, so the rod follows the straight segment between its endpoints. Positive adjacent determinant prevents it from passing through zero. Hence w = k \mathbf{w}=k w = k N − k N-k N − k h = ∣ N − 2 w ∣ h=\lvert N-2\mathbf{w}\rvert h = ∣ N − 2 w ∣
Write v N = σ v 0 \mathbf{v}_N=\sigma\mathbf{v}_0 v N = σ v 0 v N − 1 = ( p , − σ ) \mathbf{v}_{N-1}=(p,-\sigma) v N − 1 = ( p , − σ ) c = σ p c=\sigma p c = σ p c ∈ { 0 , 1 } c\in\{0,1\} c ∈ { 0 , 1 } σ \sigma σ [16 ] gives
3 h = ∣ ∑ j = 1 N − 1 d j − c ∣ ≤ N . 3h=\left\lvert\sum_{j=1}^{N-1}d_j-c\right\rvert\leq N. 3 h = j = 1 ∑ N − 1 d j − c ≤ N . Equality forces every d j = − 1 d_j=-1 d j = − 1 c = 1 c=1 c = 1 v 3 = − v 0 \mathbf{v}_3=-\mathbf{v}_0 v 3 = − v 0 N = 3 N=3 N = 3 h = N − 2 k h=N-2k h = N − 2 k
For N = 4 , 6 N=4,6 N = 4 , 6 h = 0 h=0 h = 0 h = 1 h=1 h = 1 § 4.4 also supplies the matrix identities used in the cone-convexity theorem.
Enumeration of admissible words
We call a word admissible when it passes the necessary tests in this section in its stated positive BZ sector. This term does not assert metric existence. Enumeration proceeds as follows:
Generate the ternary words with d 1 = − 1 d_1=-1 d 1 = − 1 (107) , and impose AF exterior closure.
Check every block determinant, exclude interior returns to the exterior direction, and retain the words with I W ≠ ∅ I_W\ne\varnothing I W = ∅
Compute the projective winding and the allowed reduced dressing sector. For the genuine even sector require w = N / 2 \mathbf{w}=N/2 w = N /2
Identify axis-reversal and orientation equivalences, carrying along their transformations of the marking and of β \beta β
The last step is a quotient of the candidate list, not an additional regularity inequality or a symmetry imposed on a metric. For an oriented classification, retain the orientation labels. With d 1 = − 1 d_1=-1 d 1 = − 1 W W W W W W β \beta β − 1 − β -1-\beta − 1 − β d N − 1 = − 1 d_{N-1}=-1 d N − 1 = − 1 d N − 1 = 1 d_{N-1}=1 d N − 1 = 1 W W W − - − W W W β \beta β
For example, the six-nut tests leave precisely
( − 1 , 0 , 0 , 0 , 1 ) , ( − 1 , 1 , 0 , − 1 , 1 ) , ( − 1 , 1 , 1 , 1 , − 1 ) , (-1,0,0,0,1),\qquad(-1,1,0,-1,1),\qquad(-1,1,1,1,-1), ( − 1 , 0 , 0 , 0 , 1 ) , ( − 1 , 1 , 0 , − 1 , 1 ) , ( − 1 , 1 , 1 , 1 , − 1 ) , all with necessary interval ( − 1 , 0 ) (-1,0) ( − 1 , 0 ) h = 0 h=0 h = 0 Theorem 6.6 ; their certified balance data are included in the appendix.
The enumeration classifies integral candidates in the specified positive BZ sectors. The distinction between admissibility and existence is essential: the collision criterion in § 6 is sufficient for balance, whereas the boundary-gradient obstruction excludes certain admissible words. Both tests depend on the normalized cone family, in addition to the integral recurrence. A common self-intersection block need not have the same obstruction in different ambient words, because its exterior cone normalization must be matched to the surviving geometry.
The cone potential
The regularity problem becomes a variational problem once the charges have been solved as functions of the rod lengths and directions. We first establish these coordinates and the positive Hessian, then analyze finite collision faces and large lengths. The algebraic statements use the positive reduced reconstruction of § 4.4 ; applications of canonical cone limits have the additional hypotheses stated in § 6.4 .
Unique direction matching at separated nodes
Fix ordered distinct nodes. Multiplying every positive weight ω i \omega_i ω i s > 0 s>0 s > 0 T T T (103) , it sends a ↦ s − 1 / 2 a a\mapsto s^{-1/2}a a ↦ s − 1/2 a c ↦ s 1 / 2 c c\mapsto s^{1/2}c c ↦ s 1/2 c u T v = 0 u^{T}v=0 u T v = 0 N − 1 N-1 N − 1
Fix the first rod direction and orient the normalized rods so that adjacent determinants are positive. Let α i \alpha_i α i i − 1 i-1 i − 1 i i i
0 < α i < π , ∑ i = 1 N α i = k π . (116) 0<\alpha_i<\pi,\qquad\sum_{i=1}^{N}\alpha_i=k\pi. \tag*{(116)} 0 < α i < π , i = 1 ∑ N α i = k π . (116) The prefix moment inertia calculation in § 5.5 gives winding k k k N = 2 m N=2m N = 2 m k = m k=m k = m (116) is a convex domain of dimension N − 1 N-1 N − 1
Proposition 6.1 (Direction coordinates). At fixed ordered nodes, the AF positive-charge domain maps analytically and bijectively onto the direction domain (116) . The inverse is analytic. In particular, for fixed directions in this domain and fixed L L L
Proof. Lemma C.1 supplies the coupled triangular transport, including when a principal block of T T T δ Z = 0 \delta Z=0 δ Z = 0 [ Z + v v T , T ] = 0 [Z+vv^{T},T]=0 [ Z + v v T , T ] = 0 ∥ v ∥ 2 ≤ z N − z 1 \lVert v\rVert^{2}\le z_N-z_1 ∥ v ∥ 2 ≤ z N − z 1
A normalized rod cannot vanish in such a limit. Otherwise a nonzero vector h h h T j h = 0 T_jh=0 T j h = 0 v [ j ] T h = 0 v_{[j]}^{T}h=0 v [ j ] T h = 0 T h Th T h Z + v v T Z+vv^{T} Z + v v T z j ∥ h ∥ 2 z_j\lVert h\rVert^{2} z j ∥ h ∥ 2 z j + 1 ∥ h ∥ 2 z_{j+1}\lVert h\rVert^{2} z j + 1 ∥ h ∥ 2
If relative weights escaped, the positive Cauchy–Binet sums for the polynomial projector would force a diagonal projector entry to 0 0 0 1 1 1 T T T (116) is a one-sheeted covering, proving the claim. □
A marked word belongs to this direction domain precisely when its positive signed lift has projective winding w = k \mathbf{w}=k w = k k ≤ N / 2 k\le N/2 k ≤ N /2
The proposition prevents charge escape on compact sets of separated positive lengths. Collision limits require the additional matching stated below.
Convexity from the mass differential
By Lemma C.1, every fixed-direction variation of the paired boundary data ( T , v ) (T,v) ( T , v ) K K K
δ T = K T + T K T , δ v = K v . (117) \delta T=KT+TK^{T},\qquad\delta v=Kv. \tag*{(117)} δ T = K T + T K T , δ v = K v . (117) If L \mathcal{L} L L j \mathcal{L}_j L j j j j ( det L j ) 2 (\det\mathcal{L}_j)^2 ( det L j ) 2 K j K_j K j K K K S = K + K T S=K+K^T S = K + K T
S T + T S = 0 , δ u = − K T u , δ b j = 2 tr K j . (118) ST+TS=0,\qquad\delta u=-K^T u,\qquad\delta b_j=2\operatorname{tr}K_j. \tag*{(118)} S T + T S = 0 , δ u = − K T u , δ b j = 2 tr K j . (118) The coupled transport is essential. The map K ↦ K T + T K T K\mapsto KT+TK^T K ↦ K T + T K T [3 ] , this occurs at a finite rod with zero angular component, p j − β q j = 0 p_j-\beta q_j=0 p j − β q j = 0 § 1 instead transports the auxiliary pencil T + c v v T T+cvv^T T + c v v T c c c v v v (117) for the original data. Thus the argument below also applies at these special twists.
Since Z + v v T Z+vv^T Z + v v T T T T S S S tr ( ( Z + v v T ) S ) = 0 \operatorname{tr}((Z+vv^T)S)=0 tr (( Z + v v T ) S ) = 0 (90) and summation by parts gives the first law at fixed L , β L,\beta L , β
2 δ M = v T S v = − 2 ∑ i z i K i i = ∑ j = 1 N − 1 ℓ j δ b j . (119) 2\delta M=v^T Sv=-2\sum_i z_iK_{ii}=\sum_{j=1}^{N-1}\ell_j\delta b_j. \tag*{(119)} 2 δ M = v T S v = − 2 i ∑ z i K ii = j = 1 ∑ N − 1 ℓ j δ b j . (119) It follows that
Φ W ( ℓ ; β ) = ∑ j ℓ j b j − 2 M , d ℓ Φ W = ∑ j b j d ℓ j . (120) \Phi_W(\ell;\beta)=\sum_j\ell_jb_j-2M,\qquad d_\ell\Phi_W=\sum_jb_j\,d\ell_j. \tag*{(120)} Φ W ( ℓ ; β ) = j ∑ ℓ j b j − 2 M , d ℓ Φ W = j ∑ b j d ℓ j . (120) The subscript in d ℓ d_\ell d ℓ L L L β \beta β D ℓ b D_\ell b D ℓ b
Theorem 6.2 (Strict cone convexity). Fix the marked directions in (116) , the asymptotic scale and the exterior gauge. On the full positive length orthant, Φ W \Phi_W Φ W is real analytic and strictly convex. Its Hessian H = D ℓ b H=D_\ell b H = D ℓ b satisfies
δ ℓ T H δ ℓ = ∑ i > j ( z i − z j ) K i j 2 + ∥ ( K + K T ) v ∥ 2 > 0 (121) \delta\ell^T H\delta\ell=\sum_{i>j}(z_i-z_j)K_{ij}^2+\lVert(K+K^T)v\rVert^2>0 \tag*{(121)} δ ℓ T H δ ℓ = i > j ∑ ( z i − z j ) K ij 2 + ∥( K + K T ) v ∥ 2 > 0 (121) for every nonzero length variation.
The positive identity. Differentiate [ Z + v v T , T ] = 0 [Z+vv^T,T]=0 [ Z + v v T , T ] = 0
F = δ Z + [ Z , K ] + v v T S . F=\delta Z+[Z,K]+vv^T S. F = δ Z + [ Z , K ] + v v T S . One obtains F T = T F T FT=TF^T F T = T F T T S T T = − S TST^T=-S T S T T = − S tr ( F S ) = 0 \operatorname{tr}(FS)=0 tr ( F S ) = 0
− 2 tr ( δ Z K ) = ∑ i > j ( z i − z j ) K i j 2 + ∥ S v ∥ 2 . -2\operatorname{tr}(\delta ZK)=\sum_{i>j}(z_i-z_j)K_{ij}^2+\lVert Sv\rVert^2. − 2 tr ( δ Z K ) = i > j ∑ ( z i − z j ) K ij 2 + ∥ S v ∥ 2 . By (118) , the left side is ∑ j δ ℓ j δ b j \sum_j\delta\ell_j\delta b_j ∑ j δ ℓ j δ b j K K K T T T K v = 0 Kv=0 K v = 0 K 2 K^2 K 2 T T T § 4.4 makes K 2 K^2 K 2 v ≠ 0 v\ne0 v = 0 K = 0 K=0 K = 0 δ Z \delta Z δ Z δ ℓ = 0 \delta\ell=0 δ ℓ = 0
All N − 1 N-1 N − 1 L L L ℓ \ell ℓ L L L
Mass as the dual potential
Strict convexity on a convex domain gives at most one length vector with any prescribed attainable cone vector. In particular, a balanced point is unique and is the global minimum of Φ W \Phi_W Φ W
The Legendre transform has a simple geometric interpretation. On the realized cone image,
Φ W ∗ ( b ) = 2 M , ∇ b ( 2 M ) = ℓ , D b 2 ( 2 M ) = H − 1 > 0. (122) \Phi_W^*(b) = 2M,\qquad\nabla_b(2M) = \ell,\qquad D_b^2(2M) = H^{-1} > 0. \tag*{(122)} Φ W ∗ ( b ) = 2 M , ∇ b ( 2 M ) = ℓ , D b 2 ( 2 M ) = H − 1 > 0. (122) Lengths and logarithmic cone amplitudes are dual coordinates. This does not assert that every cone vector is realized. It also does not say that the mass is minimized as a function of lengths. In fact,
2 ∇ ℓ M = H ℓ , ℓ ⋅ ∇ ℓ M = 1 2 ℓ T H ℓ > 0. (123) 2\nabla_{\ell}M = H\ell,\qquad\ell\cdot\nabla_{\ell}M = \frac{1}{2}\ell^T H\ell> 0. \tag*{(123)} 2 ∇ ℓ M = H ℓ , ℓ ⋅ ∇ ℓ M = 2 1 ℓ T H ℓ > 0. (123) The mass minima on the balanced twist curves in the appendix concern a different variation.
Canonical identification
The positive BZ reconstruction solves the canonical conical problem at separated nodes. This provides the geometric interpretation needed to apply the cone-limit results of [19 , 20 ] .
Proposition 6.3 (Canonical identification). In every positive reduced moment sector of § 4.4 , the AF-aligned BZ metric with distinct genuine nodes is the canonical conical harmonic-map metric for its marked rod directions and normalized end. For the direction lift in Proposition 6.1, the identification holds over the full positive length orthant, including rods with zero angular component.
Proof outline. The augmented bulk matrix stays positive at the axis and nuts. Its analytic rod and nut expansions place the harmonic map at bounded distance from the canonical local models. The AF expansion fixes the normalization at infinity, so uniqueness of the canonical harmonic map identifies the metrics. Appendix C.§ 2 gives the expansions and comparison argument. The same identification follows from the positive-family dictionary in § 4.5 and [4 Propositions 8.9 and 10.2] . □
Corollary 6.4 (Smooth completion at balance). A balanced positive reduced solution with distinct genuine nodes, primitive rods and unit adjacent lattice determinants defines a smooth complete AF gravitational instanton.
Cone balance removes the axis singularities, adjacent unimodularity removes the nut quotients, and the normalized end gives completeness and finite curvature energy. This also follows from [4 Proposition 11.4] . These are separated-node statements; they do not identify every collision limit with a smaller BZ model.
Opening a collision
Consider a consecutive cluster word C C C 2 π 2\pi 2 π ξ j > 0 \xi_j > 0 ξ j > 0 b C , j b_{C,j} b C , j § 3 .
The homogeneous first law gives the convex potential
Ψ C ( ξ ) = ∑ j ξ j b C , j ( ξ ) , d Ψ C = ∑ j b C , j d ξ j , μ ( C ) = inf ξ j > 0 , ∑ j ξ j = 1 Ψ C ( ξ ) . (124) \Psi_C(\xi) = \sum_j \xi_j b_{C,j}(\xi),\qquad d\Psi_C = \sum_j b_{C,j}\,d\xi_j,\qquad\mu(C) = \inf_{\xi_j>0,\ \sum_j \xi_j=1}\Psi_C(\xi). \tag*{(124)} Ψ C ( ξ ) = j ∑ ξ j b C , j ( ξ ) , d Ψ C = j ∑ b C , j d ξ j , μ ( C ) = ξ j > 0 , ∑ j ξ j = 1 inf Ψ C ( ξ ) . (124) The normalization removes the common scale; the infimum allows further collisions among the cluster’s own rods. The value of μ ( C ) \mu(C) μ ( C )
Only two facts about this invariant are needed here. A negative value supplies a positive shape with negative opening cost. Its sign can be certified at any positive direction-matched shape ξ \xi ξ ∑ j ξ j = 1 \sum_j \xi_j=1 ∑ j ξ j = 1
min j b C , j ( ξ ) ≤ μ ( C ) ≤ ∑ j ξ j b C , j ( ξ ) . (125) \min_j b_{C,j}(\xi)\le\mu(C)\le\sum_j \xi_j b_{C,j}(\xi). \tag*{(125)} j min b C , j ( ξ ) ≤ μ ( C ) ≤ j ∑ ξ j b C , j ( ξ ) . (125) Thus one negative weighted trial proves negativity, whereas a trial with every error positive proves positivity. At a positive constrained stationary point all errors are equal, and their common value is μ ( C ) \mu(C) μ ( C )
Hypotheses 6.5 (Homogeneous collision matching). For each nonparallel collision block used in the existence criterion, assume the following.
(1) At a collision with balanced surviving rods, the cluster cones converge to the homogeneous model with the inherited primitive periods and both exterior angles equal to 2 π 2\pi 2 π q q q ι C ξ \iota_C\xi ι C ξ
b j ( q + ε ι C ξ ) ⟶ b C , j ( ξ ) , j ∈ C , b_j(q+\varepsilon\iota_C\xi)\longrightarrow b_{C,j}(\xi),\qquad j\in C, b j ( q + ε ι C ξ ) ⟶ b C , j ( ξ ) , j ∈ C , locally uniformly in positive shapes and interior twist. Simultaneous disjoint blocks use the same normalization.
(2) At fixed distinct cluster nodes, the homogeneous direction map is locally invertible in positive weight ratios after fixing the exterior frame.
The first condition concerns the amplitudes of the rods, not just their projective directions. The fixed-node surjectivity theorem of [4 Section 9] does not prove either collision condition. Proposition 6.3 identifies the separated AF family but does not by itself prove this normalized collision matching.
Existence from boundary descent
Set L = 1 L=1 L = 1 Φ W \Phi_W Φ W ξ C \xi_C ξ C
lim ε ↓ 0 ∑ j ∈ C ξ j b j ( q + ε ι C ξ C ) = Ψ C ( ξ C ) . (126) \lim_{\varepsilon\downarrow0}\sum_{j\in C}\xi_j b_j(q+\varepsilon\iota_C\xi_C)=\Psi_C(\xi_C). \tag*{(126)} ε ↓ 0 lim j ∈ C ∑ ξ j b j ( q + ε ι C ξ C ) = Ψ C ( ξ C ) . (126) A negative trial therefore rules out that boundary minimum.
Theorem 6.6 (Existence from collision descent). Let W W W C C C μ ( C ) < 0 \mu(C)<0 μ ( C ) < 0 β ∈ I W \beta\in I_W β ∈ I W L > 0 L>0 L > 0
Proof outline. Negative opening costs exclude nonparallel collision minima. For parallel blocks a cone error tends to − ∞ -\infty − ∞ § 4 . □
Consequently the balanced lengths have positive lower and finite upper bounds on every compact subinterval of I W I_W I W § 8 .
Refinement by zero pairs
Inserting two adjacent zero self-intersections replaces the consecutive signed directions v − , v + \mathbf{v}_{-}, \mathbf{v}_{+} v − , v +
v − , v + , − v − , − v + , (127) \mathbf{v}_{-},\quad\mathbf{v}_{+},\quad-\mathbf{v}_{-},\quad-\mathbf{v}_{+}, \tag*{(127)} v − , v + , − v − , − v + , (127) and reverses the signs of the remaining suffix. The set of vanishing circles is unchanged, while the ordered walk acquires two genuine turning points. This operation is different from adding phantom slots.
Lemma 6.7 (Positive zero-pair insertion). Let D D D D ♯ D^{\sharp} D ♯
μ ( D ♯ ) ≤ μ ( D ) . \mu(D^{\sharp}) \leq\mu(D). μ ( D ♯ ) ≤ μ ( D ) . At an interior nut, a symmetric opening of total length ε \varepsilon ε
Ψ D ♯ ( ξ ε ) = Ψ D ( ξ ) − ε log 2 + o ( ε ) . (128) \Psi_{D^{\sharp}}(\xi_{\varepsilon}) = \Psi_D(\xi) - \varepsilon\log2 + o(\varepsilon). \tag*{(128)} Ψ D ♯ ( ξ ε ) = Ψ D ( ξ ) − ε log 2 + o ( ε ) . (128) The proof splits one positive moment node into three and evaluates the two consecutive Hankel determinants by a Schur complement; see Appendix C.§ 6 . The inequality preserves a negative sign under refinement. It does not preserve positivity, and does not prove that every sufficiently long refinement becomes negative.
Families obtained by extending zero strings
The following extension preserves both the direction arrangement and the asymptotic twist interval.
