Weil Fingerprint Orthogonality and the Flat Schmidt Spectrum of Conjugate Pairs

Jérôme Beau

Published Jun 2, 2026• 0 citations

Abstract

The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies $w_j=1$ for every admitted Weil fingerprint direction, across $q\in\{29,61,101,151\}$ and multiple generic blocks. The equality was left as an open analytic strengthening. We close it here by explicit computation. The $k{=}3$ Weil fingerprint of ${\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})$ for any block $(c_1,c_2,c_3)$ is, up to a global phase, the discrete Fourier mode ${\hat{u}}_A = (e^{2\pi iAt/q})_{t=0}^{q-1}$ at frequency $A = c_1b_1+c_2b_2+c_3b_3 \bmod q$, where $(b_1,b_2,b_3)$ are the $b$-components of the three path elements. Since distinct Fourier modes are orthogonal, any newly admitted fingerprint is already orthogonal to the current span, giving $w_j = \|w^{\mathrm{GS}}_j\|/\|v_j\| = 1$ exactly. For matched single-character blocks, the conjugate pair produces sign-reflected frequency sets, so the paired Fourier supports are canonically identified; conditional on the companion paper's supplied purity of the joint state, this fixes the diagonal Schmidt-partner pairing. For general canonical blocks, the equal-count identity is recovered, while the literal Schmidt pairing remains conditional on a canonical block identification. Combined, these results prove analytically that, conditional on the companion paper's supplied purity of the joint state, the Schmidt spectrum of the conjugate pair is flat on the residual support for matched blocks, and the reduced entropy is ${S_{\mathrm{ent}}}(n)=\log{r_{\mathrm{pair}}}(n)$, where ${r_{\mathrm{pair}}}(n)=R_\infty-R(n)$ is the residual rank.

DOI: 10.5281/zenodo.20499309

Related papers

Two Entanglement Entropies of the Conjugate Weil Pair: the Admissible Pair Sector and the Full Residual Rank

Jérôme Beau

Jun 2, 20260 citations

Residual-Rank Entanglement of Conjugate Weil Pairs in the Spectral Admissibility Cascade

Jérôme Beau

Jun 1, 20260 citations

Fourier-Support Rigidity in Conjugate Weil Sectors: A No-Go Result for Rank-Based Quantum Signatures

Jérôme Beau

Apr 14, 20260 citations

A threshold for maximal Schmidt number from spectrum

Junhyeong An, Soojoon Lee

quant-pharXiv:2609.34255Sep 29, 20260 citations

Mutually unbiased special entangled bases with Schmidt number 2 in $${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$$ C 3 ⊗ C 4 k

Yi-Fan Han, Guijun Zhang, Xin-Lei Yong +2 more

Quantum Information ProcessingJan 29, 201811 citations

A Conditional 3×3 Model on Sym²(Vρ): Equatorial Fibres from a Fixed Symmetric Square, and the Exact Decorrelation Conditions for its Spectrum

Jérôme Beau

May 3, 20260 citations

Weil Fingerprint Orthogonality and the Flat Schmidt Spectrum of Conjugate Pairs

Jérôme Beau

Published Jun 2, 2026• 0 citations

Publisher

Abstract

The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies

w_j=1

for every admitted Weil fingerprint direction, across

q\in\{29,61,101,151\}

and multiple generic blocks. The equality was left as an open analytic strengthening. We close it here by explicit computation. The

k{=}3

Weil fingerprint of

{\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})

for any block

(c_1,c_2,c_3)

is, up to a global phase, the discrete Fourier mode

{\hat{u}}_A = (e^{2\pi iAt/q})_{t=0}^{q-1}

at frequency

A = c_1b_1+c_2b_2+c_3b_3 \bmod q

, where

(b_1,b_2,b_3)

are the

b

-components of the three path elements. Since distinct Fourier modes are orthogonal, any newly admitted fingerprint is already orthogonal to the current span, giving

w_j = \|w^{\mathrm{GS}}_j\|/\|v_j\| = 1

exactly. For matched single-character blocks, the conjugate pair produces sign-reflected frequency sets, so the paired Fourier supports are canonically identified; conditional on the companion paper's supplied purity of the joint state, this fixes the diagonal Schmidt-partner pairing. For general canonical blocks, the equal-count identity is recovered, while the literal Schmidt pairing remains conditional on a canonical block identification. Combined, these results prove analytically that, conditional on the companion paper's supplied purity of the joint state, the Schmidt spectrum of the conjugate pair is flat on the residual support for matched blocks, and the reduced entropy is

