Entanglement E2 — Two Entanglement Entropies of the Conjugate Weil Pair

Two distinct objects

• Admissible pair sector. The rank-3 admissible projection $\Pi_q$ is spectrally atomic (O21 Proposition 4.1): $\mathrm{ran}(\Pi_q^{(c)}) = \mathrm{span}\{e_0, e_{\xi_c}, e_{q-\xi_c}\}$. The mean-zero lemma removes $e_0$, leaving the two-dimensional Born–Infeld parity pair $\mathcal{A}_c = \mathrm{span}\{e_{\xi_c}, e_{q-\xi_c}\} \cong V_{1/2}$.
• 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

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

The proof uses spectral atomicity (O21), the mean-zero lemma, and the Born–Infeld indiscernibility singlet identification of Q3 (Theorem 4.1) applied to the Clebsch–Gordan decomposition $V_{1/2} \otimes V_{1/2} = V_0 \oplus V_1$. The statement is unconditional for general canonical blocks, relative to the cited upstream theorems.

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; 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 — 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.
• Metaplectic dilation excluded. The dilation $\phi_\omega(a, b, z) = (\omega a, \omega^{-1} b, z)$ of O30 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.

Bell–Tsirelson sector

The entanglement that CHSH experiments measure is carried by the spin-$\tfrac12$ singlet. By the admissible pair entropy theorem, this is exactly the admissible pair sector with $S_{\mathrm{ent}}^{\mathrm{adm}} = \log 2$. The singlet correlator $E(\hat a, \hat b) = -\hat a \cdot \hat b$ and the unconditional Tsirelson bound $|S_{\mathrm{CHSH}}| \leq 2\sqrt{2}$ are derived from the same admissible structure in Q1 (Theorem 2.14, Corollary 2.16). The admissible pair entropy is therefore the entanglement entropy of the state behind those results.

The full residual-rank entropy is a finer pipeline-level observable that tracks the contraction of the Gram–Schmidt span toward saturation. It is not the quantity measured by a Bell experiment, and it should not be identified with the singlet entropy. The two coincide only in the degenerate case $r_{\mathrm{pair}}(n) = 2$.

Position in the programme

Constituent of the entanglement sub-programme. Builds on E1 (orthogonality and matched-block flat spectrum), O21 (spectral atomicity + mean-zero), O17 (equal residual dimensions), O30 (metaplectic dilation, excluded), Q3 (singlet identification), and Q1 (Tsirelson bound).

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