Theorem 6.8 (Zero-string extension). Suppose an admissible A F AF A F W W W 0 2 r + 1 0^{2r+1} 0 2 r + 1 r ≥ 0 r \geq0 r ≥ 0 0 a 0^a 0 a a a a
Proof. A zero pair repeats existing directions; it increases the nut count by two and the projective winding by one, leaving h = ∣ N − 2 w ∣ h = \lvert N - 2w\rvert h = ∣ N − 2 w ∣ § 5 . □
Theorem 6.9 (Explicit infinite families). For all nonnegative integers r , s r, s r , s
Their finite seed checks and a direct reduction for the second family appear in Appendix C.§ 7 . The first family is the Li–Sun family, whose canonical AF existence also follows from [20 Theorem 6.4] . The additional families use the explicit matching hypotheses stated here.
A positive boundary obstruction
The potential also provides a nonexistence test that will distinguish walks on the same root arrangement.
Proposition 6.10 (Positive boundary gradient). Suppose a direction-matched family q ε > 0 q_{\varepsilon} > 0 q ε > 0
q ε → q 0 , b ( q ε ) → g , q ≥ 0 , q ≠ 0 , q ⋅ q 0 = 0. q_{\varepsilon} \to q_{0},\qquad b(q_{\varepsilon}) \to g,\qquad q \geq0,\qquad q \ne0,\qquad q \cdot q_{0} = 0. q ε → q 0 , b ( q ε ) → g , q ≥ 0 , q = 0 , q ⋅ q 0 = 0. Then the positive length orthant contains no balanced point.
Table 1. Zero-string families. Here N N N h h h
Proof. If b ( x ) = 0 b(x)=0 b ( x ) = 0 b ( q ε ) ⋅ ( q ε − x ) ≥ 0 b(q_\varepsilon)\cdot(q_\varepsilon-x)\ge0 b ( q ε ) ⋅ ( q ε − x ) ≥ 0 g ⋅ x ≤ 0 g\cdot x\le0 g ⋅ x ≤ 0 x > 0 x>0 x > 0 g g g
The condition refers to a matched degeneration of the full word. A positive invariant for an isolated subword is not sufficient to apply it in an arbitrary embedding.
6.10. Variation along a balanced family. Let M ‾ = M / L \overline{M}=M/L M = M / L J ‾ = J / L 2 \overline{J}=J/L^2 J = J / L 2
d M ‾ d β = 2 J ‾ \frac{d\overline{M}}{d\beta}=2\overline{J} d β d M = 2 J on a balanced branch at fixed L L L β \beta β § 5 concern this separate variation.
Classification through generalized root systems
The rod restrictions organize the candidates by a finite set of lattice directions. This section identifies that set with a rank-two crystallographic arrangement, explains its polygon description, and tests what this description retains about cone balance. The AF and ALF catalogues test which information this organization retains; § 8 gives the complementary endpoint description. The ordered walk remains part of the data: distinct metrics can revisit the same directions in different orders and with different multiplicities.
From rod directions to a root arrangement
Use coefficient vectors v j = ( p j , q j ) ∈ Z 2 v_j=(p_j,q_j)\in\mathbb{Z}^2 v j = ( p j , q j ) ∈ Z 2 ( A , − B ) (A,-B) ( A , − B )
R ( W ) = { ± v j : 0 ≤ j ≤ N } , \mathcal{R}(W)=\{\pm v_j:0\le j\le N\}, R ( W ) = { ± v j : 0 ≤ j ≤ N } , with repetitions removed. In two dimensions, a crystallographic arrangement can be described by a finite centrally symmetric set of primitive vectors for which every pair of angularly consecutive rays forms an integral basis. It determines a finite generalized root system of a Weyl groupoid [10 ] . Here the simple-root basis changes with the chosen chamber; the groupoid records the reflection maps between these bases. A single reflection group for one fixed inner product need not preserve the entire arrangement.
Proposition 7.1 (The direction arrangement). Let a primitive unimodular rod walk start with d 1 = − 1 d_1=-1 d 1 = − 1 d j ∈ { − 1 , 0 , 1 } d_j\in\{-1,0,1\} d j ∈ { − 1 , 0 , 1 }
Proof. The first three directions are v 0 , v 1 , v 1 − v 0 v_0,v_1,v_1-v_0 v 0 , v 1 , v 1 − v 0 ± 1 \pm1 ± 1 u + v u+v u + v u , v u,v u , v u + v u+v u + v u = ( 1 , 0 ) u=(1,0) u = ( 1 , 0 ) v = ( 0 , 1 ) v=(0,1) v = ( 0 , 1 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) u + v u+v u + v u + v u+v u + v
The hypothesis d 1 = − 1 d_1=-1 d 1 = − 1
Triangulated polygons
Let s s s R ( W ) \mathcal{R}(\mathrm{W}) R ( W ) α 1 , … , α s \alpha_1,\ldots,\alpha_s α 1 , … , α s α 0 = − α s \alpha_0=-\alpha_s α 0 = − α s α s + 1 = − α 1 \alpha_{s+1}=-\alpha_1 α s + 1 = − α 1 c i c_i c i
α i − 1 + α i + 1 = c i α i , c i = det ( α i − 1 , α i + 1 ) . (129) \alpha_{i-1}+\alpha_{i+1}=c_i\alpha_i,\qquad c_i=\det(\alpha_{i-1},\alpha_{i+1}). \tag*{(129)} α i − 1 + α i + 1 = c i α i , c i = det ( α i − 1 , α i + 1 ) . (129) The cyclic sequence ( c 1 , … , c s ) (c_1,\ldots,c_s) ( c 1 , … , c s ) s s s [11 ] . Adding the sum of two neighboring directions corresponds to attaching an ear to the triangulation.
The coefficients c i c_i c i d j d_j d j d j d_j d j Figure 1 is a combinatorial encoding; it is neither the orbit space nor the rod polygon obtained by following the original ordered vectors.
Figure 1. The six triangulation types in the AF census through N = 12 N = 12 N = 12 c i c_i c i
The AF census by arrangement
Table 2 groups all 359 candidates through twelve genuine turning points by the complete direction set, including the exterior. Exactly 356 have certified midpoint balance roots; the other three are excluded by Proposition 6.10 . The analytic block tests verify the conditional full-interval criterion for the 356 retained words.
Table 2. AF candidates for 3 ≤ N ≤ 12 3 \le N \le12 3 ≤ N ≤ 12
The triangle is the Li–Sun direction arrangement; within this census it has one word at each N N N L W = ( − 1 , 0 ) \mathcal{L}_{W}=(-1,0) L W = ( − 1 , 0 )
The usual root-system names depend on whether the exterior is included. The triangle gives A 2 A_2 A 2 B 2 B_2 B 2 G 2 G_2 G 2 ± B \pm B ± B ± ( A + B ) \pm(A+B) ± ( A + B ) ± ( A + 2 B ) \pm(A+2B) ± ( A + 2 B ) A 2 A_2 A 2 ± A \pm A ± A B 2 B_2 B 2
Fixing an arrangement also turns word enumeration into a finite walk problem. Its states are ordered root pairs ( u , v ) (u,v) ( u , v ) ( u , v ) ↦ ( v , − u − d v ) (u,v) \mapsto(v,-u-dv) ( u , v ) ↦ ( v , − u − d v ) d ∈ { − 1 , 0 , 1 } d \in\{-1,0,1\} d ∈ { − 1 , 0 , 1 } G 2 G_2 G 2 N = 6 , 8 , 10 , 12 N = 6, 8, 10, 12 N = 6 , 8 , 10 , 12
An independent one-point search at N = 14 N = 14 N = 14
Different walks on the same polygon
The three excluded twelve-nut words are 12-065, 12-066 and 12-145. Their matched degenerations contain the six-nut block
E = ( − 1 , 0 , − 1 , 1 , − 1 ) , μ ( E ) = 0.09003150665 … > 0. E = (-1,0,-1,1,-1), \qquad\mu(E) = 0.09003150665\ldots> 0. E = ( − 1 , 0 , − 1 , 1 , − 1 ) , μ ( E ) = 0.09003150665 … > 0. Appendix A.7 proves the positive full boundary gradient required for exclusion. The common zigzag polygon does not by itself cause the obstruction.
To see this, define E ♯ = ( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 ) E^{\sharp}=(-1,0,0,0,-1,1,-1) E ♯ = ( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 ) E R E^{R} E R
W 0 = ( E , 0 , − E R ) , W 1 = ( E ♯ , 0 , − ( E ♯ ) R ) W_0=(E,0,-E^{R}),\qquad W_1=(E^{\sharp},0,-(E^{\sharp})^{R}) W 0 = ( E , 0 , − E R ) , W 1 = ( E ♯ , 0 , − ( E ♯ ) R ) have the same zigzag arrangement and necessary window ( − 3 / 5 , − 1 / 5 ) (-3/5,-1/5) ( − 3/5 , − 1/5 ) β = − 2 / 5 \beta=-2/5 β = − 2/5 μ ( E ♯ ) = − 0.01811463686 … < 0 \mu(E^{\sharp})=-0.01811463686\ldots<0 μ ( E ♯ ) = − 0.01811463686 … < 0
Table 3. The same six unoriented directions with different multiplicities. The exterior rods are included, so the totals are N + 1 N+1 N + 1
The root certificate for W 1 W_1 W 1 W 0 W_0 W 0
A realizability conjecture
The arrangement supplies a finite alphabet of directions. An ordered walk specifies the rod topology, and the asymptotic marking fixes which period data are to be balanced. The remaining question is which of these marked walks have an interior minimum of the cone potential.
Conjecture 7.2 (AF polygon realizability). Every rank-two crystallographic arrangement occurring in an admissible AF word in the positive reduced sectors occurs in an admissible marked word with a regular positive BZ realization at some interior twist.
The realizing walk may differ from the word that first exhibits the arrangement. The conjecture does not assert that every walk is regular, nor that every sufficiently long zero extension works. Zero-pair insertion provides a candidate construction because it preserves the arrangement and can lower the opening cost. An arbitrary arrangement may support many words, and a general realization theorem remains open.
There is a related geometric description of the direction data: the ordered matrix ( α 1 ⋯ α s ) (\alpha_1\ \cdots\ \alpha_s) ( α 1 ⋯ α s ) [11 Section 4] . This describes the integral boundary directions; it does not identify the moduli space of regular metrics with that Grassmannian. Cone balance still imposes the amplitudes of the boundary spinor descendants in one period lattice.
Classification by endpoint compositions
The polygon records which directions occur. An endpoint composition retains how successive groups of rods fit together when the AF twist approaches an endpoint of its admissible interval. Each component carries its own ordered word, normalized exterior rods and asymptotic circle marking. For an ALF component, this includes the primitive lattice direction of the circle whose length stays bounded at infinity; Appendix B explains the marking and its possible multiplicity. These additional data distinguish assemblies that have the same polygon. We first give exact compatibility rules, then separate their finite catalogue consequence from the analytic opening problem.
Marked components and the joining rule
Write the normalized rods of a component C C C R 0 , … , R m R_0,\ldots,R_m R 0 , … , R m m m m L L L ϕ i = R i ϕ \phi_i=R_i^\phi ϕ i = R i ϕ ∣ ϕ 0 ∣ = ∣ ϕ m ∣ = 1 |\phi_0|=|\phi_m|=1 ∣ ϕ 0 ∣ = ∣ ϕ m ∣ = 1
β L ( C ) = − ϕ 1 ϕ 0 , β R ( C ) = − ϕ m − 1 ϕ m . (130) \beta_L(C)=-\frac{\phi_1}{\phi_0},\qquad\beta_R(C)=-\frac{\phi_{m-1}}{\phi_m}. \tag*{(130)} β L ( C ) = − ϕ 0 ϕ 1 , β R ( C ) = − ϕ m ϕ m − 1 . (130) A component is marked when these parameters, its orientation, moment rank and primitive period lattice are retained. The name of its underlying metric alone is insufficient.
For its residue spinors u C , v C u_C,v_C u C , v C
γ C = u C T v C , g C = γ C L . (131) \gamma_C=u_C^Tv_C,\qquad g_C=\frac{\gamma_C}{L}. \tag*{(131)} γ C = u C T v C , g C = L γ C . (131) The ALF residue identities give ∥ u C ∥ 2 = ∥ v C ∥ 2 = 2 M C \lVert u_C\rVert^2=\lVert v_C\rVert^2=2M_C ∥ u C ∥ 2 = ∥ v C ∥ 2 = 2 M C R t / R ϕ R^t/R^\phi R t / R ϕ 2 γ C 2\gamma_C 2 γ C g C g_C g C [4 ] .
Suppose components C , D C,D C , D − β R ( C ) ϕ -\beta_R(C)\phi − β R ( C ) ϕ − β L ( D ) ϕ -\beta_L(D)\phi − β L ( D ) ϕ ϕ = ± 1 \phi=\pm1 ϕ = ± 1
d j o i n = β R ( C ) + β L ( D ) . (132) d_{\mathrm{join}}=\beta_R(C)+\beta_L(D). \tag*{(132)} d join = β R ( C ) + β L ( D ) . (132) Hence an admissible join must be integral and belong to { − 1 , 0 , 1 } \{-1,0,1\} { − 1 , 0 , 1 }
For an ordered assembly ( C 1 , … , C s ) (C_1,\ldots,C_s) ( C 1 , … , C s )
G j = ∑ α = 1 j g C α . G_j=\sum_{\alpha=1}^{j}g_{C_\alpha}. G j = α = 1 ∑ j g C α . AF alignment requires G s = 0 G_s=0 G s = 0 2 L G j 2LG_j 2 L G j ( − L q i , p i − q i β ) (-Lq_i,p_i-q_i\beta) ( − L q i , p i − q i β ) q i = − 2 ϕ i G j q_i=-2\phi_iG_j q i = − 2 ϕ i G j
d d β ( p i − q i β ) 2 ∣ β = β ∗ = 4 G j . \left.\frac{d}{d\beta}(p_i-q_i\beta)^2\right|_{\beta=\beta_*}=4G_j. d β d ( p i − q i β ) 2 β = β ∗ = 4 G j . The strict angular bound opens into the allowed interval precisely when
G j < 0 at a left endpoint , G j > 0 at a right endpoint , 1 ≤ j < s . (133) \begin{aligned}
G_j&<0 &&\text{at a left endpoint},\\
G_j&>0 &&\text{at a right endpoint},\qquad1\le j<s. \tag*{(133)}
\end{aligned} G j G j < 0 > 0 at a left endpoint , at a right endpoint , 1 ≤ j < s . (133) These are necessary first-order angular conditions. They do not impose cone balance inside the opened parent.
Exact reconstruction of the AF catalogue
Table 4 gives a small component library. The letters N, B, S denote Taub–NUT, Taub–bolt and Schwarzschild components, respectively [15 , 25 ] . The signs on N and B specify the sign of g g g C C C C 4 C_4 C 4 D D D g g g k k k N − k N-k N − k
Table 4. Nine marked components sufficient to generate the retained AF words through twelve turning points. The library generates words; it is not the list of all component types appearing at both ends.
Proposition 8.1 (Finite endpoint reconstruction). *Enumerate ordered assemblies from Table 4 with total turning-point number 3 ≤ N ≤ 12 3\le N\le12 3 ≤ N ≤ 12 (132) , G s = 0 G_s=0 G s = 0 (133) , and an endpoint twist in [ − 1 , 0 ] [-1,0] [ − 1 , 0 ]
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its reversal and orientation conventions. The resulting set is exactly the 356 retained AF words in Table 5 ; none of the three excluded words is generated.
Table 5. Surviving AF word classes and certified midpoint balance roots. The certificates comprise 208 genuine even-sector roots and 148 roots in reduced phantom sectors.
Finite verification. The component words and boundary parameters are integral or rational, so every operation is exact. Exhaustive generation gives, for N = 3 , … , 12 N=3,\ldots,12 N = 3 , … , 12
1 , 1 , 1 , 3 , 3 , 9 , 15 , 40 , 78 , 205. 1,\ 1,\ 1,\ 3,\ 3,\ 9,\ 15,\ 40,\ 78,\ 205. 1 , 1 , 1 , 3 , 3 , 9 , 15 , 40 , 78 , 205. Comparison of canonical words gives equality of sets, not only equality of counts. Independently, concatenating the components in all 712 recorded endpoint assemblies reconstructs their parent words. The joining formula is checked at all 3,607 joins, and all partial twist sums have the required signs. The generator and exact checks are supplied with the catalogue. □
The five classical marked components alone generate 355 of the 356 words. The remaining word, 12-135, is
( − 1 , 1 , − 1 , − 1 , − 1 , 1 , − 1 , − 1 , − 1 , 1 , − 1 ) . (-1,1,-1,-1,-1,1,-1,-1,-1,1,-1). ( − 1 , 1 , − 1 , − 1 , − 1 , 1 , − 1 , − 1 , − 1 , 1 , − 1 ) . At its left endpoint β ∗ = − 2 / 3 \beta_*=-2/3 β ∗ = − 2/3 ( D , C ) (D,C) ( D , C ) − 2 / 3 − 1 / 3 = − 1 -2/3-1/3=-1 − 2/3 − 1/3 = − 1 − 3 / 2 -3/2 − 3/2 β ∗ = − 1 / 3 \beta_*=-1/3 β ∗ = − 1/3 ( C , D ) (C,D) ( C , D )
Of the 356 words, 197 admit classical assemblies at both endpoints, 158 at one endpoint, and one at neither. Generation from one classical endpoint does not make the other endpoint classical: the endpoint atlas also contains five- and six-nut ALF components and marked Kerr components. A fully marked assembly determines the parent word, hence its necessary interval I W I_W I W ∣ p i − q i β ∣ = 1 |p_i-q_i\beta|=1 ∣ p i − q i β ∣ = 1
At fourteen turning points the earlier component library generated 1,141 words. The 18 additional numerically convergent words require fourteen further component types, three with nine nuts and eleven with ten. After these components are extracted, the same rules generate exactly the 1,159 convergent words and none of the 66 unconverged candidates. This extended agreement is a reconstruction using components identified from the data, not an independent existence test or a proof excluding the 66 failures. No fixed finite component library is asserted to suffice at arbitrary order.
The analytic opening problem. The next step is to turn compatible boundary data into a regular interior metric. Even the leading separation law retains information absent from the polygon. Put m α = M C α / L m_\alpha= M C_\alpha/L m α = M C α / L g α = g C α g_\alpha= g C_\alpha g α = g C α
K α δ = 8 m α m δ − 2 g α g δ ≥ 0. K_{\alpha\delta} = 8m_\alpha m_\delta- 2g_\alpha g_\delta\ge0. K α δ = 8 m α m δ − 2 g α g δ ≥ 0. The inequality follows from ∣ γ C ∣ ≤ 2 M C |\gamma_C| \le2M_C ∣ γ C ∣ ≤ 2 M C a α / L = X α / ε + O ( 1 ) a_\alpha/L = X_\alpha/\sqrt{\varepsilon} + O(1) a α / L = X α / ε + O ( 1 ) β = β ∗ + σ ε \beta= \beta_* + \sigma\varepsilon β = β ∗ + σ ε σ = 1 \sigma= 1 σ = 1 σ = − 1 \sigma= -1 σ = − 1
∑ α ≤ j < δ K α δ ( X δ − X α ) 2 = − 4 σ G j , 1 ≤ j < s . (134) \sum_{\alpha\le j<\delta} \frac{K_{\alpha\delta}}{(X_\delta-X_\alpha)^2} = -4\sigma G_j,\qquad1\le j<s. \tag*{(134)} α ≤ j < δ ∑ ( X δ − X α ) 2 K α δ = − 4 σ G j , 1 ≤ j < s . (134) Indeed, T 2 = I T^2=I T 2 = I T [ j ] 2 + T [ j ] , > j T [ j ] , > j T = I T^2_{[j]}+T_{[j],>j}T^T_{[j],>j}=I T [ j ] 2 + T [ j ] , > j T [ j ] , > j T = I (98) gives the cross-component entries. Their squared leading norm is L 2 K α δ / ( a δ − a α ) 2 L^2K_{\alpha\delta}/(a_\delta-a_\alpha)^2 L 2 K α δ / ( a δ − a α ) 2 4 G j 4G_j 4 G j (134) .