{S_{\mathrm{ent}}}(n)=\log{r_{\mathrm{pair}}}(n)

, where

{r_{\mathrm{pair}}}(n)=R_\infty-R(n)

is the residual rank.

DOI: 10.5281/zenodo.20499309

Related papers

Two Entanglement Entropies of the Conjugate Weil Pair: the Admissible Pair Sector and the Full Residual Rank

Jérôme Beau

The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair $\{c,q-c\}$ , which must not be conflated. The first is the entropy of the minimal admissible pair sector: the rank-3 admissible projection $\Pi_q$ is spectrally atomic, equal to $\mathrm{span}\{e_0,e_{\xi_c},e_{q-\xi_c}\}$ ; removing the zero mode leaves the two-dimensional Born–Infeld parity pair $\{e_{\xi_c},e_{q-\xi_c}\}$ , identified with the spin- $\tfrac12$ representation. The proto-state on this sector is the singlet, and its reduced state is maximally mixed, so $S_{\mathrm{ent}}^{\mathrm{adm}}=\log 2$ for general canonical blocks, unconditionally relative to the cited upstream theorems. This is the sector relevant to the singlet correlator and the Tsirelson bound. The second is the full residual fibre-level rank $r_{\mathrm{pair}}(n)=R_\infty-R(n)$ defined on the full Gram–Schmidt basis (rank $q$ at saturation, not the rank-3 admissible projection). E1 proves $S_{\mathrm{ent}}(n)=\log(r_{\mathrm{pair}}(n))$ unconditionally for matched single-character blocks; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis $[\mathrm{H}_{\mathrm{res}}]$ . The metaplectic dilation $\phi_\omega$ is excluded as a bridge for $[\mathrm{H}_{\mathrm{res}}]$ because it acts between blocks, not within the residual support of one block.The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair $\{c,q-c\}$, which must not be conflated. The first is the entropy of the minimal admissible pair sector: the rank-3 admissible projection $\Pi_q$ is spectrally atomic, equal to $\mathrm{span}\{e_0,e_{\xi_c},e_{q-\xi_c}\}$; removing the zero mode leaves the two-dimensional Born–Infeld parity pair $\{e_{\xi_c},e_{q-\xi_c}\}$, identified with the spin-$\tfrac12$ representation. The proto-state on this sector is the singlet, and its reduced state is maximally mixed, so $S_{\mathrm{ent}}^{\mathrm{adm}}=\log 2$ for general canonical blocks, unconditionally relative to the cited upstream theorems. This is the sector relevant to the singlet correlator and the Tsirelson bound. The second is the full residual fibre-level rank $r_{\mathrm{pair}}(n)=R_\infty-R(n)$ defined on the full Gram–Schmidt basis (rank $q$ at saturation, not the rank-3 admissible projection). E1 proves $S_{\mathrm{ent}}(n)=\log(r_{\mathrm{pair}}(n))$ unconditionally for matched single-character blocks; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis $[\mathrm{H}_{\mathrm{res}}]$. The metaplectic dilation $\phi_\omega$ is excluded as a bridge for $[\mathrm{H}_{\mathrm{res}}]$ because it acts between blocks, not within the residual support of one block.

