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

Jérôme Beau

Published Jun 2, 2026• 0 citations
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Abstract

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.

DOI
10.5281/zenodo.20499440

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The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair {c,q−c}\{c,q-c\}{c,q−c}, which must not be conflated. The first is the entropy of the minimal admissible pair sector: the rank-3 admissible projection Πq\Pi_qΠq​ is spectrally atomic, equal to span{e0,eξc,eq−ξc}\mathrm{span}\{e_0,e_{\xi_c},e_{q-\xi_c}\}span{e0​,eξc​​,eq−ξc​​}; removing the zero mode leaves the two-dimensional Born–Infeld parity pair {eξc,eq−ξc}\{e_{\xi_c},e_{q-\xi_c}\}{eξc​​,eq−ξc​​}, identified with the spin-12\tfrac1221​ representation. The proto-state on this sector is the singlet, and its reduced state is maximally mixed, so Sentadm=log⁡2S_{\mathrm{ent}}^{\mathrm{adm}}=\log 2Sentadm​=log2 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 rpair(n)=R∞−R(n)r_{\mathrm{pair}}(n)=R_\infty-R(n)rpair​(n)=R∞​−R(n) defined on the full Gram–Schmidt basis (rank qqq at saturation, not the rank-3 admissible projection). E1 proves Sent(n)=log⁡(rpair(n))S_{\mathrm{ent}}(n)=\log(r_{\mathrm{pair}}(n))Sent​(n)=log(rpair​(n)) unconditionally for matched single-character blocks; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis [Hres][\mathrm{H}_{\mathrm{res}}][Hres​]. The metaplectic dilation ϕω\phi_\omegaϕω​ is excluded as a bridge for [Hres][\mathrm{H}_{\mathrm{res}}][Hres​] because it acts between blocks, not within the residual support of one block.
In the Page–Wootters relational-time framework a ℤ₂ parity constraint (P_C ⊗ P_S)|Ψ⟩ = +|Ψ⟩ ties a clock C to a system S. We show that the constraint controls one specific term of the entanglement entropy — the sector (charge) entropy — which obeys H({p_e, p_o}) ≤ 1 bit with equality iff the two parity sectors are equally weighted, for every system size N, every clock size k, and every state in the constrained subspace. The familiar value I(C;S) = 2 bits is recovered exactly when the complementary configurational term vanishes, i.e. on the rank-2 family, to which the uniform ℤ₂ ground state belongs; there I = 2 bits independently of N, of k and of which qubits form the clock. We further prove an exact complementarity V² + P² = 1 between the clock–system fringe visibility and the sector predictability. On IBM quantum hardware we measure the quantity the theory actually predicts: the weight of the reduced clock state inside the rank-2 subspace. For an 8-dimensional (k = 3) clock register at N = 5 we find 0.920, against 0.250 expected if the register carried the naive 2k = 6 bits; the two surviving eigenvalues are 0.445 and 0.475. A depolarisation parameter calibrated from an independent stabilizer fidelity witness returns I = 1.998, 1.999, 1.997 bits for k = 1, 2, 3. A separable control state is included: a single-basis clock-marginal estimator returns 1.9991 bits on it, while the tomographic estimator used here returns 0.835 bits and V² + P² = 0.0005, so the protocol separates the constrained state from a classical mixture.
We compute the entanglement entropy of a closed conjugate Weil pair {c,q−c}\{c,q-c\}{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 Sent(n)=log⁡rpair(n){S_{\mathrm{ent}}}(n)=\log {r_{\mathrm{pair}}}(n)Sent​(n)=logrpair​(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, rpair(n){r_{\mathrm{pair}}}(n)rpair​(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 10−610^{-6}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\log 2log2 conditional on two explicit hypotheses (H-carrier) and (H-inv), is established separately in the companion paper ENT-E2. The remaining finite-qqq task is to measure the contraction law of rpair(n){r_{\mathrm{pair}}}(n)rpair​(n) and compare its exponent with the pair-capacity exponent δpair\delta_{\mathrm{pair}}δpair​.
The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies wj=1w_j=1wj​=1 for every admitted Weil fingerprint direction, across q∈{29,61,101,151}q\in\{29,61,101,151\}q∈{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=3k{=}3k=3 Weil fingerprint of Heis3(Z/qZ){\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})Heis3​(Z/qZ) for any block (c1,c2,c3)(c_1,c_2,c_3)(c1​,c2​,c3​) is, up to a global phase, the discrete Fourier mode u^A=(e2πiAt/q)t=0q−1{\hat{u}}_A = (e^{2\pi iAt/q})_{t=0}^{q-1}u^A​=(e2πiAt/q)t=0q−1​ at frequency A=c1b1+c2b2+c3b3 mod qA = c_1b_1+c_2b_2+c_3b_3 \bmod qA=c1​b1​+c2​b2​+c3​b3​modq, where (b1,b2,b3)(b_1,b_2,b_3)(b1​,b2​,b3​) are the bbb-components of the three path elements. Since distinct Fourier modes are orthogonal, any newly admitted fingerprint is already orthogonal to the current span, giving wj=∥wjGS∥/∥vj∥=1w_j = \|w^{\mathrm{GS}}_j\|/\|v_j\| = 1wj​=∥wjGS​∥/∥vj​∥=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 Sent(n)=log⁡rpair(n){S_{\mathrm{ent}}}(n)=\log{r_{\mathrm{pair}}}(n)Sent​(n)=logrpair​(n), where rpair(n)=R∞−R(n){r_{\mathrm{pair}}}(n)=R_\infty-R(n)rpair​(n)=R∞​−R(n) is the residual rank.
The quantum structure sub-programme comprises three companion papers on the binary icosahedral group 2I{2I}2I, plus one independently published result. Q1 proves an exact, unconditional Fourier-support rigidity theorem in the finite Weil representation of the Heisenberg group, with a precise no-go corollary for support-, rank-, and Gram-based diagnostics; it establishes no phase coherence, singlet correlator, Tsirelson bound, or Born rule. Q2 proves an exact finite-group fact: the spin-12\tfrac1221​ and spin-32\tfrac3223​ representations of 2I{2I}2I induce the same Cayley-graph Laplacian eigenvalue; it establishes no Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or SU(2){\mathrm{SU}}(2)SU(2) fixed-point/sector-selection argument. Q3 proves a theorem conditional on an explicit, undemonstrated hypothesis: given that a bipartite state on a supplied composition is invariant under the diagonal action of 2I{2I}2I — not derived from admissibility or from Born–Infeld indiscernibility — the state is uniquely the SU(2){\mathrm{SU}}(2)SU(2) singlet, with an explicit Casimir correlator, for all five admissible sectors of 2I{2I}2I; it does not derive that invariance hypothesis, phase coherence, or the Born rule. No chain currently connects these three results to each other or to the structures of quantum mechanics. The Bell paper, published independently in Quantum Reports, establishes the structural non-applicability of Bell-type factorizability in non-injective effective descriptions; the proof of that central theorem uses only non-injectivity and informational completeness. This note records the current, narrow scope of each constituent and the open problems that remain.