For gaps y j = X j + 1 − X j > 0 y_j=X_{j+1}-X_j>0 y j = X j + 1 − X j > 0
V ( y ) = ∑ α < δ K α δ y α + ⋯ + y δ − 1 − 4 σ ∑ j = 1 s − 1 G j y j . V(y)=\sum_{\alpha<\delta}\frac{K_{\alpha\delta}}{y_\alpha+\cdots+y_{\delta-1}}-4\sigma\sum_{j=1}^{s-1}G_jy_j. V ( y ) = α < δ ∑ y α + ⋯ + y δ − 1 K α δ − 4 σ j = 1 ∑ s − 1 G j y j . If every adjacent K j , j + 1 > 0 K_{j,j+1}>0 K j , j + 1 > 0 σ G j < 0 \sigma G_j<0 σ G j < 0
A gluing theorem must also solve the internal component equations, retain the common period normalization, and bound the correction to (134) . The collision hypotheses of § 6.5 concern precisely the comparison needed to continue such a family globally. The rational recursions of [12 ] and the positive moment representation of [4 ] provide explicit coordinates for this problem, but neither a rational patching matrix nor compatibility of its endpoint factors enforces cone balance by itself.
Acknowledgements
I thank Lars Andersson for pointing out a gap in an earlier version of the triangular-transport argument at rods with zero angular component.
The Belinski–Zakharov factorization presented here was obtained by the author in 2022. ChatGPT (OpenAI) was subsequently used to explore its consequences, assist with Mathematica and Python code development, and support the writing and overall presentation of the manuscript. Computations for the catalogue were performed using Mathematica locally by the author and Python within ChatGPT’s code-execution environment.
Appendix A. AF catalogue through twelve turning points
This appendix specifies the finite algebraic system and records the 356 surviving AF word classes for 3 ≤ N ≤ 12 3\le N\le12 3 ≤ N ≤ 12 [18 ] . Every listed class satisfies the proper-block collision criterion and has an interval-certified balance root at the midpoint of its strict twist interval. Numerical continuation supplies mass, angular momentum and normalized lengths; exact rod arithmetic determines the endpoint partitions. The three words excluded by the boundary-gradient argument are treated separately in Appendix A.7 and do not occur in the catalogue or its endpoint atlas.
The main distinctions are the 197 full-window words with I W = ( − 1 , 0 ) \mathcal{I}_W=(-1,0) I W = ( − 1 , 0 )
The certificates, continuation samples and endpoint partitions have distinct roles. A root certificate proves the balance equations at one parameter. Full-interval existence uses Theorem 6.6 and Hypotheses 6.5, rather than interpolation between samples. Endpoint partitions specify marked candidate components; convergence to their metrics requires a separate degeneration analysis.
Enumeration conventions
A.1. Enumeration conventions. Here N N N n n n h = n − N h=n-N h = n − N { k , N − k } \{k,N-k\} { k , N − k }
k = N − h 2 , n = N + h = 2 ( N − k ) , 1 ≤ k ≤ N / 2. k=\frac{N-h}{2}, \qquad n=N+h=2(N-k), \qquad1\leq k\leq N/2. k = 2 N − h , n = N + h = 2 ( N − k ) , 1 ≤ k ≤ N /2. The familiar two-nut Kerr family is a separate base case. The tables start with the three-nut Chen–Teo word, represented by [4 − 1 − 0], and include odd as well as even N N N
Write A = ( 0 , 1 ) A=(0,1) A = ( 0 , 1 ) B = ( L , β ) B=(L,\beta) B = ( L , β ) L > 0 L>0 L > 0
v j = p j A − q j B = ( − q j L , p j − q j β ) . v_j=p_jA-q_jB=(-q_jL,p_j-q_j\beta). v j = p j A − q j B = ( − q j L , p j − q j β ) . For a self-intersection word d = ( d 1 , … , d N − 1 ) d=(d_1,\ldots,d_{N-1}) d = ( d 1 , … , d N − 1 )
( p 0 , q 0 ) = ( 1 , 0 ) , ( p 1 , q 1 ) = ( 0 , 1 ) , ( p j + 1 , q j + 1 ) = − ( p j − 1 , q j − 1 ) − d j ( p j , q j ) . (135) (p_0,q_0)=(1,0), \qquad(p_1,q_1)=(0,1), \qquad(p_{j+1},q_{j+1})=-(p_{j-1},q_{j-1})-d_j(p_j,q_j). \tag*{(135)} ( p 0 , q 0 ) = ( 1 , 0 ) , ( p 1 , q 1 ) = ( 0 , 1 ) , ( p j + 1 , q j + 1 ) = − ( p j − 1 , q j − 1 ) − d j ( p j , q j ) . (135) Enumeration fixes d 1 = − 1 d_1=-1 d 1 = − 1
d j ∈ { − 1 , 0 , 1 } , ( p N , q N ) = ( σ , 0 ) , σ ∈ { − 1 , 1 } , d_j\in\{-1,0,1\}, \qquad(p_N,q_N)=(\sigma,0), \qquad\sigma\in\{-1,1\}, d j ∈ { − 1 , 0 , 1 } , ( p N , q N ) = ( σ , 0 ) , σ ∈ { − 1 , 1 } , q j ≠ 0 ( 1 ≤ j ≤ N − 1 ) , ∣ p a q b − q a p b ∣ < b − a ( b − a ≥ 2 ) , q_j\ne0 \quad(1\leq j\leq N-1), \qquad\lvert p_aq_b-q_ap_b\rvert<b-a \quad(b-a\geq2), q j = 0 ( 1 ≤ j ≤ N − 1 ) , ∣ p a q b − q a p b ∣ < b − a ( b − a ≥ 2 ) , I W = { β ∈ ( − 1 , 0 ) : ∣ p j − q j β ∣ < 1 for every finite rod } ≠ ∅ . I_W=\{\beta\in(-1,0):\lvert p_j-q_j\beta\rvert<1\text{ for every finite rod}\}\ne\varnothing. I W = { β ∈ ( − 1 , 0 ) : ∣ p j − q j β ∣ < 1 for every finite rod } = ∅ . All interval endpoints and integer tests are computed exactly. The projective winding w w w h = ∣ N − 2 w ∣ h=\lvert N-2w\rvert h = ∣ N − 2 w ∣ k = min ( w , N − w ) k=\min(w,N-w) k = min ( w , N − w ) h ≡ N ( m o d 2 ) h\equiv N\pmod{2} h ≡ N ( mod 2 ) 3 h < N 3h<N 3 h < N N > 3 N>3 N > 3 h = 1 h=1 h = 1
The final enumeration step is a quotient, not a symmetry constraint on the unknown metric. For d N − 1 = − 1 d_{N-1}=-1 d N − 1 = − 1 d d d β \beta β − 1 − β -1-\beta − 1 − β d N − 1 = 1 d_{N-1}=1 d N − 1 = 1 d d d β \beta β
The full system solved
A.2. The full system solved. We use the outer-node normalization z 1 = − 1 z_1=-1 z 1 = − 1 z N = 1 z_N=1 z N = 1 L L L N N N u i u_i u i N − 2 N-2 N − 2
D ( x ) = ∏ i = 1 N ( x − z i ) , ω i = u i ∣ D ′ ( z i ) ∣ > 0 , μ a [ j ] = ∑ i = 1 N c i [ j ] ω i z i a , c i [ j ] = { − 1 i ≤ j , 1 i > j . D(x)=\prod_{i=1}^{N}(x-z_i), \qquad\omega_i=\frac{u_i}{\lvert D'(z_i)\rvert}>0, \qquad\mu_a^{[j]}=\sum_{i=1}^{N}c_i^{[j]}\omega_i z_i^a, \qquad c_i^{[j]}=\begin{cases}-1 & i\leq j,\\1 & i>j.\end{cases} D ( x ) = i = 1 ∏ N ( x − z i ) , ω i = ∣ D ′ ( z i )∣ u i > 0 , μ a [ j ] = i = 1 ∑ N c i [ j ] ω i z i a , c i [ j ] = { − 1 1 i ≤ j , i > j . For j = 0 , … , N j=0,\ldots,N j = 0 , … , N
H j = ( μ α + b [ j ] ) b = 0 k − 1 , c j = ( μ 0 [ j ] , … , μ 2 k − 1 [ j ] ) T , e = ( 0 , … , 0 , 1 ) T H_j=(\mu_{\alpha+b}^{[j]})_{b=0}^{k-1}, \qquad c_j=(\mu_0^{[j]},\ldots,\mu_{2k-1}^{[j]})^T, \qquad e=(0,\ldots,0,1)^T H j = ( μ α + b [ j ] ) b = 0 k − 1 , c j = ( μ 0 [ j ] , … , μ 2 k − 1 [ j ] ) T , e = ( 0 , … , 0 , 1 ) T and the two-component homogeneous rod
t j = ( det ( H j c j + e ( c j + e ) T μ 2 k [ j ] + ∑ i z i ) , det H j ) . (136) \mathbf{t}_j=\left(\det\begin{pmatrix}H_j & c_j+e\\(c_j+e)^T & \mu_{2k}^{[j]}+\sum_i z_i\end{pmatrix},\det H_j\right). \tag*{(136)} t j = ( det ( H j ( c j + e ) T c j + e μ 2 k [ j ] + ∑ i z i ) , det H j ) . (136) These are the reduced moment rods. They incorporate the cone normalization, so equality of the complete vectors, rather than only their directions, is essential. If w ≠ k w \ne k w = k d s = − d d^{s}=-d d s = − d β s = − β \beta_{s}=-\beta β s = − β d s = d d^{s}=d d s = d β s = β \beta_{s}=\beta β s = β ε = ± 1 \varepsilon=\pm1 ε = ± 1 ( p j s , q j s ) (p_{j}^{s},q_{j}^{s}) ( p j s , q j s ) d s d^{s} d s 2 N − 2 2N-2 2 N − 2
t j = p j s t 0 + q j s t 1 , j = 2 , … , N − 1 , t N t = σ s t 0 t , σ s = ( − 1 ) k , t 1 ϕ + β s t 0 ϕ = 0. (137) \begin{aligned}
t_j &= p_j^s t_0+q_j^s t_1,\qquad j=2,\ldots,N-1,\\
t_N^t &= \sigma_s t_0^t,\qquad\sigma_s=(-1)^k,\\
t_1^\phi+\beta_s t_0^\phi&= 0. \tag*{(137)}
\end{aligned} t j t N t t 1 ϕ + β s t 0 ϕ = p j s t 0 + q j s t 1 , j = 2 , … , N − 1 , = σ s t 0 t , σ s = ( − 1 ) k , = 0. (137) The other exterior component is automatic: t N ϕ = ( − 1 ) k t 0 ϕ t_N^\phi=(-1)^k t_0^\phi t N ϕ = ( − 1 ) k t 0 ϕ
L = det ( t 0 , t 1 ) ( t 0 ϕ ) 2 > 0 , x j = z j + 1 − z j L , M ‾ = M L , J ‾ = J L 2 . (138) L=\frac{\det(\mathbf{t}_0,\mathbf{t}_1)}{(t_0^\phi)^2}>0,\qquad x_j=\frac{z_{j+1}-z_j}{L},\qquad\overline{M}=\frac{M}{L},\qquad\overline{J}=\frac{J}{L^2}. \tag*{(138)} L = ( t 0 ϕ ) 2 det ( t 0 , t 1 ) > 0 , x j = L z j + 1 − z j , M = L M , J = L 2 J . (138) The resulting rods are exactly ( − q j s L , p j s − q j s β s ) (-q_j^sL,p_j^s-q_j^s\beta_s) ( − q j s L , p j s − q j s β s ) ε \varepsilon ε
The numerical search uses log u i \log u_i log u i N − 2 N-2 N − 2 2 N − 2 2N-2 2 N − 2 I W I_W I W
Reinserting phantoms. A root of the reduced system has an explicit even-slot representative. Choose h h h a s = 3 + s a_s=3+s a s = 3 + s s = 0 , … , h − 1 s=0,\ldots,h-1 s = 0 , … , h − 1
y i = u i ∏ s ( a s − z i ) , P i = 1 − ( − 1 ) i − 1 y i , Q i = 1 + ( − 1 ) i − 1 y i . y_i=\frac{u_i}{\prod_s(a_s-z_i)},\qquad P_i=1-(-1)^{i-1}y_i,\qquad Q_i=1+(-1)^{i-1}y_i. y i = ∏ s ( a s − z i ) u i , P i = 1 − ( − 1 ) i − 1 y i , Q i = 1 + ( − 1 ) i − 1 y i . The genuine affine BZ charge is C i = P i / Q i C_i=P_i/Q_i C i = P i / Q i Q i ≠ 0 Q_i\ne0 Q i = 0 ( P , Q ) = ( − 1 , 1 ) (P,Q)=(-1,1) ( P , Q ) = ( − 1 , 1 ) C = − 1 C=-1 C = − 1 (136) . Across each phantom the homogeneous rod changes sign, while its vanishing circle is unchanged. The independent checks use this sign and compare all genuine full rods and the recovered scale L L L n n n
What a balance certificate proves
What a balance certificate proves. For each accepted midpoint, Newton refinement at 90 decimal digits produces a centre in the variables
( u 1 , … , u N , z 2 , … , z N − 1 ) . (u_1,\ldots,u_N,z_2,\ldots,z_{N-1}). ( u 1 , … , u N , z 2 , … , z N − 1 ) . Outward-rounded ball arithmetic then treats a box of radius 10 − 32 10^{-32} 1 0 − 32 F F F (137) , a fixed decimal matrix B B B x 0 x_0 x 0
sup x ∈ X ∥ I − B D F ( x ) ∥ ∞ ≤ q < 1 , ∥ B F ( x 0 ) ∥ ∞ + q 10 − 32 < 10 − 32 . \sup_{x\in X}\lVert I-B\,DF(x)\rVert_\infty\le q<1,\qquad\lVert BF(x_0)\rVert_\infty+q\,10^{-32}<10^{-32}. x ∈ X sup ∥ I − B D F ( x ) ∥ ∞ ≤ q < 1 , ∥ B F ( x 0 ) ∥ ∞ + q 1 0 − 32 < 1 0 − 32 . It also verifies positive weights, strictly ordered nodes and every denominator used in the evaluation. The contraction map x ↦ x − B F ( x ) x\mapsto x-BF(x) x ↦ x − B F ( x )
All 356 midpoint roots passed. The largest contraction bound is below 4.49 × 10 − 17 4.49\times10^{-17} 4.49 × 1 0 − 17 3.40 × 10 − 74 3.40\times10^{-74} 3.40 × 1 0 − 74
The certificates establish exact balanced data in the positive reduced representation. Bulk positivity follows from the matrix reconstruction, and the primitive rods and unit adjacent lattice determinants are exact properties of the prescribed word. Corollary 6.4 therefore gives smooth complete metrics at the certified parameters; this conclusion does not require the homogeneous collision hypotheses. The finite contraction calculation supplies the exact balanced data, while the analytic reconstruction supplies completion. Independent whole-axis interval audits are available for the five types 6-002, 8-002, 8-006, 8-007 and 8-009. They are additional checks, not an audit of every row. Global uniqueness in the positive reduced sector follows from Theorem 6.2.
Explicit low-order solutions
Explicit low-order solutions. The finite moment system gives exact solutions without requiring a radical parametrization: a polynomial system, a specified real branch and a reconstruction map are sufficient. The following examples make this description concrete. Here N N N
κ = z N − z 1 2 \kappa=\frac{z_N-z_1}{2} κ = 2 z N − z 1 for half the outer node span. This coordinate scale differs from the asymptotic period scale L L L M / κ M/\kappa M / κ M / L M/L M / L
Four turning points: the Teo family. For d = ( − 1 , 0 , 1 ) d=(-1,0,1) d = ( − 1 , 0 , 1 )
z = κ ( − 1 , − r , r , 1 ) , C = ( a , b , − b − 1 , − a − 1 ) , 0 < r < 1. (139) z=\kappa(-1,-r,r,1),\qquad C=(a,b,-b^{-1},-a^{-1}),\qquad0<r<1. \tag*{(139)} z = κ ( − 1 , − r , r , 1 ) , C = ( a , b , − b − 1 , − a − 1 ) , 0 < r < 1. (139) The reciprocal charges satisfy AF alignment. The chamber containing the regular branch has 0 < a < 1 < b 0<a<1<b 0 < a < 1 < b
φ ( a , b ) = ( a − 1 ) 2 ( 1 + b 2 ) + 4 a b , U = φ ( a , b ) , V = φ ( b , a ) , \varphi(a,b)=(a-1)^2(1+b^2)+4ab,\qquad U=\varphi(a,b),\qquad V=\varphi(b,a), φ ( a , b ) = ( a − 1 ) 2 ( 1 + b 2 ) + 4 ab , U = φ ( a , b ) , V = φ ( b , a ) , and its two derivatives
N c = 1 2 ∂ a φ = ( a − 1 ) ( 1 + b 2 ) + 2 b , D c = 1 2 ∂ b φ = b ( a − 1 ) 2 + 2 a . N_c=\frac{1}{2}\partial_a\varphi=(a-1)(1+b^2)+2b,\qquad D_c=\frac{1}{2}\partial_b\varphi=b(a-1)^2+2a. N c = 2 1 ∂ a φ = ( a − 1 ) ( 1 + b 2 ) + 2 b , D c = 2 1 ∂ b φ = b ( a − 1 ) 2 + 2 a . The normalized rod equations reduce to
( 1 + r 2 ) U − 2 r V = 0 , N c − r D c = 0. (140) (1+r^2)U-2rV=0,\qquad N_c-rD_c=0. \tag*{(140)} ( 1 + r 2 ) U − 2 r V = 0 , N c − r D c = 0. (140) Since D c > 0 D_c>0 D c > 0 r r r
U ( D c 2 + N c 2 ) − 2 N c D c V = 0 , r = N c / D c . (141) U(D_c^2+N_c^2)-2N_cD_cV=0,\qquad r=N_c/D_c. \tag*{(141)} U ( D c 2 + N c 2 ) − 2 N c D c V = 0 , r = N c / D c . (141) This is a compact charge-curve description of the four-nut AF solution of [26 ] . The inequalities 0 < r < 1 0<r<1 0 < r < 1
A useful alternative chart is
a = 1 − u 1 + u , b = 1 + v 1 − v , r = 1 − t 1 + t , 0 < u < v < 1 , 0 < t < 1. a=\frac{1-u}{1+u},\qquad b=\frac{1+v}{1-v},\qquad r=\frac{1-t}{1+t},\qquad0<u<v<1,\qquad0<t<1. a = 1 + u 1 − u , b = 1 − v 1 + v , r = 1 + t 1 − t , 0 < u < v < 1 , 0 < t < 1. Equation (140) becomes the polynomial pair
u 2 − v 2 + t 2 + 3 u 2 v 2 t 2 = 0 , u 2 { 1 + v ( 1 − t ) + 3 v 2 t } + 2 u v 2 ( 1 + t ) + ( v − 1 ) ( v + t ) = 0. (142) \begin{aligned}
u^2-v^2+t^2+3u^2v^2t^2&=0,\\
u^2\{1+v(1-t)+3v^2t\}+2uv^2(1+t)+(v-1)(v+t)&=0. \tag*{(142)}
\end{aligned} u 2 − v 2 + t 2 + 3 u 2 v 2 t 2 u 2 { 1 + v ( 1 − t ) + 3 v 2 t } + 2 u v 2 ( 1 + t ) + ( v − 1 ) ( v + t ) = 0 , = 0. (142) For example, setting v = m v=m v = m u = m γ u=m^\gamma u = m γ
t = m 1 − γ 2 1 + 3 m 4 γ 2 , 0 < γ < 1 , t=\frac{m\sqrt{1-\gamma^2}}{\sqrt{1+3m^{4}\gamma^2}},\qquad0<\gamma<1, t = 1 + 3 m 4 γ 2 m 1 − γ 2 , 0 < γ < 1 , and leaves one equation A + t B = 0 A+tB=0 A + tB = 0
A = m 2 γ ( 1 + m ) + 2 m 3 γ + m ( m − 1 ) , A=m^{2\gamma}(1+m)+2m^{3\gamma}+m(m-1), A = m 2 γ ( 1 + m ) + 2 m 3 γ + m ( m − 1 ) , B = m 3 γ ( 3 m − 1 ) + 2 m 3 γ + m − 1. B=m^{3\gamma}(3m-1)+2m^{3\gamma}+m-1. B = m 3 γ ( 3 m − 1 ) + 2 m 3 γ + m − 1. The eliminated equation A 2 ( 1 + 3 m 4 γ 2 ) − m 2 ( 1 − γ 2 ) B 2 = 0 A^2(1+3m^{4\gamma^2})-m^2(1-\gamma^2)B^2=0 A 2 ( 1 + 3 m 4 γ 2 ) − m 2 ( 1 − γ 2 ) B 2 = 0 A + t B = 0 A+tB=0 A + tB = 0
A numerical zero-spin representative is
( a , b , r ) ≃ ( 0.5228834958296844 , 3.4674923457823892 , 0.3930322274582969 ) . (a,b,r)\simeq(0.5228834958296844,\ 3.4674923457823892,\ 0.3930322274582969). ( a , b , r ) ≃ ( 0.5228834958296844 , 3.4674923457823892 , 0.3930322274582969 ) . The residue formulas give M / κ ≃ 0.6415956930225703 M/\kappa\simeq0.6415956930225703 M / κ ≃ 0.6415956930225703 L / κ ≃ 1.6003207199276681 L/\kappa\simeq1.6003207199276681 L / κ ≃ 1.6003207199276681 M / L ≃ 0.4009169443557347 M/L\simeq0.4009169443557347 M / L ≃ 0.4009169443557347 β ≃ − 0.4448362860057288 \beta\simeq-0.4448362860057288 β ≃ − 0.4448362860057288 Table 6 . The numerical location of this spin zero is separate from the certified midpoint existence result.