Jun 2, 20260 citations

Residual-Rank Entanglement of Conjugate Weil Pairs in the Spectral Admissibility Cascade

Jérôme Beau

We compute the entanglement entropy of a closed conjugate Weil pair $\{c,q-c\}$ along the spectral admissibility cascade of the Cosmochrony programme, where entanglement is the infra-projectable invariant of a non-decomposable joint fibre of the non-injective projection. The bipartite state lives on the residual fibre-level support of the Gram–Schmidt span of Weil fingerprints on the Heisenberg Cayley graph. We show that the normalized admissibility residual of each newly admitted independent direction equals unity: the admitted Weil fingerprints are mutually orthogonal (proved analytically in the companion paper ENT-E1), so the reduced Schmidt spectrum is flat and the reduced entropy is exactly ${S_{\mathrm{ent}}}(n)=\log {r_{\mathrm{pair}}}(n)$ , the logarithm of the residual Gram–Schmidt rank. No Born–Infeld spectral correction survives: the admissibility-weight deficit is identically zero. Because the cascade only adjoins independent directions, ${r_{\mathrm{pair}}}(n)$ is non-increasing structurally, so the residual conjugate-pair entanglement is maximal at the earliest admissible stage and decreases monotonically as correlated capacity is stabilized into projected structure. The capacity-level statement (non-negativity of the stabilization rate) is a conjecture of the projective temporal ordering analysis, supported numerically only within a relative tolerance of $10^{-6}$ . The entropy computed here is the full residual-rank observable on the complete Gram–Schmidt basis; its relation to the minimal admissible pair sector, whose entropy is $\log 2$ conditional on two explicit hypotheses (H-carrier) and (H-inv), is established separately in the companion paper ENT-E2. The remaining finite- $q$ task is to measure the contraction law of ${r_{\mathrm{pair}}}(n)$ and compare its exponent with the pair-capacity exponent $\delta_{\mathrm{pair}}$ .We compute the entanglement entropy of a closed conjugate Weil pair $\{c,q-c\}$ along the spectral admissibility cascade of the Cosmochrony programme, where entanglement is the infra-projectable invariant of a non-decomposable joint fibre of the non-injective projection. The bipartite state lives on the residual fibre-level support of the Gram–Schmidt span of Weil fingerprints on the Heisenberg Cayley graph. We show that the normalized admissibility residual of each newly admitted independent direction equals unity: the admitted Weil fingerprints are mutually orthogonal (proved analytically in the companion paper ENT-E1), so the reduced Schmidt spectrum is flat and the reduced entropy is exactly ${S_{\mathrm{ent}}}(n)=\log {r_{\mathrm{pair}}}(n)$, the logarithm of the residual Gram–Schmidt rank. No Born–Infeld spectral correction survives: the admissibility-weight deficit is identically zero. Because the cascade only adjoins independent directions, ${r_{\mathrm{pair}}}(n)$ is non-increasing structurally, so the residual conjugate-pair entanglement is maximal at the earliest admissible stage and decreases monotonically as correlated capacity is stabilized into projected structure. The capacity-level statement (non-negativity of the stabilization rate) is a conjecture of the projective temporal ordering analysis, supported numerically only within a relative tolerance of $10^{-6}$. The entropy computed here is the full residual-rank observable on the complete Gram–Schmidt basis; its relation to the minimal admissible pair sector, whose entropy is $\log 2$ conditional on two explicit hypotheses (H-carrier) and (H-inv), is established separately in the companion paper ENT-E2. The remaining finite-$q$ task is to measure the contraction law of ${r_{\mathrm{pair}}}(n)$ and compare its exponent with the pair-capacity exponent $\delta_{\mathrm{pair}}$.

Jun 1, 20260 citations

Fourier-Support Rigidity in Conjugate Weil Sectors: A No-Go Result for Rank-Based Quantum Signatures

Jérôme Beau

We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ , $q$ an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}), S_q)$ , the orbit vectors generated by conjugate Weil sectors $c$ and $q-c$ span identical subspaces of $\mathbb{C}^q$ , for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by $1{,}548$ exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms $\rho_{q-c}=\overline{\rho_c}$ is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This paper corrects an earlier version (v1.1.2): independent review found its central-theorem proof unsound and its separate Born-rule proof independently defective; both are withdrawn, and the singlet correlator and Tsirelson bound survive only as a standard, conditional remark. What remains is the rigidity result above and its precisely scoped no-go corollaries. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, $q$ an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}), S_q)$, the orbit vectors generated by conjugate Weil sectors $c$ and $q-c$ span identical subspaces of $\mathbb{C}^q$, for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by $1{,}548$ exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms $\rho_{q-c}=\overline{\rho_c}$ is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This paper corrects an earlier version (v1.1.2): independent review found its central-theorem proof unsound and its separate Born-rule proof independently defective; both are withdrawn, and the singlet correlator and Tsirelson bound survive only as a standard, conditional remark. What remains is the rigidity result above and its precisely scoped no-go corollaries. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.

