Entanglement E2 — Two Entanglement Entropies of the Conjugate Weil Pair

Two distinct objects

• Admissible pair sector. The sector is defined by the explicit two-clause hypothesis (H-carrier): (i) the admissible pair sector is carried by the atomic Fourier triple $\mathrm{span}\{e_0, e_{\xi_c}, e_{q-\xi_c}\}$; (ii) its pair plane $\mathcal{A}_c = \mathrm{span}\{e_{\xi_c}, e_{q-\xi_c}\}$ carries the spin-$\tfrac12$ module $V_{1/2}$ of the binary group acting in (H-inv) (a genuine assumption: the natural swap on the plane has eigenvalues $\pm 1$ while $-1 \in 2I$ must act as $-\mathbf{1}$). (H-carrier) is stated by E2 itself and derived nowhere in the corpus; removal of the zero mode $e_0$ is part of the sector definition.
• Full residual fibre-level support. The full Gram–Schmidt basis is a unitary $q \times q$ matrix whose rows span $\mathbb{C}^q$ at saturation; the residual rank is $r_{\mathrm{pair}}(n) = R_\infty - R(n)$ with $R_\infty = q$. This is the object of the entanglement-note. It is not the theoretical admissible projection.

Core result 1 — Admissible pair entropy under (H-carrier) and (H-inv)

Under two explicit, independent hypotheses — (H-carrier) and (H-inv) — the proto-state on $\mathcal{A}_c \otimes \mathcal{A}_c$ is the spin-$\tfrac12$ singlet $$ |\Omega_c\rangle = \frac{1}{\sqrt{2}} \bigl( |\xi_c\rangle \otimes |q-\xi_c\rangle - |q-\xi_c\rangle \otimes |\xi_c\rangle \bigr), $$ its reduced state is $\rho^{\mathrm{adm}}_c = \tfrac12 \mathbf{1}_2$, and $$ S_{\mathrm{ent}}^{\mathrm{adm}} = \log 2. $$

(H-carrier), stated by E2 itself, is the two-clause hypothesis that the admissible pair sector is carried by the atomic Fourier triple and that its pair plane carries the spin-$\tfrac12$ module $V_{1/2}$ of $2I$; it is derived nowhere in the corpus. (H-inv), stated and left open by Q3, is diagonal $2I$-invariance of the proto-state; under it, Q3's Theorem 3.2 identifies the unique invariant vector as the singlet via Schur's lemma applied to the Clebsch–Gordan decomposition $V_{1/2} \otimes V_{1/2} = V_0 \oplus V_1$. Neither hypothesis is derived from Born–Infeld indiscernibility, from admissibility, or from any other principle in the corpus. The $\log 2$ statement is conditional on both (H-carrier) and (H-inv) for general canonical blocks.

Core result 2 — Full residual-rank entropy

The residual-rank statement $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ has a stratified status:

• Matched single-character blocks. Proved unconditionally by E1 (Theorem 7): the sign-reflected Fourier supports $S_{q-c}^{\mathrm{res}}(n) = -S_c^{\mathrm{res}}(n)$ canonically identify the residual modes and yield a maximally entangled residual state.
• Independently sampled canonical blocks. The literal mode-by-mode identification is not automatic. O17 gives equality of residual dimensions within its own scalar toy model — O17's own scope section states no argument connects that model to the real pipeline, so this is carried as a standing gap; universal admission orthogonality (E1 Theorem 4) gives orthonormal residual bases. Together they do not fix the joint state or force a flat marginal.
• Conditional bridge $[\mathrm{H}_{\mathrm{res}}]$. Under residual admissibility indiscernibility — no invariant carried by the admissible generators distinguishes the residual fingerprint directions — and assuming O17's equal-dimension result transfers to the real pipeline, the reduced state is $\rho_c^{\mathrm{res}}(n) = r_{\mathrm{pair}}(n)^{-1} \mathbf{1}_{r_{\mathrm{pair}}(n)}$ and $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ for general canonical blocks.
• Candidate metaplectic dilation excluded. The map $\phi_\omega(a, b, z) = (\omega a, \omega^{-1} b, z)$, considered directly here, acts between blocks (relating $c$ to $\omega c$), not within the residual support of a single block. It cannot make the residual modes admissibility-indiscernible and is therefore not a bridge for $[\mathrm{H}_{\mathrm{res}}]$.

$[\mathrm{H}_{\mathrm{res}}]$ has a status parallel to the conditional hypotheses $[\mathrm{H\text{-}color}]$ and $[\mathrm{H\text{-}ext}]$ elsewhere in the programme.

Position in the programme

Constituent of the entanglement sub-programme. Builds on E1 (orthogonality and matched-block flat spectrum), Q5a-O2 (pure-Fourier-mode structure of the pre-saturation pipeline), O23 (conditional adjoint-dimension theorem on a supplied carrier), O18 (equivariant parity and fixed-probe no-go), O17 (equal residual dimensions, own toy model), the candidate metaplectic dilation formerly used in O31 (excluded here without relying on O31), and Q3 (singlet identification, conditional on (H-inv)); (H-carrier) is E2's own explicit hypothesis.

References

Beau, J. Two Entanglement Entropies of the Conjugate Weil Pair: the Admissible Pair Sector and the Full Residual Rank. Working paper, 2026. https://doi.org/10.5281/zenodo.20499440