Table 6. Numerical representatives with vanishing total angular momentum.
Five turning points: a one-phantom representative. The sole five-nut word in the census is
d = ( − 1 , 0 , 0 , − 1 ) , ( v 0 , … , v 5 ) = ( ( 1 , 0 ) , ( 0 , 1 ) , ( − 1 , 1 ) , ( 0 , − 1 ) , ( 1 , − 1 ) , ( 1 , 0 ) ) . d=(-1,0,0,-1),\qquad(v_0,\ldots,v_5)=((1,0),(0,1),(-1,1),(0,-1),(1,-1),(1,0)). d = ( − 1 , 0 , 0 , − 1 ) , ( v 0 , … , v 5 ) = (( 1 , 0 ) , ( 0 , 1 ) , ( − 1 , 1 ) , ( 0 , − 1 ) , ( 1 , − 1 ) , ( 1 , 0 )) . Its reduced rank is k = 2 k=2 k = 2 β = − 1 / 2 \beta=-1/2 β = − 1/2 κ = 1 \kappa=1 κ = 1 (136) :
z = ( − 1 , − r , 0 , r , 1 ) , ( u 1 , … , u 5 ) = ( a 5 , b 5 , c 5 , b 5 , a 5 ) . z=(-1,-r,0,r,1),\qquad(u_1,\ldots,u_5)=(a_5,b_5,c_5,b_5,a_5). z = ( − 1 , − r , 0 , r , 1 ) , ( u 1 , … , u 5 ) = ( a 5 , b 5 , c 5 , b 5 , a 5 ) . Here a 5 , b 5 , c 5 a_5,b_5,c_5 a 5 , b 5 , c 5 t 5 − j t = − t j t t^t_{5-j}=-t^t_j t 5 − j t = − t j t t 5 − j ϕ = t j ϕ t^\phi_{5-j}=t^\phi_j t 5 − j ϕ = t j ϕ
t 0 t = 0 , t 1 ϕ = 1 2 t 0 ϕ , t 2 t = t 1 t , t 2 ϕ = − 1 2 t 0 ϕ . (143) t^t_0=0,\qquad t^\phi_1=\frac{1}{2}t^\phi_0,\qquad t^t_2=t^t_1,\qquad t^\phi_2=-\frac{1}{2}t^\phi_0. \tag*{(143)} t 0 t = 0 , t 1 ϕ = 2 1 t 0 ϕ , t 2 t = t 1 t , t 2 ϕ = − 2 1 t 0 ϕ . (143) Each t j t_j t j 2 × 2 2\times2 2 × 2 3 × 3 3\times3 3 × 3 (136) . The certificate for 5-001 selects the positive solution near
r = 0.5221304683685786 , a 5 = 0.4994219409569822 , r=0.5221304683685786,\qquad a_5=0.4994219409569822, r = 0.5221304683685786 , a 5 = 0.4994219409569822 , b 5 = 0.7106895457957642 , c 5 = 0.9800704111573928. b_5=0.7106895457957642,\qquad c_5=0.9800704111573928. b 5 = 0.7106895457957642 , c 5 = 0.9800704111573928. Its dimensionless invariants are M / L ≃ 0.4533846765481244 M/L\simeq0.4533846765481244 M / L ≃ 0.4533846765481244 J = 0 J=0 J = 0 a a a c c c (103) , and u i v i = a i 2 − c i 2 u_iv_i=a_i^2-c_i^2 u i v i = a i 2 − c i 2
To recover ordinary BZ data, place the phantom at z 6 = 3 z_6=3 z 6 = 3 y i = u i / ( 3 − z i ) y_i=u_i/(3-z_i) y i = u i / ( 3 − z i )
C i = 1 − ( − 1 ) i − 1 y i 1 + ( − 1 ) i − 1 y i , 1 ≤ i ≤ 5 , C 6 = − 1. C_i=\frac{1-(-1)^{i-1}y_i}{1+(-1)^{i-1}y_i},\qquad1\leq i\leq5,\qquad C_6=-1. C i = 1 + ( − 1 ) i − 1 y i 1 − ( − 1 ) i − 1 y i , 1 ≤ i ≤ 5 , C 6 = − 1. Thus the reduced determinant solution gives explicit six-slot data as well as the five genuine rods. The auxiliary sixth position changes the representation, not the geometry.
Six turning points beyond the Li–Sun word. The six-nut census contains three words,
( − 1 , 0 , 0 , 0 , 1 ) , ( − 1 , 1 , 0 , − 1 , 1 ) , ( − 1 , 1 , 1 , 1 , − 1 ) . (-1,0,0,0,1),\qquad(-1,1,0,-1,1),\qquad(-1,1,1,1,-1). ( − 1 , 0 , 0 , 0 , 1 ) , ( − 1 , 1 , 0 , − 1 , 1 ) , ( − 1 , 1 , 1 , 1 , − 1 ) . We give a charge chart for the second, 6-002. With nodes in ascending order, take
z = κ ( − 1 , − r , − s , s , r , 1 ) , C = ( a , b , c , − c − 1 , − b − 1 , − a − 1 ) , 0 < s < r < 1. (144) z=\kappa(-1,-r,-s,s,r,1),\qquad C=(a,b,c,-c^{-1},-b^{-1},-a^{-1}),\qquad0<s<r<1. \tag*{(144)} z = κ ( − 1 , − r , − s , s , r , 1 ) , C = ( a , b , c , − c − 1 , − b − 1 , − a − 1 ) , 0 < s < r < 1. (144) The relevant positive chamber has 0 < a < 1 0<a<1 0 < a < 1 b > 1 b>1 b > 1 − 1 < c < 1 -1<c<1 − 1 < c < 1 c ≠ 0 c\ne0 c = 0
R = 1 + r 1 − r , S = 1 + s 1 − s , 1 < S < R , R=\frac{1+r}{1-r},\qquad S=\frac{1+s}{1-s},\qquad1<S<R, R = 1 − r 1 + r , S = 1 − s 1 + s , 1 < S < R , and retain U = φ ( a , b ) U=\varphi(a,b) U = φ ( a , b ) V = φ ( b , a ) V=\varphi(b,a) V = φ ( b , a )
f 1 = ( − 1 2 ( 1 + c 2 ) U − 1 2 ( 1 + c 2 ) V 1 2 ( 1 − c 2 ) ( a 2 − 1 ) ( b 2 + 1 ) + c { V − 2 a ( b + 1 ) 2 } 1 2 ( 1 − c 2 ) ( a − 1 ) 2 ( b 2 − 1 ) − 2 c ( a 2 b + a b 2 − 2 a b + a + b ) ) , f 2 = ( a { ( 1 + c 2 ) ( b + 1 ) 2 + 4 b c } ( 1 + c 2 ) ( a 2 + b ) ( b + 1 ) + 2 c ( a 2 + b 2 ) 2 a 2 b + c ( 1 + a 2 ) ( b + 1 ) 2 + 2 b c 2 a { ( 1 + c 2 ) ( b 2 + 1 ) + 2 c ( b + 1 ) 2 } ) . \mathbf{f}_1=
\begin{pmatrix}
-\frac{1}{2}(1+c^2)U\\
-\frac{1}{2}(1+c^2)V\\
\frac{1}{2}(1-c^2)(a^2-1)(b^2+1)+c\{V-2a(b+1)^2\}\\
\frac{1}{2}(1-c^2)(a-1)^2(b^2-1)-2c(a^2b+ab^2-2ab+a+b)
\end{pmatrix},
\qquad
\mathbf{f}_2=
\begin{pmatrix}
a\{(1+c^2)(b+1)^2+4bc\}\\
(1+c^2)(a^2+b)(b+1)+2c(a^2+b^2)\\
2a^2b+c(1+a^2)(b+1)^2+2bc^2\\
a\{(1+c^2)(b^2+1)+2c(b+1)^2\}
\end{pmatrix}. f 1 = − 2 1 ( 1 + c 2 ) U − 2 1 ( 1 + c 2 ) V 2 1 ( 1 − c 2 ) ( a 2 − 1 ) ( b 2 + 1 ) + c { V − 2 a ( b + 1 ) 2 } 2 1 ( 1 − c 2 ) ( a − 1 ) 2 ( b 2 − 1 ) − 2 c ( a 2 b + a b 2 − 2 ab + a + b ) , f 2 = a {( 1 + c 2 ) ( b + 1 ) 2 + 4 b c } ( 1 + c 2 ) ( a 2 + b ) ( b + 1 ) + 2 c ( a 2 + b 2 ) 2 a 2 b + c ( 1 + a 2 ) ( b + 1 ) 2 + 2 b c 2 a {( 1 + c 2 ) ( b 2 + 1 ) + 2 c ( b + 1 ) 2 } . The spectral vector w = ( W 0 , W 1 , W 2 , W 3 ) T \mathbf{w}=(W_0,W_1,W_2,W_3)^{\mathsf{T}} w = ( W 0 , W 1 , W 2 , W 3 ) T
W 0 = ( R − S ) 2 ( 1 + R 2 S 2 ) + ( R S − 1 ) 2 ( R 2 + S 2 ) , W_0=(R-S)^2(1+R^2S^2)+(RS-1)^2(R^2+S^2), W 0 = ( R − S ) 2 ( 1 + R 2 S 2 ) + ( R S − 1 ) 2 ( R 2 + S 2 ) , W 1 = − 2 S ( R 2 − 1 ) ( R − S ) ( R S − 1 ) , W_1=-2S(R^2-1)(R-S)(RS-1), W 1 = − 2 S ( R 2 − 1 ) ( R − S ) ( R S − 1 ) , W 2 = 2 R ( R − S ) ( S 2 − 1 ) ( R S − 1 ) , W_2=2R(R-S)(S^2-1)(RS-1), W 2 = 2 R ( R − S ) ( S 2 − 1 ) ( R S − 1 ) , W 3 = − 2 R S ( R 2 − 1 ) ( S 2 − 1 ) . W_3=-2RS(R^2-1)(S^2-1). W 3 = − 2 R S ( R 2 − 1 ) ( S 2 − 1 ) . For the derivative equations introduce
Ω a = 4 R S ( ( 1 + R 2 ) ( 1 + S 2 ) − 4 R S , 0 , 0 , − ( R 2 − 1 ) ( S 2 − 1 ) ) T , \boldsymbol{\Omega}_a=4RS\left((1+R^2)(1+S^2)-4RS,\ 0,\ 0,\ -(R^2-1)(S^2-1)\right)^{\mathsf{T}}, Ω a = 4 R S ( ( 1 + R 2 ) ( 1 + S 2 ) − 4 R S , 0 , 0 , − ( R 2 − 1 ) ( S 2 − 1 ) ) T , Ω b = − 4 R ( R − 1 ) ( R − S ) ( R S − 1 ) R + 1 ( 1 + S 2 , 0 , − ( S 2 − 1 ) , 0 ) T , \boldsymbol{\Omega}_b=-\frac{4R(R-1)(R-S)(RS-1)}{R+1}(1+S^2,\ 0,\ -(S^2-1),\ 0)^{\mathsf{T}}, Ω b = − R + 1 4 R ( R − 1 ) ( R − S ) ( R S − 1 ) ( 1 + S 2 , 0 , − ( S 2 − 1 ) , 0 ) T , Ω c = 4 S ( S − 1 ) ( R − S ) ( R S − 1 ) S + 1 ( 1 + R 2 , − ( R 2 − 1 ) , 0 , 0 ) T . \boldsymbol{\Omega}_c=\frac{4S(S-1)(R-S)(RS-1)}{S+1}(1+R^2,\ -(R^2-1),\ 0,\ 0)^{\mathsf{T}}. Ω c = S + 1 4 S ( S − 1 ) ( R − S ) ( R S − 1 ) ( 1 + R 2 , − ( R 2 − 1 ) , 0 , 0 ) T . Substitution into the normalized rod equations gives the four conditions
w T f i = 0 , ∑ q ∈ { a , b , c } Ω q T ∂ q f i = 0 , i = 1 , 2. (145) \mathbf{w}^{\mathsf{T}}\mathbf{f}_i=0,\qquad
\sum_{q\in\{a,b,c\}}\boldsymbol{\Omega}_q^{\mathsf{T}}\partial_q\mathbf{f}_i=0,\qquad i=1,2. \tag*{(145)} w T f i = 0 , q ∈ { a , b , c } ∑ Ω q T ∂ q f i = 0 , i = 1 , 2. (145) The derivatives act only on the charge variables. The denominators R + 1 R+1 R + 1 S + 1 S+1 S + 1
The signed primitive rods of the intended branch are
( 1 , 0 ) , ( 0 , 1 ) , ( − 1 , 1 ) , ( 1 , − 2 ) , ( 1 , − 1 ) , ( 0 , 1 ) , ( − 1 , 0 ) . (1,0),\ (0,1),\ (-1,1),\ (1,-2),\ (1,-1),\ (0,1),\ (-1,0). ( 1 , 0 ) , ( 0 , 1 ) , ( − 1 , 1 ) , ( 1 , − 2 ) , ( 1 , − 1 ) , ( 0 , 1 ) , ( − 1 , 0 ) . A numerical zero-spin representative is
a ≃ 0.5220893460504066 , b ≃ 13.0668692937716688 , a \simeq0.5220893460504066,\qquad b \simeq13.0668692937716688, a ≃ 0.5220893460504066 , b ≃ 13.0668692937716688 , c ≃ − 0.3477147817372575 , r ≃ 0.7010419372452687 , c \simeq-0.3477147817372575,\qquad r \simeq0.7010419372452687, c ≃ − 0.3477147817372575 , r ≃ 0.7010419372452687 , s ≃ 0.4049395967930934. s \simeq0.4049395967930934. s ≃ 0.4049395967930934. It has M / κ ≃ 0.6976106882824007 M/\kappa\simeq0.6976106882824007 M / κ ≃ 0.6976106882824007 L / κ ≃ 1.2838780930652332 L/\kappa\simeq1.2838780930652332 L / κ ≃ 1.2838780930652332 M / L ≃ 0.5433620933720189 M/L \simeq0.5433620933720189 M / L ≃ 0.5433620933720189 β ≃ − 0.5033316240992153 \beta\simeq-0.5033316240992153 β ≃ − 0.5033316240992153 r = 0.65 r=0.65 r = 0.65 r = 0.75 r=0.75 r = 0.75 10 − 60 10^{-60} 1 0 − 60 § 6.5 .
The exact endpoint partitions for this word are, from left to right,
β = − 1 ∣ ∅ , ( 1 ) , ( − 1 ) , ∅ \beta=-1\quad\mid\quad\varnothing,\ (1),\ (-1),\ \varnothing β = − 1 ∣ ∅ , ( 1 ) , ( − 1 ) , ∅ β = 0 ∣ ( − 1 ) , ∅ , ∅ , ( 1 ) . \beta=0\quad\mid\quad(-1),\ \varnothing,\ \varnothing,\ (1). β = 0 ∣ ( − 1 ) , ∅ , ∅ , ( 1 ) . Here ∅ \varnothing ∅ § 8 retain this ordering.
Asymptotic invariants along the branches
For every certified word, continuation was corrected at
β = β − + s ( β + − β − ) , s ∈ { 0.01 , 0.05 , 0.20 , 0.35 , 0.50 , 0.65 , 0.80 , 0.95 , 0.99 } . \beta=\beta_-+s(\beta_+-\beta_-),\qquad s\in\{0.01,0.05,0.20,0.35,0.50,0.65,0.80,0.95,0.99\}. β = β − + s ( β + − β − ) , s ∈ { 0.01 , 0.05 , 0.20 , 0.35 , 0.50 , 0.65 , 0.80 , 0.95 , 0.99 } . All 3204 standard samples passed the full-rod test. The maximum residual in the integral rod coordinates was below 4.18 × 10 − 11 4.18\times10^{-11} 4.18 × 1 0 − 11 7.10 × 10 − 5 7.10\times10^{-5} 7.10 × 1 0 − 5
The invariants are evaluated directly from the positive moment data. Put H = H 0 H=H_0 H = H 0 V i a = z i a V_{ia}=z_i^a V ia = z i a a = 0 , … , k − 1 a=0,\ldots,k-1 a = 0 , … , k − 1
W = diag ( ω i ) V , b = V T ( ω i z i k ) i = 1 N , a = W H − 1 e , c = ( ω i z i k ) i = 1 N − W H − 1 b . W=\operatorname{diag}(\sqrt{\omega_i})V,\qquad b=V^{T}(\omega_i z_i^k)_{i=1}^{N},\qquad a=WH^{-1}e,\qquad c=(\sqrt{\omega_i}z_i^k)_{i=1}^{N}-WH^{-1}b. W = diag ( ω i ) V , b = V T ( ω i z i k ) i = 1 N , a = W H − 1 e , c = ( ω i z i k ) i = 1 N − W H − 1 b . With residue columns u r e s = a + c u^{\mathrm{res}}=a+c u res = a + c v r e s = a − c v^{\mathrm{res}}=a-c v res = a − c Z = diag ( z i ) Z=\operatorname{diag}(z_i) Z = diag ( z i )
M = 1 2 ∥ u r e s ∥ 2 , J = − 1 2 ( u r e s ) T Z v r e s . (146) M=\frac{1}{2}\lVert u^{\mathrm{res}}\rVert^2,\qquad J=-\frac{1}{2}(u^{\mathrm{res}})^T Z v^{\mathrm{res}}. \tag*{(146)} M = 2 1 ∥ u res ∥ 2 , J = − 2 1 ( u res ) T Z v res . (146) The superscript distinguishes residue columns from the positive scalar weights u i u_i u i a ⊥ c a\perp c a ⊥ c ∥ a ∥ 2 = ∥ c ∥ 2 \lVert a\rVert^2=\lVert c\rVert^2 ∥ a ∥ 2 = ∥ c ∥ 2
For a stable second evaluation, define for r = k − 1 , k , k + 1 r=k-1,k,k+1 r = k − 1 , k , k + 1
A r = ∑ ∣ I ∣ = r ∏ i ∈ I ω i ∏ i < j , i , j ∈ I ( z j − z i ) 2 , S r = 1 A r ∑ ∣ I ∣ = r ( ∑ i ∈ I z i ) ∏ i ∈ I ω i ∏ i < j , i , j ∈ I ( z j − z i ) 2 . A_r=\sum_{\lvert I\rvert=r}\prod_{i\in I}\omega_i\prod_{i<j,\ i,j\in I}(z_j-z_i)^2,\qquad S_r=\frac{1}{A_r}\sum_{\lvert I\rvert=r}\left(\sum_{i\in I}z_i\right)\prod_{i\in I}\omega_i\prod_{i<j,\ i,j\in I}(z_j-z_i)^2. A r = ∣ I ∣ = r ∑ i ∈ I ∏ ω i i < j , i , j ∈ I ∏ ( z j − z i ) 2 , S r = A r 1 ∣ I ∣ = r ∑ ( i ∈ I ∑ z i ) i ∈ I ∏ ω i i < j , i , j ∈ I ∏ ( z j − z i ) 2 . For the empty subset take A 0 = 1 A_0=1 A 0 = 1 S 0 = 0 S_0=0 S 0 = 0
M = A k − 1 + A k + 1 2 A k , J = A k + 1 ( S k + 1 − S k ) − A k − 1 ( S k − S k − 1 ) 2 A k . M=\frac{A_{k-1}+A_{k+1}}{2A_k},\qquad J=\frac{A_{k+1}(S_{k+1}-S_k)-A_{k-1}(S_k-S_{k-1})}{2A_k}. M = 2 A k A k − 1 + A k + 1 , J = 2 A k A k + 1 ( S k + 1 − S k ) − A k − 1 ( S k − S k − 1 ) . These identities follow from the Gram determinants of consecutive monic orthogonal polynomials. They use positive sums, remain valid when a signed prefix determinant vanishes, and agree with the independent high-precision matrix evaluation at every certified centre: the largest double-precision differences in M ˉ \bar{M} M ˉ J ˉ \bar{J} J ˉ 1.20 × 10 − 14 1.20 \times10^{-14} 1.20 × 1 0 − 14 4.70 × 10 − 14 4.70 \times10^{-14} 4.70 × 1 0 − 14
Figure 2. Balanced AF invariant curves. Left: all three six-nut words. Right: the three-nut one-phantom, eight-nut two-phantom and eleven-nut three-phantom words. Each line joins 51 corrected samples; endpoint values are not extrapolated. The spin panels are clipped to [ − 3 , 3 ] [-3,3] [ − 3 , 3 ]
As checks of scale and marking, the Chen–Teo midpoint gives M ˉ = 6 − 4 2 \bar{M}=6-4\sqrt{2} M ˉ = 6 − 4 2 (138) . Central differences on five representatives, spanning the genuine and phantom sectors, agree with d M ˉ / d β = 2 J ˉ \mathrm{d}\bar{M}/\mathrm{d}\beta=2\bar{J} d M ˉ / d β = 2 J ˉ 3.0 × 10 − 10 3.0\times10^{-10} 3.0 × 1 0 − 10 (171) .