Apr 14, 20260 citations

A threshold for maximal Schmidt number from spectrum

Junhyeong An, Soojoon Lee

The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on $\mathbb{C}^d \otimes \mathbb{C}^d$ alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for $(d-1)$ -block positive operators, we prove that $\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}$ for $p\leq \lfloor d^2/2\rfloor$ , where $\mathrm{LS}_p$ denotes the set of states whose largest eigenvalue does not exceed the sum of their $p$ smallest eigenvalues, and $\mathrm{ASN}_{d-1}$ denotes the set of states whose Schmidt number remains at most $d-1$ under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to $\mathrm{ASN}_{d-1}$ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to $\mathrm{ASN}_{d-1}$ . The global unitary requirement is nevertheless essential: for every $d\geq3$ , we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on $\mathbb{C}^d \otimes \mathbb{C}^d$ alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for $(d-1)$-block positive operators, we prove that $\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}$ for $p\leq \lfloor d^2/2\rfloor$, where $\mathrm{LS}_p$ denotes the set of states whose largest eigenvalue does not exceed the sum of their $p$ smallest eigenvalues, and $\mathrm{ASN}_{d-1}$ denotes the set of states whose Schmidt number remains at most $d-1$ under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to $\mathrm{ASN}_{d-1}$ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to $\mathrm{ASN}_{d-1}$. The global unitary requirement is nevertheless essential: for every $d\geq3$, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.

quant-pharXiv:2609.34255Sep 29, 20260 citations

Mutually unbiased special entangled bases with Schmidt number 2 in ${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$ C 3 ⊗ C 4 kMutually unbiased special entangled bases with Schmidt number 2 in $${\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}$$ C 3 ⊗ C 4 k

Yi-Fan Han, Guijun Zhang, Xin-Lei Yong +2 more

Quantum Information ProcessingJan 29, 201811 citations

A Conditional 3×3 Model on Sym²(Vρ): Equatorial Fibres from a Fixed Symmetric Square, and the Exact Decorrelation Conditions for its Spectrum