Each of the 356 sampled curves has negative spin at its leftmost standard sample and positive spin at its rightmost sample. A corrected one-dimensional solve locates one zero-spin representative for each word. Table 6 gives all words through N = 8 N=8 N = 8
Two examples illustrate the positive phantom sectors. At β = − 1 / 2 \beta= -1/2 β = − 1/2
8 8 8 d = ( − 1 , 0 , − 1 , − 1 , − 1 , 0 , − 1 ) d = (-1,0,-1,-1,-1,0,-1) d = ( − 1 , 0 , − 1 , − 1 , − 1 , 0 , − 1 ) [ n − h − 0 ] = [ 10 − 2 − 0 ] [n-h-0] = [10-2-0] [ n − h − 0 ] = [ 10 − 2 − 0 ]
has M ≃ 0.609363826117 M \simeq0.609363826117 M ≃ 0.609363826117
( x 1 , … , x 7 ) ≃ ( 0.247301498 , 0.277587993 , 0.253411943 , (x_1,\ldots,x_7) \simeq(0.247301498,0.277587993,0.253411943, ( x 1 , … , x 7 ) ≃ ( 0.247301498 , 0.277587993 , 0.253411943 ,
0.470134767 , 0.253411943 , 0.277587993 , 0.247301498 ) 0.470134767,0.253411943,0.277587993,0.247301498) 0.470134767 , 0.253411943 , 0.277587993 , 0.247301498 )
The word
11 11 11 d = ( − 1 , 0 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , 0 , − 1 ) d = (-1,0,-1,-1,-1,-1,-1,-1,0,-1) d = ( − 1 , 0 , − 1 , − 1 , − 1 , − 1 , − 1 , − 1 , 0 , − 1 ) [ 14 − 3 − 0 ] [14-3-0] [ 14 − 3 − 0 ]
has M ≃ 0.723634122631 M \simeq0.723634122631 M ≃ 0.723634122631
( 0.211393072 , 0.237170999 , 0.218740736 , 0.336329201 , 0.266762863 , (0.211393072,0.237170999,0.218740736,0.336329201,0.266762863, ( 0.211393072 , 0.237170999 , 0.218740736 , 0.336329201 , 0.266762863 ,
0.266762863 , 0.336329201 , 0.218740736 , 0.237170999 , 0.211393072 ) 0.266762863,0.336329201,0.218740736,0.237170999,0.211393072) 0.266762863 , 0.336329201 , 0.218740736 , 0.237170999 , 0.211393072 )
At these parameters the solution has equal paired lengths, although the system treats all gaps as independent variables.
Endpoint components
The finite rod spheres have the tridiagonal intersection matrix
Q W = ( d 1 1 0 1 d 2 ⋱ ⋱ ⋱ 1 1 d N − 1 ) . Q_W =
\begin{pmatrix}
d_1 & 1 & & & 0 \\
1 & d_2 & \ddots& & \\
& \ddots& \ddots& 1 & \\
& & 1 & d_{N-1}
\end{pmatrix}. Q W = d 1 1 1 d 2 ⋱ ⋱ ⋱ 1 1 d N − 1 0 . Exact rational congruence reduction was performed for every candidate. Each matrix has a one-dimensional radical and signature of absolute value h h h
( b + , b − , b 0 ) = ( k − 1 , N − k − 1 , 1 ) . (b_{+},b_{-},b_{0})=(k-1,N-k-1,1). ( b + , b − , b 0 ) = ( k − 1 , N − k − 1 , 1 ) . For the standard simply connected toric AF completion, the rod topology gives χ = N \chi=N χ = N b 2 = N − 1 b_{2}=N-1 b 2 = N − 1
At either rational wall β ∗ \beta_{*} β ∗ I W \mathcal{I}_{W} I W
∣ p j − q j β ∗ ∣ = 1 , \lvert p_{j}-q_{j}\beta_{*}\rvert=1, ∣ p j − q j β ∗ ∣ = 1 , including j = 0 , N j=0,N j = 0 , N a < b a<b a < b
nuts = b − a , d b l o c k = ( d a + 1 , … , d b − 1 ) , p b l o c k = ∣ p a q b − q a p b ∣ . (147) \operatorname{nuts}=b-a,\qquad d^{\mathrm{block}}=(d_{a+1},\ldots,d_{b-1}),\qquad p^{\mathrm{block}}=\lvert p_{a}q_{b}-q_{a}p_{b}\rvert. \tag*{(147)} nuts = b − a , d block = ( d a + 1 , … , d b − 1 ) , p block = ∣ p a q b − q a p b ∣ . (147) This construction uses only the ordered marked rods. To obtain a pointed metric limit, one must additionally establish the divergence of separating lengths, finite nondegenerate internal geometry, and the matching of periods and local twist. The nine-point sweep above does not establish those limiting assertions.
The 33 rod types occurring for surviving words are listed in Table 7 . A number in the endpoint atlas refers to that table; repeated numbers denote successive blocks. Type 1 is the one-nut Taub–NUT type, type 3 the two-nut Taub–bolt type, and type 2 the zero-square two-nut AF type. The latter is Kerr in general and Schwarzschild only when its inherited local twist is zero. Types 4 and 5 have the three- and four-nut AF rod data. Type 6 is the component C 4 C_{4} C 4 C 5 a , C 5 b C_{5}^{a},C_{5}^{b} C 5 a , C 5 b C 6 a , … , C 6 e C_{6}^{a},\ldots,C_{6}^{e} C 6 a , … , C 6 e
Table 7. Rod types occurring in the endpoint partitions, modulo reversal and orientation.
Endpoint partitions of the zero-string family. For W r = ( − 1 , 1 , 0 2 r + 1 , − 1 , 1 ) W_r=(-1,1,0^{2r+1},-1,1) W r = ( − 1 , 1 , 0 2 r + 1 , − 1 , 1 )
β = − 1 : ∅ , ( 1 ) , ( 0 ) , … , ( 0 ) , ( − 1 ) , ∅ , ⏟ r b l o c k s β = 0 : ( − 1 ) , ∅ , … , ∅ , ( 1 ) . ⏟ 2 r + 2 b l o c k s \begin{aligned}
\beta=-1:\quad&\varnothing,\ (1),\ (0),\ldots,(0),\ (-1),\ \varnothing, \qquad\underbrace{\hspace{2.2cm}}_{r\ \mathrm{blocks}}\\
\beta=0:\quad&(-1),\ \varnothing,\ldots,\varnothing,\ (1). \qquad\underbrace{\hspace{2.2cm}}_{2r+2\ \mathrm{blocks}}
\end{aligned} β = − 1 : β = 0 : ∅ , ( 1 ) , ( 0 ) , … , ( 0 ) , ( − 1 ) , ∅ , r blocks ( − 1 ) , ∅ , … , ∅ , ( 1 ) . 2 r + 2 blocks An empty word denotes a one-nut component. Thus adding two zeros introduces one zero-square two-nut block at the left wall and two one-nut blocks at the right wall. These are exact marked rod partitions. The one-nut and ± 1 \pm1 ± 1
Metric evidence at selected endpoints. High-precision tails and local comparisons are available for 4-001, 6-001, 6-002, 8-002, 8-003, 8-006, 8-007 and 8-009. For the latter four eight-nut branches, the tails reach within 1.08 × 10 − 9 1.08\times10^{-9} 1.08 × 1 0 − 9 7.3 × 10 − 39 7.3\times10^{-39} 7.3 × 1 0 − 39
For example, the right wall of 8-006 is − 1 / 3 -1/3 − 1/3 ( 6 , 6 ) (6,6) ( 6 , 6 ) C 4 C_4 C 4
r = 7 − 2 10 3 , u = r 3 + 4 3 r , v = 3 u , a = 1 − u 1 + u , c = 1 − v 1 + v , s 0 = 12 r u ( 1 − r 2 ) ( 1 + u ) 2 , z = s 0 ( − 1 , − r , r , 1 ) , C = ( a , c − 1 , c , a − 1 ) . (148) \begin{aligned}
r&=\frac{7-2\sqrt{10}}{3},\qquad u=\frac{r}{3+4\sqrt{3r}},\qquad v=3u,\qquad a=\frac{1-u}{1+u},\qquad c=\frac{1-v}{1+v},\\
s_0&=\frac{12ru}{(1-r^2)(1+u)^2},\qquad z=s_0(-1,-r,r,1),\qquad C=(a,c^{-1},c,a^{-1}). \tag*{(148)}
\end{aligned} r s 0 = 3 7 − 2 10 , u = 3 + 4 3 r r , v = 3 u , a = 1 + u 1 − u , c = 1 + v 1 − v , = ( 1 − r 2 ) ( 1 + u ) 2 12 r u , z = s 0 ( − 1 , − r , r , 1 ) , C = ( a , c − 1 , c , a − 1 ) . (148) Its three finite lengths in the retained local normalization are
s 0 ( 1 − r ) , 2 s 0 r , s 0 ( 1 − r ) ≃ 0.073603373799 , 0.042773778540 , 0.073603373799. s_0(1-r),\qquad2s_0r,\qquad s_0(1-r)\simeq0.073603373799,\quad0.042773778540,\quad0.073603373799. s 0 ( 1 − r ) , 2 s 0 r , s 0 ( 1 − r ) ≃ 0.073603373799 , 0.042773778540 , 0.073603373799. The parent tail approaches these values while its central separating gap diverges. Independently computed balanced points for component types 9, 11, 12 and 13 have original-rod residuals between 1.43 × 10 − 84 1.43\times10^{-84} 1.43 × 1 0 − 84 1.86 × 10 − 83 1.86\times10^{-83} 1.86 × 1 0 − 83
Effective length enclosures. The finite supporting-plane calculation for 6-002 proves a balanced solution for every ∣ β + 0.5033316240992 ∣ ≤ 9 × 10 − 7 \lvert\beta+ 0.5033316240992\rvert\le9 \times10^{-7} ∣ β + 0.5033316240992 ∣ ≤ 9 × 1 0 − 7 min j x j > 0.2002 \min_j x_j > 0.2002 min j x j > 0.2002 min j x j > 0.03690 \min_j x_j > 0.03690 min j x j > 0.03690 ∣ β + 0.4 ∣ ≤ 9 × 10 − 11 \lvert\beta+ 0.4\rvert\le9 \times10^{-11} ∣ β + 0.4 ∣ ≤ 9 × 1 0 − 11 Theorem 6.6 .
The three excluded twelve-nut words
Let E = ( − 1 , 0 , − 1 , 1 , − 1 ) E=(-1,0,-1,1,-1) E = ( − 1 , 0 , − 1 , 1 , − 1 ) E R E^R E R
Table 1 (PDF p. 46).
Their exclusion uses a certified conical model for E E E
A positive common-cone model. The coefficient rods of E E E
( 1 , 0 ) , ( 0 , 1 ) , ( − 1 , 1 ) , ( 0 , − 1 ) , ( 1 , − 2 ) , ( − 1 , 3 ) , ( − 2 , 5 ) . (1,0),(0,1),(-1,1),(0,-1),(1,-2),(-1,3),(-2,5). ( 1 , 0 ) , ( 0 , 1 ) , ( − 1 , 1 ) , ( 0 , − 1 ) , ( 1 , − 2 ) , ( − 1 , 3 ) , ( − 2 , 5 ) . For six nodes ξ i \xi_i ξ i ζ i \zeta_i ζ i ω i = ζ i / ∏ a ≠ i ∣ ξ i − ξ a ∣ \omega_i=\zeta_i/\prod_{a\ne i}\lvert\xi_i-\xi_a\rvert ω i = ζ i / ∏ a = i ∣ ξ i − ξ a ∣ t j = ( H 3 [ j ] , H 2 [ j ] ) t_j=(H_3^{[j]},H_2^{[j]}) t j = ( H 3 [ j ] , H 2 [ j ] ) (160) . With ξ 1 = − 1 \xi_1=-1 ξ 1 = − 1 ξ 6 = 1 \xi_6=1 ξ 6 = 1 ζ 6 = 1 \zeta_6=1 ζ 6 = 1
5 t j = λ [ ( 5 p j + 2 q j ) t 0 + q j t 6 ] , j = 1 , … , 5. (149) 5t_j=\lambda\left[(5p_j+2q_j)t_0+q_jt_6\right],\qquad j=1,\ldots,5. \tag*{(149)} 5 t j = λ [ ( 5 p j + 2 q j ) t 0 + q j t 6 ] , j = 1 , … , 5. (149) An interval certificate isolates an exact root near
ξ = ( − 1 , − 0.6043803602402567 , 0.2317344053019192 , 0.4639277115162177 , 0.6405233684366971 , 1 ) , \xi=(-1,-0.6043803602402567,0.2317344053019192,
0.4639277115162177,0.6405233684366971,1), ξ = ( − 1 , − 0.6043803602402567 , 0.2317344053019192 , 0.4639277115162177 , 0.6405233684366971 , 1 ) , log ζ = ( 0.6561651153410297 , 1.213438511134147 , 0.1801659598474378 , 1.0700871944828563 , 1.372320385032065 , 0 ) . \log\zeta=(0.6561651153410297,1.213438511134147,
0.1801659598474378,1.0700871944828563,1.372320385032065,0). log ζ = ( 0.6561651153410297 , 1.213438511134147 , 0.1801659598474378 , 1.0700871944828563 , 1.372320385032065 , 0 ) . The decisive bounds are
1.09420875801306906 < λ ∗ < 1.09420875801306907 , 1.09420875801306906<\lambda_*<1.09420875801306907, 1.09420875801306906 < λ ∗ < 1.09420875801306907 , 0.09003150664870304 < c ∗ : = log λ ∗ < 0.09003150664870305. (150) 0.09003150664870304<c_*:=\log\lambda_*<0.09003150664870305. \tag*{(150)} 0.09003150664870304 < c ∗ := log λ ∗ < 0.09003150664870305. (150) The certificate uses a box of radius 10 − 30 10^{-30} 1 0 − 30 2.244 × 10 − 23 2.244\times10^{-23} 2.244 × 1 0 − 23 4.168 × 10 − 75 4.168\times10^{-75} 4.168 × 1 0 − 75 (164) give the required equivalent words, with the same multiplier and local invertibility.
Matching the two clusters. Place the left nodes at − 1 + ε ( ξ i + 1 ) -1+\varepsilon(\xi_i+1) − 1 + ε ( ξ i + 1 ) 1 + ε ( η i − 1 ) 1+\varepsilon(\eta_i-1) 1 + ε ( η i − 1 ) 2 ε 2\varepsilon 2 ε
Z m [ j ] = ∑ ∣ I ∣ = m ( − 1 ) ∣ I ∩ { 1 , … , j } ∣ Δ ( ξ I ) Δ ( ξ I c ) ∏ i ∈ I ζ i . Z_m^{[j]}=\sum_{\lvert I\rvert=m}(-1)^{\lvert I\cap\{1,\ldots,j\}\rvert}\Delta(\xi_I)\Delta(\xi_{I^c})\prod_{i\in I}\zeta_i. Z m [ j ] = ∣ I ∣ = m ∑ ( − 1 ) ∣ I ∩ { 1 , … , j }∣ Δ ( ξ I ) Δ ( ξ I c ) i ∈ I ∏ ζ i . Here, each Vandermonde uses positive differences in the ordered subset. These sums differ from the corresponding Hankel determinants by one common positive factor. On the left write ( Y , X ) = ( Z 3 , Z 2 ) (Y, X) = (Z_3, Z_2) ( Y , X ) = ( Z 3 , Z 2 ) ( Q , P ) = ( Z 3 , Z 2 ) (Q, P) = (Z_3, Z_2) ( Q , P ) = ( Z 3 , Z 2 ) ( Q , P ) = ( Z 4 , Z 3 ) (Q, P) = (Z_4, Z_3) ( Q , P ) = ( Z 4 , Z 3 )
For rank five, give the two clusters occupations ε A ζ i \varepsilon A\zeta_i ε A ζ i ε B ζ ~ i \varepsilon B\tilde{\zeta}_i εB ζ ~ i
t t = A B Y Q − 4 X P − 2 B X Q + 2 A Y P , t ϕ = B X Q + A Y P . (151) t^t = ABYQ - 4XP - 2BXQ + 2AYP,\qquad t^\phi= BXQ + AYP. \tag*{(151)} t t = A B Y Q − 4 X P − 2 B X Q + 2 A Y P , t ϕ = B X Q + A Y P . (151) For rank six, the occupations are ε A ζ i \varepsilon A\zeta_i ε A ζ i ε − 1 B ζ ~ i \varepsilon^{-1}B\tilde{\zeta}_i ε − 1 B ζ ~ i
t t = A B Y Q − X P − B X Q , t ϕ = A Y P + 1 4 B X Q . (152) t^t = ABYQ - XP - BXQ,\qquad t^\phi= AYP + \frac{1}{4}BXQ. \tag*{(152)} t t = A B Y Q − X P − B X Q , t ϕ = A Y P + 4 1 B X Q . (152) These identities follow by retaining the leading occupations in each cluster; the cross-cluster factor of a term with degrees r , s r,s r , s 2 r s + ( 6 − r ) ( 6 − s ) 2^{rs+(6-r)(6-s)} 2 r s + ( 6 − r ) ( 6 − s )
Put α = − 2 − 5 β \alpha= -2 - 5\beta α = − 2 − 5 β α = − 3 − 5 β \alpha= -3 - 5\beta α = − 3 − 5 β s = 1 − α 2 s = \sqrt{1-\alpha^2} s = 1 − α 2 ∣ α ∣ < 1 |\alpha| < 1 ∣ α ∣ < 1 h L = A Y 0 / X 0 h_L = AY_0/X_0 h L = A Y 0 / X 0 h R = B Q 0 / P 0 h_R = BQ_0/P_0 h R = B Q 0 / P 0
( h L , h R ) = { ( 2 ( 1 − α ) / s , 2 ( 1 + α ) / s ) , k = 5 , ( ( 1 − α ) / ( 2 s ) , 2 ( 1 + α ) / s ) , k = 6. (h_L,h_R) =
\begin{cases}
(2(1-\alpha)/s,\ 2(1+\alpha)/s), & k=5,\\
((1-\alpha)/(2s),\ 2(1+\alpha)/s), & k=6.
\end{cases} ( h L , h R ) = { ( 2 ( 1 − α ) / s , 2 ( 1 + α ) / s ) , (( 1 − α ) / ( 2 s ) , 2 ( 1 + α ) / s ) , k = 5 , k = 6. Dividing by t 0 ϕ t^\phi_0 t 0 ϕ
r 0 = ( 0 , 1 ) , r 6 = ( − 2 s ( 1 + s ) , α ) , L = 2 s ( 1 + s ) 5 . r_0 = (0,1),\qquad r_6 = (-2s(1+s),\alpha),\qquad L = \frac{2s(1+s)}{5}. r 0 = ( 0 , 1 ) , r 6 = ( − 2 s ( 1 + s ) , α ) , L = 5 2 s ( 1 + s ) . The surviving rods are balanced in this lattice, while all ten internal cluster rods have multiplier λ ∗ \lambda_* λ ∗ (151) and (152) restrict to fixed linear frame maps of each cluster’s two-component rods, preserving (149) . The contracted nuts have lattice index five; no smoothness of the contracted background is required.
After extracting the common powers of ε \varepsilon ε α / [ s ( 1 + s ) ] \alpha/[s(1+s)] α / [ s ( 1 + s )] ( − 1 , 1 ) (-1,1) ( − 1 , 1 ) ε \varepsilon ε
In normalized length coordinates the resulting path satisfies
q ε ⟶ q 0 = ( 0 , 0 , 0 , 0 , 0 , x 6 , ∗ , 0 , 0 , 0 , 0 , 0 ) , x 6 , ∗ = 5 s ( 1 + s ) , q_\varepsilon\longrightarrow q_0 = (0,0,0,0,0,x_{6,*},0,0,0,0,0),\qquad x_{6,*} = \frac{5}{s(1+s)}, q ε ⟶ q 0 = ( 0 , 0 , 0 , 0 , 0 , x 6 , ∗ , 0 , 0 , 0 , 0 , 0 ) , x 6 , ∗ = s ( 1 + s ) 5 , b ( q ε ) ⟶ c ∗ ( 1 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 1 , 1 , 1 ) . (153) b(q_\varepsilon) \longrightarrow c_*(1,1,1,1,1,0,1,1,1,1,1). \tag*{(153)} b ( q ε ) ⟶ c ∗ ( 1 , 1 , 1 , 1 , 1 , 0 , 1 , 1 , 1 , 1 , 1 ) . (153) Proposition 6.10 excludes balance for every twist in the respective interval. More quantitatively, put S ( x ) = ∑ j ≠ 6 x j S(x) = \sum_{j\ne6} x_j S ( x ) = ∑ j = 6 x j c ∗ > 9 / 100 c_* > 9/100 c ∗ > 9/100
max j ∣ b j ( x ) ∣ ≥ 9 100 S ( x ) S ( x ) + ∣ x 6 − x 6 , ∗ ∣ > 0 ( x > 0 ) . (154) \max_j |b_j(x)| \ge\frac{9}{100}\frac{S(x)}{S(x)+|x_6-x_{6,*}|} > 0\qquad(x>0). \tag*{(154)} j max ∣ b j ( x ) ∣ ≥ 100 9 S ( x ) + ∣ x 6 − x 6 , ∗ ∣ S ( x ) > 0 ( x > 0 ) . (154) This estimate is pointwise on the full positive orthant; it need not remain uniformly positive at its boundary. Neither the index five nor the occurrence of E E E (153) .
Verification of collision-block coverage
Every proper consecutive block of the 356 remaining words is enumerated exactly, with reversal and orientation identified. The resulting counts are Thus all 1,746 nonparallel blocks have negative invariant by Appendix C.§ 5 . Independent interval trials also verify a normalized opening cost at most − log ( 16 / 15 ) -\log(16/15) − log ( 16/15 )
Table 14.
C = ( − 1 , 1 , − 1 ) , z = ( − 1 , − 1 / 5 , 1 / 5 , 1 ) , ζ = ( 1 , 3 , 3 , 1 ) , a 1 = a 2 = a 3 = 15 / 16. C=(-1,1,-1),\quad z=(-1,-1/5,1/5,1),\quad\zeta=(1,3,3,1),\quad a_1=a_2=a_3=15/16. C = ( − 1 , 1 , − 1 ) , z = ( − 1 , − 1/5 , 1/5 , 1 ) , ζ = ( 1 , 3 , 3 , 1 ) , a 1 = a 2 = a 3 = 15/16. The uniform opening-cost bound is a finite-catalogue statement, not a lower bound for the rod lengths. The existence proof needs only the strict signs, which the exact reduction census supplies without interval evaluation of every block.
Organization by zero strings
Compress each nonempty zero run to one zero if its length is odd and to two zeros if its length is even, then quotient by reversal and orientation. The 356 surviving words have exactly 250 distinct compressed seeds, all themselves surviving catalogue members. Their counts by turning-point order are
Theorem 6.8 applies to every such seed. Ten seeds have no zero entries and hence have no nontrivial extension by this operation. The other 240 generate infinite families, with one nonnegative insertion parameter for each zero run. This is an exact organization of the finite catalogue; it is not a bounded-seed theorem for arbitrary order.
A coarser combinatorial compression deletes every even zero run entirely and retains one zero from each odd run. It produces 126 parity cores, again all surviving words in the catalogue. The inverse operation can create a zero run where none existed. Theorem 6.8 only extends existing runs; preservation of all proper-block signs under creation of a new run requires an additional argument. The 126-core organization is therefore recorded as combinatorial data, not as a stronger existence theorem. Exact membership maps and the verification script accompany the catalogue.
A higher-order test
The AF counterexample 16-3583 has a separate 100-digit interval certificate at β = − 2 / 5 \beta=-2/5 β = − 2/5 1.913 ⋅ 10 − 5 1.913\cdot10^{-5} 1.913 ⋅ 1 0 − 5
The electronic catalogue
In the word tables, the letters m m m z z z p p p − 1 -1 − 1 + 1 +1 + 1 entire self-intersection word, not a half-word. The ID starts with the genuine nut count. The integer h h h β = ( β − + β + ) / 2 \beta=(\beta_-+\beta_+)/2 β = ( β − + β + ) /2
The electronic catalogue contains the complete words and rods, exact rational windows and equivalence maps, dressing degrees, certified midpoint data, nine-point continuations, zero-spin sections, asymptotic invariants, and exact component partitions. The solver and certificate scripts are included. The catalogue CSV provides a compact index; JSON files retain the full numerical state and evidence. The supplementary verification also includes the proper-block census and the exact zero-string insertion checks.
Table 8. AF N = 3 N = 3 N = 3
Table 9. AF N = 4 N = 4 N = 4
Table 10. AF N = 5 N = 5 N = 5
Table 11. AF N = 6 N = 6 N = 6
Table 12. AF N = 7 N = 7 N = 7
Table 13. AF N = 8 N = 8 N = 8
Table 15. AF N = 10 N=10 N = 10
Table 16.
Table 16 (PDF p. 51). AF N = 11 N = 11 N = 11
Table 17. AF N = 12 N = 12 N = 12
Endpoint-partition atlas
The left and right walls are those in the corresponding word table. Every entry is an exact rod-data prediction by (147) for a surviving parent word. Component numbers refer to Table 7 ; the atlas does not assign metric existence or convergence to a component number.
Table 18. Exact saturated-rod partitions for N = 3 N = 3 N = 3 Table 7 ; these are candidate limit data.
Table 19. Exact saturated-rod partitions for N = 4 N = 4 N = 4 Table 7 ; these are candidate limit data.
Table 20. Exact saturated-rod partitions for N = 5 N = 5 N = 5 Table 7 ; these are candidate limit data.
Table 21. Exact saturated-rod partitions for N = 6 N = 6 N = 6 Table 7 ; these are candidate limit data.
Table 22. Exact saturated-rod partitions for N = 7 N = 7 N = 7 Table 7 ; these are candidate limit data.
Table 23. Exact saturated-rod partitions for N = 8 N = 8 N = 8 Table 7 ; these are candidate limit data.
Table 24. Exact saturated-rod partitions for N = 9 N = 9 N = 9 Table 7 ; these are candidate limit data.
Table 25. Exact saturated-rod partitions for N = 10 N = 10 N = 10 Table 7 ; these are candidate limit data.
Table 26. Exact saturated-rod partitions for N = 11 N = 11 N = 11 Table 7 ; these are candidate limit data.
Continued on next page
Table 26 (PDF p. 61).
Table 27. Exact saturated-rod partitions for N = 12 N = 12 N = 12 Table 7 ; these are candidate limit data.
Appendix B. ALF catalogue through eight turning points
This appendix records 52 certified isolated regular solutions on 49 admissible rod words through eight turning points. Three eight-nut words support two different asymptotic circle markings. All 49 words have negative homogeneous cluster invariant, while the sixteen remaining candidates have positive invariant; the general implication from this sign to ALF existence remains open. The catalogue tests the polygon organization of § 7 and supplies marked components for the endpoint construction in § 8 .
The nonparallel exterior problem retains the positive moment reconstruction and changes the asymptotic marking. Here ALF is used in the finite-cover sense: the end has a circle direction of bounded length, and exceptional orbits of that circle are permitted. This is also the convention used in [12 , 4 ] ; the census is not restricted to principal circle bundles over a smooth two-sphere.
B.1. The bounded circle at infinity. Keep the lattice marking of § 5 . Let the last coefficient vector be v N = ( p N , q N ) v_N = (p_N, q_N) v N = ( p N , q N ) k k k σ = ( − 1 ) k \sigma= (-1)^k σ = ( − 1 ) k q N ≠ 0 q_N \ne0 q N = 0
p N − q N β = σ , β = p N − σ q N . (155) p_N-q_N\beta=\sigma,\qquad\beta=\frac{p_N-\sigma}{q_N}. \tag*{(155)} p N − q N β = σ , β = q N p N − σ . (155) Thus β \beta β
The primitive bounded circle is represented in coefficient coordinates by
f ∞ = prim ( v N − σ v 0 ) , (156) f_\infty=\operatorname{prim}(v_N-\sigma v_0), \tag*{(156)} f ∞ = prim ( v N − σ v 0 ) , (156) where prim divides an integral vector by the greatest common divisor of its components and its overall sign is immaterial. Its physical angular component vanishes by (155) . The pair consisting of the ordered rod word and this marked fibre distinguishes sectors that the direction polygon alone forgets.
For example, the word
( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 ) , v N = ( 2 , − 5 ) , (-1,0,0,0,-1,1,-1),\qquad v_N=(2,-5), ( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 ) , v N = ( 2 , − 5 ) , has certified balance roots in sectors ( k , h ) = ( 3 , 2 ) (k,h)=(3,2) ( k , h ) = ( 3 , 2 ) ( 4 , 0 ) (4,0) ( 4 , 0 ) − 3 / 5 -3/5 − 3/5 − 1 / 5 -1/5 − 1/5 ( 3 , − 5 ) (3,-5) ( 3 , − 5 ) ( 1 , − 5 ) (1,-5) ( 1 , − 5 ) ( − 1 , 0 , − 1 , 1 ) (-1,0,-1,1) ( − 1 , 0 , − 1 , 1 ) ( − 1 , 1 , 1 , − 1 ) (-1,1,1,-1) ( − 1 , 1 , 1 , − 1 )
B.2. The finite census. The ALF enumeration imposes the ternary rule, all block determinant inequalities, nonparallel exteriors, the strict angular bounds at (155) , and the prefix winding test in each positive reduced sector k ≥ 1 k \ge1 k ≥ 1 Table 28 is exhaustive for these tests through eight genuine turning points.
Table 28. The positive reduced ALF census. Three eight-nut words have roots in two sectors, giving 52 certified word–sector points. The last column does not assert nonexistence.
Each certificate proves an exact root of the normalized rod equations, positive weights, separated nodes and nonzero required frame factors. Its invertible normalized Jacobian proves local isolation after translation and scale are removed. The dictionary in Section 4.5 and [4 Proposition 11.4] then give smooth complete ALF metrics in the stated finite-cover sense. Local isolation does not imply global uniqueness in a sector. The equations and scope of the verification are given below.
These isolated roots do not imply that ALF moduli spaces are discrete. Collinear multi–Taub–NUT metrics retain continuous centre-separation parameters; they occupy the excluded maximal-phantom rank-zero sector. Their − 2 -2 − 2 [15 ] also explain why the ternary restriction cannot be extended to arbitrary phantom limits or to all ALF geometries.
The two five-nut words are ( − 1 , 0 , − 1 , 1 ) (-1, 0, -1, 1) ( − 1 , 0 , − 1 , 1 ) ( − 1 , 1 , 1 , − 1 ) (-1, 1, 1, -1) ( − 1 , 1 , 1 , − 1 ) [12 Section 7] have these same words, and their printed nodes and residues agree with our points to the stated precision. That work checks regularity numerically; here the normalized balance equations have interval certificates and completion follows from the positive-family theorem above. The four-nut component also appears in explicit radical form in [4 Section 11.5] . These overlaps provide independent comparisons of the twistor, BZ and moment descriptions.
The polygon structure of the ALF data
All 65 words give crystallographic arrangements by the exact rod audit. Their polygon distribution is summarized in Table 29 . The original four- and five-nut endpoint components have the pentagon arrangement.
Table 29. ALF direction arrangements through N = 8 N = 8 N = 8
The homogeneous sign test provides an additional distinction. All 49 words with balance roots have certified μ < 0 \mu< 0 μ < 0 μ > 0 \mu> 0 μ > 0 μ \mu μ
The finite equations
Use the reduced moment rods t j t_j t j (136) . Fix the outer nodes at z 1 = − 1 z_1=-1 z 1 = − 1 z N = 1 z_N=1 z N = 1 N N N N − 2 N-2 N − 2
t j = p j t 0 + q j t 1 , j = 2 , … , N . (157) t_j=p_jt_0+q_jt_1,\qquad j=2,\ldots,N. \tag*{(157)} t j = p j t 0 + q j t 1 , j = 2 , … , N . (157) There are 2 N − 2 2N-2 2 N − 2 2 N − 2 2N-2 2 N − 2 (155) ; no separate AF alignment equation is imposed. The same common exterior shear and normalization as in (138) recovers L > 0 L>0 L > 0 ( z j + 1 − z j ) / L (z_{j+1}-z_j)/L ( z j + 1 − z j ) / L
The search uses logarithmic weights and gap ratios. Outward-rounded interval contraction boxes, evaluated with Arb [18 ] , certify exact roots and an invertible Jacobian. Positivity of the occupations, separation of the nodes and every denominator used in the equations are enclosed on each box. These checks prove local isolation after the two stated coordinate normalizations. They do not exclude a second root elsewhere in the same sector.
Scope of the catalogue
The supplement contains all 65 words, the allowed reduced sectors, 52 balance certificates, the primitive rod and fibre data, and the separate homogeneous sign certificates. The sign comparison uses (125) ; a failed common-cone search is not assigned a sign without a separate trial certificate. In particular, both positive boundary-minimization cases in the eight-nut census have independently certified positive lower bounds.
The original endpoint labels C 4 C_4 C 4 C 5 a C_5^a C 5 a C 5 b C_5^b C 5 b C 6 a , … , C 6 e C_6^a,\ldots,C_6^e C 6 a , … , C 6 e [4 Section 11.5] . The symmetric five-nut example has exact algebraic balance data specified by a degree-sixteen polynomial and an isolating interval. Such algebraic specifications describe exact solutions without requiring radical expressions. The later finite census supplies interval balance certificates for the original eight endpoint types.
The whole-axis and end analysis is supplied analytically by [4 Propositions 7.5, 8.7–8.9 and 11.4] , through the normalization dictionary in § 4.5 . Applied to the certified positive, separated roots of (157) , it gives smooth complete ALF metrics. Thus bulk grid sampling is not used to establish their global positivity or completion. The interval certificates prove existence and local isolation; they do not prove global uniqueness in each marked sector. The census excludes rank-zero multi–Taub–NUT and the classical one- and two-nut families, and therefore is not an exhaustive classification of ALF instantons.
Certified word–sector points
Table 30 lists all 52 certified points. The identifier includes the number of genuine nuts. The rank k k k h = N − 2 k h=N-2k h = N − 2 k 2 ( N − k ) 2(N-k) 2 ( N − k ) ( A , − B ) (A,-B) ( A , − B ) M / L M/L M / L k k k β \beta β f ∞ f_\infty f ∞
Table 30. The ALF balance catalogue through eight genuine turning points.
The three doubled words are ALF8-019, ALF8-025 and ALF8-028. Their two values of β \beta β
B.7. Candidates without an ALF balance certificate. Table 31 lists the other sixteen words. Each has certified positive invariant for its normalized homogeneous model. These are not certified nonexistence statements for the full ALF problem: the general implication from the homogeneous sign to finite ALF balance remains open.
Table 31. ALF candidates with positive homogeneous invariant and no certified balance root.
The twelve-nut heptagon refinement
B.8. The twelve-nut heptagon refinement. A targeted refinement tests the last row beyond the exhaustive census. The eight-nut word
C 8 = ( − 1 , 0 , − 1 , 1 , − 1 , − 1 , 1 ) C_{8}=(-1,0,-1,1,-1,-1,1) C 8 = ( − 1 , 0 , − 1 , 1 , − 1 , − 1 , 1 ) has the non-fan heptagon arrangement and 0.09454 < μ ( C 8 ) < 0.09541 0.09454 < \mu(C_{8}) < 0.09541 0.09454 < μ ( C 8 ) < 0.09541
C 10 = ( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 , − 1 , 1 ) , C_{10}=(-1,0,0,0,-1,1,-1,-1,1), C 10 = ( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 , − 1 , 1 ) , C 12 = ( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 , 0 , 0 , − 1 , 1 ) . C_{12}=(-1,0,0,0,-1,1,-1,0,0,-1,1). C 12 = ( − 1 , 0 , 0 , 0 , − 1 , 1 , − 1 , 0 , 0 , − 1 , 1 ) . The certified homogeneous values change from μ ( C 10 ) = 0.07155305867 … \mu(C_{10})=0.07155305867\ldots μ ( C 10 ) = 0.07155305867 … μ ( C 12 ) = − 0.01968371494 … \mu(C_{12})=-0.01968371494\ldots μ ( C 12 ) = − 0.01968371494 … β = − 4 / 7 \beta=-4/7 β = − 4/7 − 2 / 7 -2/7 − 2/7
Together with the AF zigzag example, this refinement supports studying realizability at the level of marked walks on a polygon. It leaves open which refinements succeed, whether an arrangement always supports a regular walk, and whether the same criterion governs the AF and non-GH ALF sectors.
For the word C 12 C_{12} C 12 ( A , − B ) (A,-B) ( A , − B )
Table 32. Two isolated positive ALF balance roots with the same twelve-nut word and different asymptotic fibres.
Figure 3. The two heptagon triangulations in the ALF census. The right-hand type has no certified balance root through eight nuts but admits the twelve-nut refinement C 12 C_{12} C 12 (129) .
dent evaluation of the original homogeneous BZ determinants at the refined centres gives rod residuals below 1.7 ⋅ 10 − 73 1.7 \cdot10^{-73} 1.7 ⋅ 1 0 − 73 4.0 ⋅ 10 − 35 4.0 \cdot10^{-35} 4.0 ⋅ 1 0 − 35 − 0.0196837149414386576970427241280012 -0.0196837149414386576970427241280012 − 0.0196837149414386576970427241280012
Appendix C. Supplementary analytic arguments
This appendix proves the coupled transport lemma and supplies the local comparison, collision, and insertion arguments used in § 6 . It also records finite enclosures and derivative formulas used to interpret the numerical catalogue.