Jérôme Beau

Papers O28--O29 of the Cosmochrony spectral admissibility sub-programme established that the per-pair covariance $\mathcal{C}_c$ in $\operatorname{Sym}^2(V_\rho)$ has rank $r_{\mathrm{eff}} = 3$ with invariant eigenvalue structure $[\lambda_1 : \lambda_2 : \lambda_3] = [1 : \tfrac{1}{2} : \tfrac{1}{2}]$ , and identified this as an open problem requiring an analytical explanation. The present paper provides the complete derivation. This is the first analytical derivation of the invariant covariance spectrum observed in O28: the ratio $\lambda_1/\lambda_2 = 2$ is not dynamical, not statistical, but a purely structural consequence of three compatible structures---BI parity, the $\mathrm{Sym}^2(V_\rho)$ identification, and the Heisenberg grading. We prove that the Born--Infeld parity anti-linearity $\rho_{q-c} = \bar{\rho}_c$ (O18) forces every admissible trajectory vector $w_j = \pi_c(v_j) \in V_\rho \cong \mathbb{C}^2$ to satisfy $|\alpha_j| = |\beta_j|$ , where $(\alpha_j, \beta_j)$ are the components in the canonical basis $\{v_+, v_-\}$ of $V_\rho$ aligned with the BI involution. This is the equatorial constraint: every $w_j$ lives on an equatorial $S^1 \subset S^3$ in $V_\rho$ , admitting the canonical form $w_j = (r_j/\!\sqrt{2})(e^{i\phi_j} v_+ + e^{-i\phi_j} v_-)$ . In the equatorial regime, the central component $\sqrt{2}\,\alpha_j \beta_j$ of $\operatorname{vec}(M_j)$ is phase-independent (equal to $r_j^2/\!\sqrt{2}$ for all $j$ ), while the two diagonal components $\alpha_j^2$ and $\beta_j^2$ oscillate as $e^{\pm 2i\phi_j}$ . The covariance then diagonalises immediately under the single condition $\langle e^{2i\phi_j} \rangle = 0$ (phase decorrelation), yielding $\mathcal{C}_c = \operatorname{diag}\!\bigl(\tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{4}\bigr) \langle r_j^4 \rangle$ and spectral ratio $[2:1:1]$ , normalised $[1:\tfrac{1}{2}:\tfrac{1}{2}]$ . The factor of $2$ is traced to the degree- $2$ Carnot weight of the central generator $Z = [X, Y]$ in $\mathrm{Heis}_3$ : the bilinear term $\sqrt{2}\,\alpha_j\beta_j$ concentrates exactly twice the covariance of each pure-degree- $1$ term $\alpha_j^2$ or $\beta_j^2$ . The correction to the equatorial condition is $O(q^{-1/2})$ from the universality rate of U1, consistent with the symmetry-ratio measurements of O29. This closes the open direction explicitly stated in O28 and O29.Papers O28--O29 of the Cosmochrony spectral admissibility sub-programme established that the per-pair covariance $\mathcal{C}_c$ in $\operatorname{Sym}^2(V_\rho)$ has rank $r_{\mathrm{eff}} = 3$ with invariant eigenvalue structure $[\lambda_1 : \lambda_2 : \lambda_3] = [1 : \tfrac{1}{2} : \tfrac{1}{2}]$, and identified this as an open problem requiring an analytical explanation. The present paper provides the complete derivation. This is the first analytical derivation of the invariant covariance spectrum observed in O28: the ratio $\lambda_1/\lambda_2 = 2$ is not dynamical, not statistical, but a purely structural consequence of three compatible structures---BI parity, the $\mathrm{Sym}^2(V_\rho)$ identification, and the Heisenberg grading. We prove that the Born--Infeld parity anti-linearity $\rho_{q-c} = \bar{\rho}_c$ (O18) forces every admissible trajectory vector $w_j = \pi_c(v_j) \in V_\rho \cong \mathbb{C}^2$ to satisfy $|\alpha_j| = |\beta_j|$, where $(\alpha_j, \beta_j)$ are the components in the canonical basis $\{v_+, v_-\}$ of $V_\rho$ aligned with the BI involution. This is the equatorial constraint: every $w_j$ lives on an equatorial $S^1 \subset S^3$ in $V_\rho$, admitting the canonical form $w_j = (r_j/\!\sqrt{2})(e^{i\phi_j} v_+ + e^{-i\phi_j} v_-)$. In the equatorial regime, the central component $\sqrt{2}\,\alpha_j \beta_j$ of $\operatorname{vec}(M_j)$ is phase-independent (equal to $r_j^2/\!\sqrt{2}$ for all $j$), while the two diagonal components $\alpha_j^2$ and $\beta_j^2$ oscillate as $e^{\pm 2i\phi_j}$. The covariance then diagonalises immediately under the single condition $\langle e^{2i\phi_j} \rangle = 0$ (phase decorrelation), yielding $\mathcal{C}_c = \operatorname{diag}\!\bigl(\tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{4}\bigr) \langle r_j^4 \rangle$ and spectral ratio $[2:1:1]$, normalised $[1:\tfrac{1}{2}:\tfrac{1}{2}]$. The factor of $2$ is traced to the degree-$2$ Carnot weight of the central generator $Z = [X, Y]$ in $\mathrm{Heis}_3$: the bilinear term $\sqrt{2}\,\alpha_j\beta_j$ concentrates exactly twice the covariance of each pure-degree-$1$ term $\alpha_j^2$ or $\beta_j^2$. The correction to the equatorial condition is $O(q^{-1/2})$ from the universality rate of U1, consistent with the symmetry-ratio measurements of O29. This closes the open direction explicitly stated in O28 and O29.

May 3, 20260 citations

The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies

w_j=1

for every admitted Weil fingerprint direction, across

q\in\{29,61,101,151\}

and multiple generic blocks. The equality was left as an open analytic strengthening. We close it here by explicit computation. The

k{=}3

Weil fingerprint of

{\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})

for any block

(c_1,c_2,c_3)

is, up to a global phase, the discrete Fourier mode

{\hat{u}}_A = (e^{2\pi iAt/q})_{t=0}^{q-1}

at frequency

A = c_1b_1+c_2b_2+c_3b_3 \bmod q

, where

(b_1,b_2,b_3)

are the

b

-components of the three path elements. Since distinct Fourier modes are orthogonal, any newly admitted fingerprint is already orthogonal to the current span, giving

w_j = \|w^{\mathrm{GS}}_j\|/\|v_j\| = 1

{S_{\mathrm{ent}}}(n)=\log{r_{\mathrm{pair}}}(n)

, where

{r_{\mathrm{pair}}}(n)=R_\infty-R(n)

is the residual rank.

The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair

\{c,q-c\}

, which must not be conflated. The first is the entropy of the minimal admissible pair sector: the rank-3 admissible projection

\Pi_q

is spectrally atomic, equal to

\mathrm{span}\{e_0,e_{\xi_c},e_{q-\xi_c}\}

; removing the zero mode leaves the two-dimensional Born–Infeld parity pair

\{e_{\xi_c},e_{q-\xi_c}\}

, identified with the spin-

\tfrac12

representation. The proto-state on this sector is the singlet, and its reduced state is maximally mixed, so

S_{\mathrm{ent}}^{\mathrm{adm}}=\log 2

for general canonical blocks, unconditionally relative to the cited upstream theorems. This is the sector relevant to the singlet correlator and the Tsirelson bound. The second is the full residual fibre-level rank

r_{\mathrm{pair}}(n)=R_\infty-R(n)

defined on the full Gram–Schmidt basis (rank

q

at saturation, not the rank-3 admissible projection). E1 proves

S_{\mathrm{ent}}(n)=\log(r_{\mathrm{pair}}(n))

unconditionally for matched single-character blocks; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis

[\mathrm{H}_{\mathrm{res}}]

. The metaplectic dilation

\phi_\omega

is excluded as a bridge for

[\mathrm{H}_{\mathrm{res}}]

because it acts between blocks, not within the residual support of one block.

We compute the entanglement entropy of a closed conjugate Weil pair

\{c,q-c\}

along the spectral admissibility cascade of the Cosmochrony programme, where entanglement is the infra-projectable invariant of a non-decomposable joint fibre of the non-injective projection. The bipartite state lives on the residual fibre-level support of the Gram–Schmidt span of Weil fingerprints on the Heisenberg Cayley graph. We show that the normalized admissibility residual of each newly admitted independent direction equals unity: the admitted Weil fingerprints are mutually orthogonal (proved analytically in the companion paper ENT-E1), so the reduced Schmidt spectrum is flat and the reduced entropy is exactly

{S_{\mathrm{ent}}}(n)=\log {r_{\mathrm{pair}}}(n)

, the logarithm of the residual Gram–Schmidt rank. No Born–Infeld spectral correction survives: the admissibility-weight deficit is identically zero. Because the cascade only adjoins independent directions,

{r_{\mathrm{pair}}}(n)

is non-increasing structurally, so the residual conjugate-pair entanglement is maximal at the earliest admissible stage and decreases monotonically as correlated capacity is stabilized into projected structure. The capacity-level statement (non-negativity of the stabilization rate) is a conjecture of the projective temporal ordering analysis, supported numerically only within a relative tolerance of

10^{-6}

. The entropy computed here is the full residual-rank observable on the complete Gram–Schmidt basis; its relation to the minimal admissible pair sector, whose entropy is

\log 2

conditional on two explicit hypotheses (H-carrier) and (H-inv), is established separately in the companion paper ENT-E2. The remaining finite-

q

task is to measure the contraction law of

{r_{\mathrm{pair}}}(n)

and compare its exponent with the pair-capacity exponent

\delta_{\mathrm{pair}}

We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group

\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})

q

an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph

\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}), S_q)

, the orbit vectors generated by conjugate Weil sectors

c

and

q-c

span identical subspaces of

\mathbb{C}^q

, for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by

1{,}548

exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms

\rho_{q-c}=\overline{\rho_c}

is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This paper corrects an earlier version (v1.1.2): independent review found its central-theorem proof unsound and its separate Born-rule proof independently defective; both are withdrawn, and the singlet correlator and Tsirelson bound survive only as a standard, conditional remark. What remains is the rigidity result above and its precisely scoped no-go corollaries. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.