Transport through singular principal blocks
The following finite-matrix statement supplies (117) without a nonvanishing assumption on the principal minors of T T T
Lemma C.1 (Coupled triangular transport). Let T T T N × N N \times N N × N v ∈ R N v \in\mathbb{R}^{N} v ∈ R N
D j = det T j , A j = v [ j ] T adj ( T j ) v [ j ] , r j = ( − 1 ) j ( 2 A j , D j ) , r 0 = ( 0 , 1 ) . D_j = \det T_j,\qquad A_j = v_{[j]}^{T}\operatorname{adj}(T_j)v_{[j]},\qquad r_j = (-1)^j(2A_j,D_j),\qquad r_0 = (0,1). D j = det T j , A j = v [ j ] T adj ( T j ) v [ j ] , r j = ( − 1 ) j ( 2 A j , D j ) , r 0 = ( 0 , 1 ) . Suppose consecutive rods are nonparallel. Every infinitesimal variation ( δ T , δ v ) (\delta T,\delta v) ( δ T , δ v ) [ r j ] [r_j] [ r j ] 1 ≤ j ≤ N 1 \le j \le N 1 ≤ j ≤ N K K K
δ T = K T + T K T , δ v = K v , δ r j = 2 tr ( K j ) r j . \delta T = KT + TK^{T},\qquad\delta v = Kv,\qquad\delta r_j = 2\operatorname{tr}(K_j)r_j. δ T = K T + T K T , δ v = K v , δ r j = 2 tr ( K j ) r j . The matrix K K K D j = 0 D_j = 0 D j = 0
Proof. Choose a real scalar c c c Q = T + c v v T Q = T + cvv^{T} Q = T + c v v T
det Q j = D j + c A j , adj ( Q j ) v [ j ] = adj ( T j ) v [ j ] . (158) \det Q_j = D_j + cA_j,\qquad\operatorname{adj}(Q_j)v_{[j]} = \operatorname{adj}(T_j)v_{[j]}. \tag*{(158)} det Q j = D j + c A j , adj ( Q j ) v [ j ] = adj ( T j ) v [ j ] . (158) The second identity follows first for invertible T j T_j T j ( A j , D j ) (A_j,D_j) ( A j , D j ) c c c Q Q Q L D L T LDL^{T} L D L T K K K δ Q = K Q + Q K T \delta Q = KQ + QK^{T} δ Q = K Q + Q K T e = δ v − K v e=\delta v-Kv e = δ v − K v A j A_j A j T j T_j T j Q j Q_j Q j
δ A j = 2 tr ( K j ) A j + 2 e [ j ] T adj ( Q j ) v [ j ] , δ ( D j + c A j ) = 2 tr ( K j ) ( D j + c A j ) . \begin{aligned}
\delta A_j&=2\operatorname{tr}(K_j)A_j+2e_{[j]}^{T}\operatorname{adj}(Q_j)v_{[j]},\\
\delta(D_j+cA_j)&=2\operatorname{tr}(K_j)(D_j+cA_j).
\end{aligned} δ A j δ ( D j + c A j ) = 2 tr ( K j ) A j + 2 e [ j ] T adj ( Q j ) v [ j ] , = 2 tr ( K j ) ( D j + c A j ) . The shifted pair ( 2 A j , D j + c A j ) (2A_j,D_j+cA_j) ( 2 A j , D j + c A j ) ( 2 A j , D j ) (2A_j,D_j) ( 2 A j , D j )
e [ j ] T adj ( Q j ) v [ j ] = 0 , 1 ≤ j ≤ N . e_{[j]}^{T}\operatorname{adj}(Q_j)v_{[j]}=0,\qquad1\leq j\leq N. e [ j ] T adj ( Q j ) v [ j ] = 0 , 1 ≤ j ≤ N . These equations form a lower triangular system for e e e j j j
g j = ( adj ( T j ) v [ j ] ) j , 2 g j 2 = det ( r j − 1 , r j ) > 0. g_j=\left(\operatorname{adj}(T_j)v_{[j]}\right)_j,\qquad2g_j^2=\det(r_{j-1},r_j)>0. g j = ( adj ( T j ) v [ j ] ) j , 2 g j 2 = det ( r j − 1 , r j ) > 0. The last equality is the polynomial form of (110) ; it also holds when T j − 1 T_{j-1} T j − 1 e = 0 e=0 e = 0 c δ ( v v T ) c\,\delta(vv^T) c δ ( v v T ) δ Q \delta Q δ Q δ T \delta T δ T Q Q Q c c c
The pencil is only an auxiliary coordinate choice; it is not required to satisfy the involution or displacement identities. Those identities are applied to the recovered transport of the original ( T , v ) (T,v) ( T , v ) β \beta β
Canonical identification
The following proves Proposition 6.3.
Proof. We first show that B ^ \widehat B B (101) is positive definite everywhere on the closed orbit half-plane. Away from nuts all R i R_i R i z i z_i z i i i i T e i Te_i T e i e i e_i e i P = ( I + T ) / 2 P=(I+T)/2 P = ( I + T ) /2 0 < P i i < 1 0<P_{ii}<1 0 < P ii < 1 i i i T e i Te_i T e i
On a rod interior, (102) is analytic in ρ 2 \rho^2 ρ 2 z z z G = O ( ρ 2 ) G=O(\rho^2) G = O ( ρ 2 ) det G = ρ 2 \det G=\rho^2 det G = ρ 2 ρ 2 \rho^2 ρ 2 O ( ρ 2 ) O(\rho^2) O ( ρ 2 ) G / ρ G/\rho G / ρ det T j = 0 \det T_j=0 det T j = 0
Near a nut put ζ = z − z i \zeta=z-z_i ζ = z − z i r = ( ρ 2 + ζ 2 ) 1 / 2 r=(\rho^2+\zeta^2)^{1/2} r = ( ρ 2 + ζ 2 ) 1/2 x = r − ζ x=r-\zeta x = r − ζ y = r + ζ y=r+\zeta y = r + ζ x y = ρ 2 xy=\rho^2 x y = ρ 2 x + y = 2 r x+y=2r x + y = 2 r (102) are analytic in ( x , y ) (x,y) ( x , y ) S = ( r i − 1 r i ) S=(r_{i-1}\ r_i) S = ( r i − 1 r i ) δ = det S > 0 \delta=\det S>0 δ = det S > 0
S T G S = ( y A ( x , y ) x y E ( x , y ) x y E ( x , y ) x D ( x , y ) ) , A D − x y E 2 = δ 2 . S^TGS=
\begin{pmatrix}
yA(x,y)&xyE(x,y)\\
xyE(x,y)&xD(x,y)
\end{pmatrix},
\qquad
AD-xyE^2=\delta^2. S T GS = ( y A ( x , y ) x y E ( x , y ) x y E ( x , y ) x D ( x , y ) ) , A D − x y E 2 = δ 2 . Here A ( 0 , 0 ) , D ( 0 , 0 ) > 0 A(0,0),D(0,0)>0 A ( 0 , 0 ) , D ( 0 , 0 ) > 0 e 2 ν = F ( x , y ) / ( x + y ) e^{2\nu}=F(x,y)/(x+y) e 2 ν = F ( x , y ) / ( x + y ) F F F A ( x , 0 ) = F ( x , 0 ) A(x,0)=F(x,0) A ( x , 0 ) = F ( x , 0 ) D ( 0 , y ) = F ( 0 , y ) D(0,y)=F(0,y) D ( 0 , y ) = F ( 0 , y ) A ( 0 , 0 ) = D ( 0 , 0 ) = F ( 0 , 0 ) = δ A(0,0)=D(0,0)=F(0,0)=\delta A ( 0 , 0 ) = D ( 0 , 0 ) = F ( 0 , 0 ) = δ
δ − 1 S T G S = ( y [ 1 + O ( r ) ] O ( x y ) O ( x y ) x [ 1 + O ( r ) ] ) , e 2 ν = δ 2 r [ 1 + O ( r ) ] . (159) \delta^{-1}S^TGS=
\begin{pmatrix}
y[1+O(r)]&O(xy)\\
O(xy)&x[1+O(r)]
\end{pmatrix},
\qquad
e^{2\nu}=\frac{\delta}{2r}[1+O(r)]. \tag*{(159)} δ − 1 S T GS = ( y [ 1 + O ( r )] O ( x y ) O ( x y ) x [ 1 + O ( r )] ) , e 2 ν = 2 r δ [ 1 + O ( r )] . (159) The relative Gram error is O ( r ) O(r) O ( r )
At infinity, B = 2 r I + O ( 1 ) \mathcal{B}=2rI+O(1) B = 2 r I + O ( 1 ) (100) , using u T v = 0 u^{T}v=0 u T v = 0
f = 1 − 2 M r + O ( r − 2 ) , ω = O ( ρ 2 / r 3 ) , e 2 k = 1 + O ( ρ 2 / r 4 ) . f=1-\frac{2M}{r}+O(r^{-2}),\qquad\omega=O(\rho^{2}/r^{3}),\qquad e^{2k}=1+O(\rho^{2}/r^{4}). f = 1 − r 2 M + O ( r − 2 ) , ω = O ( ρ 2 / r 3 ) , e 2 k = 1 + O ( ρ 2 / r 4 ) . These estimates are uniform in angle; they follow by expanding in r − 1 r^{-1} r − 1 O ( r − 1 ) O(r^{-1}) O ( r − 1 ) [20 Theorem 4.24] now identifies the harmonic maps. The augmentation equations determine ν \nu ν [20 Proposition 4.26] . Finally, Proposition 6.1 supplies the direction-matched data at every positive length vector. □
For Corollary 6.4, the completion argument is as follows.
Proof. The rod and nut estimates above put the metric in the scope of the local regularity results [20 Propositions 4.15 and 4.23] . Cone balance removes the rod singularities, and the lattice-basis condition removes orbifold quotients at nuts. The normalized AF expansion gives a complete end and finite curvature energy. □
The homogeneous moment model
Let a cluster have r r r z 1 < ⋯ < z r z_{1}<\cdots<z_{r} z 1 < ⋯ < z r ω i \omega_{i} ω i k k k
H a [ j ] = det ( ∑ i = 1 r ϵ i [ j ] ω i z i s + t ) s , t = 0 a − 1 , H 0 [ j ] = 1 , t j = ( H k + 1 [ j ] , H k [ j ] ) . (160) H_{a}^{[j]}=\det\left(\sum_{i=1}^{r}\epsilon_{i}^{[j]}\omega_{i}z_{i}^{s+t}\right)_{s,t=0}^{a-1},\qquad H_{0}^{[j]}=1,\qquad t_{j}=(H_{k+1}^{[j]},H_{k}^{[j]}). \tag*{(160)} H a [ j ] = det ( i = 1 ∑ r ϵ i [ j ] ω i z i s + t ) s , t = 0 a − 1 , H 0 [ j ] = 1 , t j = ( H k + 1 [ j ] , H k [ j ] ) . (160) where ϵ i [ j ] = − 1 \epsilon_{i}^{[j]}=-1 ϵ i [ j ] = − 1 i ≤ j i\leq j i ≤ j + 1 +1 + 1 a j > 0 a_{j}>0 a j > 0 r j = a j v j r_{j}=a_{j}v_{j} r j = a j v j b C , j = log a j b_{C,j}=\log a_{j} b C , j = log a j k k k
For completeness, the homogeneous first law and convexity have a direct moment interpretation. Let Z = diag ( z i ) Z=\operatorname{diag}(z_{i}) Z = diag ( z i ) P a P_{a} P a ( ω i p ( z i ) ) i (\sqrt{\omega_{i}}p(z_{i}))_{i} ( ω i p ( z i ) ) i deg p < a \deg p<a deg p < a z z z
P k Z ( I − P k + 1 ) = 0. P_{k}Z(I-P_{k+1})=0. P k Z ( I − P k + 1 ) = 0. Write T − = I − 2 P k T_{-}=I-2P_{k} T − = I − 2 P k T + = I − 2 P k + 1 T_{+}=I-2P_{k+1} T + = I − 2 P k + 1 P k + 1 − P k = q q T P_{k+1}-P_{k}=qq^{T} P k + 1 − P k = q q T ∥ q ∥ = 1 \lVert q\rVert=1 ∥ q ∥ = 1 H a [ j ] / H a [ 0 ] = det ( I − 2 P a ) [ j ] H_{a}^{[j]}/H_{a}^{[0]}=\det(I-2P_{a})_{[j]} H a [ j ] / H a [ 0 ] = det ( I − 2 P a ) [ j ] T + T_{+} T + T − T_{-} T − T + = T − − 2 q q T T_{+}=T_{-}-2qq^{T} T + = T − − 2 q q T ( T − , q ) (T_{-},q) ( T − , q ) Lemma C.1 . The exterior frame is held fixed here, as in the definition of the direction-matched model. The lemma gives δ T − = K T − + T − K T \delta T_{-}=KT_{-}+T_{-}K^{T} δ T − = K T − + T − K T δ q = K q \delta q=Kq δ q = K q δ T + = K T + + T + K T \delta T_{+}=KT_{+}+T_{+}K^{T} δ T + = K T + + T + K T δ b j = 2 tr K [ j ] \delta b_{j}=2\operatorname{tr}K_{[j]} δ b j = 2 tr K [ j ] K [ j ] K_{[j]} K [ j ] j × j j\times j j × j S = K + K T S=K+K^{T} S = K + K T S T ± + T ± S = 0 S T_{\pm}+T_{\pm}S=0 S T ± + T ± S = 0 ∥ q ∥ = 1 \lVert q\rVert=1 ∥ q ∥ = 1 S q = 0 Sq=0 S q = 0 ε j = z j + 1 − z j \varepsilon_{j}=z_{j+1}-z_{j} ε j = z j + 1 − z j
tr ( Z S ) = 0 , ∑ j ε j δ b j = 0 , δ ξ T ( D ξ b ) δ ξ = ∑ i > j ( z i − z j ) K i j 2 ≥ 0. \operatorname{tr}(ZS)=0,\qquad\sum_{j}\varepsilon_{j}\,\delta b_{j}=0,\qquad\delta\xi^{T}(D_{\xi}b)\delta\xi=\sum_{i>j}(z_{i}-z_{j})K_{ij}^{2}\geq0. tr ( Z S ) = 0 , j ∑ ε j δ b j = 0 , δ ξ T ( D ξ b ) δ ξ = i > j ∑ ( z i − z j ) K ij 2 ≥ 0. These identities establish the homogeneous potential and its convexity on the positive direction-matched family. The local direction-coordinate hypothesis in § 6.4 specifies the family on which the transport is used. In rank zero the potential can be linear; no strict convexity assertion is needed for the sign tests.
Proof of the collision criterion
Parallel blocks require a bound preventing other cone errors from canceling the divergent negative error. The involution and AF residue identity give
∥ v ∥ 2 ≤ S : = z N − z 1 , ∣ det T j ∣ ≤ 1 , ∥ adj ( T j ) ∥ ≤ 1. \lVert v\rVert^2 \le S := z_N-z_1,\qquad\lvert\det T_j\rvert\le1,\qquad\lVert\operatorname{adj}(T_j)\rVert\le1. ∥ v ∥ 2 ≤ S := z N − z 1 , ∣ det T j ∣ ≤ 1 , ∥ adj ( T j )∥ ≤ 1. Equation (99) implies
∣ r j t ∣ ≤ 2 S , ∣ r j ϕ ∣ ≤ 1. (161) \lvert r_j^t\rvert\le2S,\qquad\lvert r_j^\phi\rvert\le1. \tag*{(161)} ∣ r j t ∣ ≤ 2 S , ∣ r j ϕ ∣ ≤ 1. (161) Since r j = e b j ( − q j , p j − β q j ) r_j=e^{b_j}(-q_j,p_j-\beta q_j) r j = e b j ( − q j , p j − β q j ) b j b_j b j q j = 0 q_j=0 q j = 0 ∣ p j ∣ = 1 \lvert p_j\rvert=1 ∣ p j ∣ = 1 b j ≤ 0 b_j\le0 b j ≤ 0 b j ≤ log ( 2 S / ∣ q j ∣ ) b_j\le\log(2S/\lvert q_j\rvert) b j ≤ log ( 2 S / ∣ q j ∣) b j → − ∞ b_j\to-\infty b j → − ∞ − ∞ -\infty − ∞ (161) , since S → 0 S\to0 S → 0
Proof. At a finite stratum minimum, open each nonparallel block in a shape of negative cost and assign the blocks positive total-length fractions. Equation (126) gives a negative total derivative. If a parallel block occurs, (161) instead gives derivative − ∞ -\infty − ∞
For the upper side, suppose a sequence has total span tending to infinity. Rescale to span one and pass to a subsequence. At least one rod retains a positive fraction of the span. The large-span cone limit [20 Proposition 5.15] , together with max j ∣ p j − β q j ∣ < 1 \max_j\lvert p_j-\beta q_j\rvert<1 max j ∣ p j − β q j ∣ < 1 2 π 2\pi 2 π b j > 0 b_j>0 b j > 0
Minimize Φ W \Phi_W Φ W [ δ , R ] N − 1 [\delta,R]^{N-1} [ δ , R ] N − 1 0 < δ ≤ 1 < R 0<\delta\le1<R 0 < δ ≤ 1 < R x j > δ x_j>\delta x j > δ b j ≤ 0 b_j\le0 b j ≤ 0 M ( W , β ) M(W,\beta) M ( W , β ) δ , R \delta,R δ , R R > M + 1 R>M+1 R > M + 1 δ ↓ 0 \delta\downarrow0 δ ↓ 0 q ∈ [ 0 , M ] N − 1 q\in[0,M]^{N-1} q ∈ [ 0 , M ] N − 1
Φ ‾ W ( q ) ≤ lim inf Φ W ( x δ ) ≤ Φ W ( y ) for every y ∈ ( 0 , R ) N − 1 . \overline{\Phi}_W(q)\le\liminf\Phi_W(x_\delta)\le\Phi_W(y)\qquad\text{for every }y\in(0,R)^{N-1}. Φ W ( q ) ≤ lim inf Φ W ( x δ ) ≤ Φ W ( y ) for every y ∈ ( 0 , R ) N − 1 . Thus q q q q q q b ( q ) = 0 b(q)=0 b ( q ) = 0
Negative opening tests
Three elementary tests suffice for the families below. First, write the internal rods in the real exterior basis,
v j = α j v 0 + γ j v n , D ( C ) = max 0 < j < n ( ∣ α j ∣ + ∣ γ j ∣ ) . v_j=\alpha_jv_0+\gamma_jv_n,\qquad D(C)=\max_{0<j<n}(\lvert\alpha_j\rvert+\lvert\gamma_j\rvert). v j = α j v 0 + γ j v n , D ( C ) = 0 < j < n max (∣ α j ∣ + ∣ γ j ∣) . Cauchy–Binet expresses H m [ j ] / H m [ 0 ] H_m^{[j]}/H_m^{[0]} H m [ j ] / H m [ 0 ] ( 1 , 1 ) (1,1) ( 1 , 1 ) ( ( − 1 ) k + 1 , ( − 1 ) k ) ((-1)^{k+1},(-1)^k) (( − 1 ) k + 1 , ( − 1 ) k )
b C , j ≤ − log ( ∣ α j ∣ + ∣ γ j ∣ ) , μ ( C ) ≤ − log D ( C ) . (162) b_{C,j}\le-\log(\lvert\alpha_j\rvert+\lvert\gamma_j\rvert),\qquad\mu(C)\le-\log D(C). \tag*{(162)} b C , j ≤ − log (∣ α j ∣ + ∣ γ j ∣) , μ ( C ) ≤ − log D ( C ) . (162) The second inequality follows by concentrating positive normalized lengths near an index maximizing D D D D > 1 D>1 D > 1 μ < 0 \mu<0 μ < 0
Second, deletion of an entry d i = σ ∈ { − 1 , 1 } d_i=\sigma\in\{-1,1\} d i = σ ∈ { − 1 , 1 } d ↦ d − σ d\mapsto d-\sigma d ↦ d − σ − 1 -1 − 1 W W W s W sW s W ( 1 − s ) W (1-s)W ( 1 − s ) W
t n e w = ( 1 − s ) t − + s t + . t_{\mathrm{new}}=(1-s)t_-+st_+. t new = ( 1 − s ) t − + s t + . Since v n e w = v − + v + v_{\mathrm{new}}=v_-+v_+ v new = v − + v +
s = a − a − + a + , a n e w = a − a + a − + a + . (163) s=\frac{a_-}{a_-+a_+},\qquad a_{\mathrm{new}}=\frac{a_-a_+}{a_-+a_+}. \tag*{(163)} s = a − + a + a − , a new = a − + a + a − a + . (163) The new direction equation has nonzero derivative in s s s s s s + 1 +1 + 1
H m [ j ] ( ω ) = ( − 1 ) j A H n − m [ j ] ( ω ~ ) , ω ~ i = 1 ω i D ′ ( z i ) 2 , A = Δ ( z ) 2 ∏ i ω i , (164) H_m^{[j]}(\omega)=(-1)^j A H_{n-m}^{[j]}(\widetilde{\omega}),\qquad\widetilde{\omega}_i=\frac{1}{\omega_iD'(z_i)^2},\qquad A=\Delta(z)^2\prod_i\omega_i, \tag*{(164)} H m [ j ] ( ω ) = ( − 1 ) j A H n − m [ j ] ( ω ) , ω i = ω i D ′ ( z i ) 2 1 , A = Δ ( z ) 2 i ∏ ω i , (164) where D ( z ) = ∏ i ( z − z i ) D(z)=\prod_i(z-z_i) D ( z ) = ∏ i ( z − z i ) Δ ( z ) = ∏ i < j ( z j − z i ) \Delta(z)=\prod_{i<j}(z_j-z_i) Δ ( z ) = ∏ i < j ( z j − z i )
μ ( C ) ≤ μ ( C d e l e t e d ) . \mu(C)\leq\mu(C_{\mathrm{deleted}}). μ ( C ) ≤ μ ( C deleted ) . A nonempty Gibbons–Hawking chain ( − 2 , … , − 2 ) (-2,\ldots,-2) ( − 2 , … , − 2 ) b j = 0 b_j=0 b j = 0 − log 2 -\log2 − log 2 μ < 0 \mu<0 μ < 0
Third, an even-length all-zero word has a negative Weyl trial. For n = 2 k + 1 n=2k+1 n = 2 k + 1 ω i = 1 / ∣ D ′ ( z i ) ∣ \omega_i=1/|D'(z_i)| ω i = 1/∣ D ′ ( z i ) ∣ (164) gives H k + 1 [ j ] = ( − 1 ) j H k [ j ] H_{k+1}^{[j]}=(-1)^jH_k^{[j]} H k + 1 [ j ] = ( − 1 ) j H k [ j ] 0 < ∣ H k [ j ] ∣ / H k [ 0 ] < 1 0<|H_k^{[j]}|/H_k^{[0]}<1 0 < ∣ H k [ j ] ∣/ H k [ 0 ] < 1
Ψ ( 0 , 0 ) ( s , 1 − s ) = s log s + ( 1 − s ) log ( 1 − s ) , 0 < s < 1. (165) \Psi_{(0,0)}(s,1-s)=s\log s+(1-s)\log(1-s),\qquad0<s<1. \tag*{(165)} Ψ ( 0 , 0 ) ( s , 1 − s ) = s log s + ( 1 − s ) log ( 1 − s ) , 0 < s < 1. (165) Odd-length all-zero words have parallel exterior rods and belong to the separate descent argument.
The confluent insertion calculation
This proves Lemma 6.7 ; D D D
Proof. Use positive occupation variables ζ i = ω i ∣ D ′ ( z i ) ∣ \zeta_i=\omega_i|D'(z_i)| ζ i = ω i ∣ D ′ ( z i ) ∣ ( c , ζ ∗ ) (c,\zeta_*) ( c , ζ ∗ ) c − ε A , c , c + ε B c-\varepsilon A,c,c+\varepsilon B c − ε A , c , c + εB A , B > 0 A,B>0 A , B > 0 ζ ∗ \zeta_* ζ ∗ c c c W W W ε − 2 λ i \varepsilon^{-2}\lambda_i ε − 2 λ i
( λ 1 , λ 2 , λ 3 ) = W ( 1 A ( A + B ) , 1 A B , 1 B ( A + B ) ) . (\lambda_1,\lambda_2,\lambda_3)=W\left(\frac{1}{A(A+B)},\frac{1}{AB},\frac{1}{B(A+B)}\right). ( λ 1 , λ 2 , λ 3 ) = W ( A ( A + B ) 1 , A B 1 , B ( A + B ) 1 ) . An outside mass ω i \omega_i ω i ω i / ( z i − c ) 2 \omega_i/(z_i-c)^2 ω i / ( z i − c ) 2 j = 0 , 1 , 2 , 3 j=0,1,2,3 j = 0 , 1 , 2 , 3 ξ = ( − A , 0 , B ) \xi=(-A,0,B) ξ = ( − A , 0 , B )
M j = ∑ i ϵ i [ j ] λ i , S j = ∑ i ϵ i [ j ] λ i ξ i , Q j = ∑ i ϵ i [ j ] λ i ξ i 2 , V j = Q j − S j 2 / M j . M_j=\sum_i\epsilon_i^{[j]}\lambda_i,\qquad S_j=\sum_i\epsilon_i^{[j]}\lambda_i\xi_i,\qquad Q_j=\sum_i\epsilon_i^{[j]}\lambda_i\xi_i^2,\qquad V_j=Q_j-S_j^2/M_j. M j = i ∑ ϵ i [ j ] λ i , S j = i ∑ ϵ i [ j ] λ i ξ i , Q j = i ∑ ϵ i [ j ] λ i ξ i 2 , V j = Q j − S j 2 / M j . Direct calculation gives
( V 0 , V 1 , V 2 , V 3 ) = ( W , − W , W , − W ) , ( M 0 , M 1 , M 2 , M 3 ) M 0 = ( 1 , s , − ( 1 − s ) , − 1 ) , s = A A + B . (166) (V_0,V_1,V_2,V_3)=(W,-W,W,-W),\qquad\frac{(M_0,M_1,M_2,M_3)}{M_0}=(1,s,-(1-s),-1),\qquad s=\frac{A}{A+B}. \tag*{(166)} ( V 0 , V 1 , V 2 , V 3 ) = ( W , − W , W , − W ) , M 0 ( M 0 , M 1 , M 2 , M 3 ) = ( 1 , s , − ( 1 − s ) , − 1 ) , s = A + B A . (166) Use the monic basis 1 , ( z − c ) , ( z − c ) z , … , ( z − c ) z m − 1 1,(z-c),(z-c)z,\ldots,(z-c)z^{m-1} 1 , ( z − c ) , ( z − c ) z , … , ( z − c ) z m − 1 ε − 2 M j \varepsilon^{-2}M_j ε − 2 M j ε − 1 S j ( 1 , c , … , c m − 1 ) + O ( 1 ) \varepsilon^{-1}S_j(1,c,\ldots,c^{m-1})+O(1) ε − 1 S j ( 1 , c , … , c m − 1 ) + O ( 1 ) V j V_j V j
ε 2 H m + 1 n e w , [ j ] ⟶ M j H m o l d ( V j ) . (167) \varepsilon^2 H_{m+1}^{\mathrm{new},[j]} \longrightarrow M_j H_m^{\mathrm{old}}(V_j). \tag*{(167)} ε 2 H m + 1 new , [ j ] ⟶ M j H m old ( V j ) . (167) The common factor is the same in the two consecutive degrees. After exterior normalization, the surviving rods converge to the old rods, with the signs in (127) , and the two new rods tend to s r + sr_+ s r + − ( 1 − s ) r − -(1-s)r_- − ( 1 − s ) r −
To obtain exact directions at positive separation, use the effective mass V 0 V_0 V 0 ε = 0 \varepsilon=0 ε = 0 V 1 = − V 0 V_1=-V_0 V 1 = − V 0 V 2 = V 0 V_2=V_0 V 2 = V 0 x , y x,y x , y x = B / ( A + B ) x=B/(A+B) x = B / ( A + B ) y = A / ( A + B ) y=A/(A+B) y = A / ( A + B )
∂ ( V 1 + V 0 , V 2 − V 0 ) ∂ ( x , y ) = ( − B ( 2 A + B ) B 2 − A 2 A ( A + 2 B ) ) , det = − 2 A B ( A + B ) 2 ≠ 0. \frac{\partial(V_1+V_0,V_2-V_0)}{\partial(x,y)}
=
\begin{pmatrix}
-B(2A+B) & B^2 \\
-A^2 & A(A+2B)
\end{pmatrix},
\qquad
\det=-2AB(A+B)^2\ne0. ∂ ( x , y ) ∂ ( V 1 + V 0 , V 2 − V 0 ) = ( − B ( 2 A + B ) − A 2 B 2 A ( A + 2 B ) ) , det = − 2 A B ( A + B ) 2 = 0. Fixing V 0 = W > 0 V_0=W>0 V 0 = W > 0 ζ ∗ \zeta_* ζ ∗
The new errors have finite limits and their lengths vanish, so Ψ D ♯ ( h ε ) → Ψ D ( h ) \Psi_{D^\sharp}(h_\varepsilon)\to\Psi_D(h) Ψ D ♯ ( h ε ) → Ψ D ( h ) (124) gives
Ψ D ♯ ′ ( 0 + ) = 1 2 log ( a + / 2 ) + 1 2 log ( a − / 2 ) − 1 2 log a − − 1 2 log a + = − log 2. \Psi'_{D^\sharp}(0+)=\frac{1}{2}\log(a_+/2)+\frac{1}{2}\log(a_-/2)-\frac{1}{2}\log a_- -\frac{1}{2}\log a_+=-\log2. Ψ D ♯ ′ ( 0 + ) = 2 1 log ( a + /2 ) + 2 1 log ( a − /2 ) − 2 1 log a − − 2 1 log a + = − log 2. Integration gives (128) . The remainder is local in the original data; no uniform finite-opening estimate is required. □
Verification of the explicit families
Proof. For the five base words at r = s = 0 r=s=0 r = s = 0 D ( C ) > 1 D(C)>1 D ( C ) > 1 ± 1 \pm1 ± 1 r , s r,s r , s
For the second family there is also a direct description at every order. In lattice coefficient coordinates its finite unoriented directions are ( 0 , 1 ) , ( − 1 , 1 ) , ( 1 , − 2 ) (0,1),(-1,1),(1,-2) ( 0 , 1 ) , ( − 1 , 1 ) , ( 1 , − 2 ) ( 1 , 0 ) (1,0) ( 1 , 0 ) − β , − 1 − β , 1 + 2 β -\beta,-1-\beta,1+2\beta − β , − 1 − β , 1 + 2 β ( − 1 , 0 ) (-1,0) ( − 1 , 0 ) ( 1 , − 2 ) (1,-2) ( 1 , − 2 )
Its proper blocks, up to reversal and orientation, are all-zero blocks, a single end entry ± 1 \pm1 ± 1 ( − 1 , 1 , 0 2 a ) (-1,1,0^{2a}) ( − 1 , 1 , 0 2 a ) ( − 1 , 1 , 0 2 a + 1 ) (-1,1,0^{2a+1}) ( − 1 , 1 , 0 2 a + 1 ) ( 1 , 0 2 r + 1 , − 1 ) (1,0^{2r+1},-1) ( 1 , 0 2 r + 1 , − 1 ) ( − 1 , 1 , 0 2 r + 1 , − 1 ) (-1,1,0^{2r+1},-1) ( − 1 , 1 , 0 2 r + 1 , − 1 ) B a = ( − 1 , 1 , 0 2 a ) B_a=(-1,1,0^{2a}) B a = ( − 1 , 1 , 0 2 a )
B a ⟶ ( − 2 , − 1 , 0 2 a − 1 ) ⟶ B a − 1 , B 0 = ( − 1 , 1 ) ⟶ ( − 2 ) . B_a\longrightarrow(-2,-1,0^{2a-1})\longrightarrow B_{a-1},
\qquad
B_0=(-1,1)\longrightarrow(-2). B a ⟶ ( − 2 , − 1 , 0 2 a − 1 ) ⟶ B a − 1 , B 0 = ( − 1 , 1 ) ⟶ ( − 2 ) . This proves all its block signs directly. □
Finite interior certificates
A complementary finite certificate gives effective enclosures of balanced lengths without separate collision calculations. Normalize L = 1 L=1 L = 1 x = ℓ / L x=\ell/L x = ℓ / L Φ = Φ W \Phi=\Phi_W Φ = Φ W H = D x b H=D_xb H = D x b q 0 , q 1 , … , q s q_0,q_1,\ldots,q_s q 0 , q 1 , … , q s g k = b ( q k ; β ) g_k=b(q_k;\beta) g k = b ( q k ; β )
Φ ( x ) − Φ ( q 0 ) ≥ g k ⋅ ( x − q 0 ) − ( g k − g 0 ) ⋅ ( q k − q 0 ) . (168) \Phi(x)-\Phi(q_0) \ge g_k\cdot(x-q_0)-(g_k-g_0)\cdot(q_k-q_0). \tag*{(168)} Φ ( x ) − Φ ( q 0 ) ≥ g k ⋅ ( x − q 0 ) − ( g k − g 0 ) ⋅ ( q k − q 0 ) . (168) Define
P = { x ∈ R N − 1 : g k ⋅ ( x − q 0 ) ≤ ( g k − g 0 ) ⋅ ( q k − q 0 ) for all k } . (169) \mathcal{P}=\left\{x\in\mathbb{R}^{N-1}:g_k\cdot(x-q_0)\le(g_k-g_0)\cdot(q_k-q_0)\ \text{ for all }k\right\}. \tag*{(169)} P = { x ∈ R N − 1 : g k ⋅ ( x − q 0 ) ≤ ( g k − g 0 ) ⋅ ( q k − q 0 ) for all k } . (169) Theorem C.2 (Finite interior certificate). If P \mathcal{P} P P \mathcal{P} P
x j ≥ min y ∈ P y j > 0. x_j\ge\min_{y\in\mathcal{P}}y_j>0. x j ≥ y ∈ P min y j > 0. If the same conditions hold uniformly over a twist interval, the bounds and the existence conclusion hold throughout that interval.
Proof. Equation (168) places the nonempty sublevel set { Φ ≤ Φ ( q 0 ) } \{\Phi\le\Phi(q_0)\} { Φ ≤ Φ ( q 0 )} P \mathcal{P} P
The argument controls arbitrary ratios of collapsing lengths and arbitrary sizes of the other lengths. It can also give a cone-error estimate away from balance. For example, let E E E α > 0 \alpha>0 α > 0 B 0 ≥ 0 B_0\ge0 B 0 ≥ 0
Φ ( x ) − Φ ( q 0 ) ≥ α ∥ E − 1 ( x − q 0 ) ∥ ∞ − B 0 , α > 0 , \Phi(x)-\Phi(q_0)\ge\alpha\lVert E^{-1}(x-q_0)\rVert_\infty-B_0,\qquad\alpha>0, Φ ( x ) − Φ ( q 0 ) ≥ α ∥ E − 1 ( x − q 0 ) ∥ ∞ − B 0 , α > 0 , and the box q 0 + E [ − B 0 / α , B 0 / α ] N − 1 q_0+E[-B_0/\alpha,B_0/\alpha]^{N-1} q 0 + E [ − B 0 / α , B 0 / α ] N − 1 x x x ∥ b ( x ) ∥ ∞ \lVert b(x)\rVert_\infty ∥ b ( x ) ∥ ∞
This criterion is useful when explicit quantitative bounds are required. The collision argument proves existence for the families above without first constructing such an enclosure; the supporting planes can subsequently yield effective bounds on selected parameter intervals.
Twist derivatives
At fixed β \beta β c c c Φ ( x ) − c ⋅ x \Phi(x)-c\cdot x Φ ( x ) − c ⋅ x δ x = H − 1 δ c \delta x=H^{-1}\delta c δ x = H − 1 δ c x ˙ = − H − 1 b \dot{x}=-H^{-1}b x ˙ = − H − 1 b
b ˙ = − b , d Φ d t = − b T H − 1 b . \dot{b}=-b,\qquad\frac{d\Phi}{dt}=-b^{T}H^{-1}b. b ˙ = − b , d t d Φ = − b T H − 1 b . It balances nearby cones and converges locally around a minimum. Global convergence still requires the trajectory to remain in the domain. This is finite-dimensional cone stability, rather than stability under arbitrary metric perturbations.
Changing β \beta β τ \tau τ L = 1 L=1 L = 1 ( r t , r ϕ ) ↦ ( r t , r ϕ + τ r t ) (r^t,r^\phi)\mapsto(r^t,r^\phi+\tau r^t) ( r t , r ϕ ) ↦ ( r t , r ϕ + τ r t ) (99) , the shear is represented exactly by
( T , v ) ↦ ( T + 2 τ v v T , v ) . (T,v)\mapsto(T+2\tau vv^T,v). ( T , v ) ↦ ( T + 2 τ v v T , v ) . The physical deformation restores T 2 = I T^2=I T 2 = I d / d β d/d\beta d / d β
T ′ = K T + T K T + 2 v v T , v ′ = K v , u ′ = − K T u , K i i = 0. (170) T'=KT+TK^T+2vv^T,\qquad v'=Kv,\qquad u'=-K^Tu,\qquad K_{ii}=0. \tag*{(170)} T ′ = K T + T K T + 2 v v T , v ′ = K v , u ′ = − K T u , K ii = 0. (170) The same trace argument as above gives the extended first law
2 d M = x ⋅ d b + 4 J d β , d Φ = b ⋅ d x − 4 J d β . (171) 2\,dM=x\cdot db+4J\,d\beta,\qquad d\Phi=b\cdot dx-4J\,d\beta. \tag*{(171)} 2 d M = x ⋅ d b + 4 J d β , d Φ = b ⋅ d x − 4 J d β . (171) For arbitrary L L L M ˉ = M / L \bar{M}=M/L M ˉ = M / L J ˉ = J / L 2 \bar{J}=J/L^2 J ˉ = J / L 2 L L L d M ˉ / d β = 2 J ˉ d\bar{M}/d\beta=2\bar{J} d M ˉ / d β = 2 J ˉ
The relative length response is
R = diag ( x ) − 1 H − 1 ∂ β b = − d d β log x . (172) R = \operatorname{diag}(x)^{-1}H^{-1}\partial_{\beta}b = -\frac{d}{d\beta}\log x. \tag*{(172)} R = diag ( x ) − 1 H − 1 ∂ β b = − d β d log x . (172) The BZ representation refines this generic implicit-derivative formula. In the positive polynomial row gauge, T j , j + 1 > 0 T_{j,j+1} > 0 T j , j + 1 > 0
x j = v j u j + 1 − u j v j + 1 T j , j + 1 , v j u j + 1 − u j v j + 1 > 0. (173) x_j = \frac{v_j u_{j+1} - u_j v_{j+1}}{T_{j,j+1}}, \qquad v_j u_{j+1} - u_j v_{j+1} > 0. \tag*{(173)} x j = T j , j + 1 v j u j + 1 − u j v j + 1 , v j u j + 1 − u j v j + 1 > 0. (173) Thus R j R_j R j ( T ∗ , u ∗ , v ∗ ) (T_*,u_*,v_*) ( T ∗ , u ∗ , v ∗ ) (98) and u ∗ T v ∗ = 0 u_*^T v_* = 0 u ∗ T v ∗ = 0
( T ∗ + 2 τ v ∗ v ∗ T ) − 1 = T ∗ − 2 τ u ∗ u ∗ T . \left(T_* + 2\tau v_*v_*^T\right)^{-1} = T_* - 2\tau u_*u_*^T. ( T ∗ + 2 τ v ∗ v ∗ T ) − 1 = T ∗ − 2 τ u ∗ u ∗ T . These identities isolate concrete objects to estimate. A quantitative response estimate additionally requires control of the compensating triangular transport and the relative node motion. If a branch has candidate interval ( β − , β + ) (\beta_-,\beta_+) ( β − , β + )
w ( β ) = ( β − β − ) ( β + − β ) β + − β − , s ( β ) = log β − β − β + − β . w(\beta) = \frac{(\beta-\beta_-)(\beta_+-\beta)}{\beta_+-\beta_-}, \qquad s(\beta) = \log\frac{\beta-\beta_-}{\beta_+-\beta}. w ( β ) = β + − β − ( β − β − ) ( β + − β ) , s ( β ) = log β + − β β − β − . A bound ∥ w R ∥ ∞ ≤ C \lVert wR\rVert_\infty\le C ∥ w R ∥ ∞ ≤ C
x j ( β 0 ) e − C ∣ s ( β ) − s ( β 0 ) ∣ ≤ x j ( β ) ≤ x j ( β 0 ) e C ∣ s ( β ) − s ( β 0 ) ∣ . (174) x_j(\beta_0)e^{-C\lvert s(\beta)-s(\beta_0)\rvert} \le x_j(\beta) \le x_j(\beta_0)e^{C\lvert s(\beta)-s(\beta_0)\rvert}. \tag*{(174)} x j ( β 0 ) e − C ∣ s ( β ) − s ( β 0 )∣ ≤ x j ( β ) ≤ x j ( β 0 ) e C ∣ s ( β ) − s ( β 0 )∣ . (174) These are uniform upper and lower length bounds on compact subintervals. These estimates would provide quantitative continuation bounds. For words satisfying Theorem 6.6 , interior continuation already follows from existence and uniqueness; a bound on the relative response would additionally control its rate. No uniform estimate of this kind at the twist endpoints is asserted.
References 