The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on

\mathbb{C}^d \otimes \mathbb{C}^d

alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for

(d-1)

-block positive operators, we prove that

\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}

for

p\leq \lfloor d^2/2\rfloor

, where

\mathrm{LS}_p

denotes the set of states whose largest eigenvalue does not exceed the sum of their

p

smallest eigenvalues, and

\mathrm{ASN}_{d-1}

denotes the set of states whose Schmidt number remains at most

d-1

under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to

\mathrm{ASN}_{d-1}

in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to

\mathrm{ASN}_{d-1}

. The global unitary requirement is nevertheless essential: for every

d\geq3

, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.

Mutually unbiased special entangled bases with Schmidt number 2 in

{\mathbb {C}}^3 \otimes {\mathbb {C}}^{4k}

C 3 ⊗ C 4 k

Papers O28--O29 of the Cosmochrony spectral admissibility sub-programme established that the per-pair covariance

\mathcal{C}_c

\operatorname{Sym}^2(V_\rho)

has rank

r_{\mathrm{eff}} = 3

with invariant eigenvalue structure

[\lambda_1 : \lambda_2 : \lambda_3] = [1 : \tfrac{1}{2} : \tfrac{1}{2}]

, and identified this as an open problem requiring an analytical explanation. The present paper provides the complete derivation. This is the first analytical derivation of the invariant covariance spectrum observed in O28: the ratio

\lambda_1/\lambda_2 = 2

is not dynamical, not statistical, but a purely structural consequence of three compatible structures---BI parity, the

\mathrm{Sym}^2(V_\rho)

identification, and the Heisenberg grading. We prove that the Born--Infeld parity anti-linearity

\rho_{q-c} = \bar{\rho}_c

(O18) forces every admissible trajectory vector

w_j = \pi_c(v_j) \in V_\rho \cong \mathbb{C}^2

to satisfy

|\alpha_j| = |\beta_j|

, where

(\alpha_j, \beta_j)

are the components in the canonical basis

\{v_+, v_-\}

V_\rho

aligned with the BI involution. This is the equatorial constraint: every

w_j

lives on an equatorial

S^1 \subset S^3

V_\rho

, admitting the canonical form

w_j = (r_j/\!\sqrt{2})(e^{i\phi_j} v_+ + e^{-i\phi_j} v_-)

. In the equatorial regime, the central component

\sqrt{2}\,\alpha_j \beta_j

\operatorname{vec}(M_j)

is phase-independent (equal to

r_j^2/\!\sqrt{2}

for all

j

), while the two diagonal components

\alpha_j^2

and

\beta_j^2

oscillate as

e^{\pm 2i\phi_j}

. The covariance then diagonalises immediately under the single condition

\langle e^{2i\phi_j} \rangle = 0

(phase decorrelation), yielding

\mathcal{C}_c = \operatorname{diag}\!\bigl(\tfrac{1}{4}, \tfrac{1}{2}, \tfrac{1}{4}\bigr) \langle r_j^4 \rangle

and spectral ratio

[2:1:1]

, normalised

[1:\tfrac{1}{2}:\tfrac{1}{2}]

. The factor of

2

is traced to the degree-

2

Carnot weight of the central generator

Z = [X, Y]

\mathrm{Heis}_3

: the bilinear term

\sqrt{2}\,\alpha_j\beta_j

concentrates exactly twice the covariance of each pure-degree-

1

term

\alpha_j^2

\beta_j^2

. The correction to the equatorial condition is

O(q^{-1/2})

from the universality rate of U1, consistent with the symmetry-ratio measurements of O29. This closes the open direction explicitly stated in O28 and O29.