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

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, and which now carry different epistemic status. The first is the entropy of the minimal admissible pair sector. The carrier of that sector is fixed by an explicit hypothesis, here labelled (H-carrier): the admissible sector of each factor is the atomic Fourier triple $\mathrm{span}\{e_0,e_{\xi_c},e_{q-\xi_c}\}$. The hypothesis is motivated but not derived. The pre-saturation pipeline proves that every selected Gram–Schmidt direction is a pure Fourier mode (Q5a-O2), so an atomic carrier is the natural candidate; but the rank-three input is a supplied selection rule, not a derived constant (O23), the pipeline's three-coordinate reduction is a truncation fixed in advance, and no mean-zero theorem removes the zero mode – the passage to the pair sector is part of the sector definition. Under (H-carrier) — which includes, as an explicit clause, the identification of the pair sector $\{e_{\xi_c},e_{q-\xi_c}\}$ with the spin-$\tfrac12$ module $V_{1/2}$ of the binary group acting in (H-inv) — the pair sector carries the singlet analysis below. Under the further, independent hypothesis (H-inv) that the proto-state on this sector is invariant under the diagonal $2I$-action – stated and left open, not derived, by Q3 – the proto-state is the singlet, its reduced state is maximally mixed, and $S_{\mathrm{ent}}^{\mathrm{adm}}=\log 2$ for general canonical blocks. This admissible pair entropy is therefore conditional on both (H-carrier) and (H-inv), not unconditional. The second object 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 – this is the one unconditional result of the present note; for independently sampled canonical blocks it remains conditional on the residual admissibility indiscernibility hypothesis $[\mathrm{H}_{\mathrm{res}}]$, and on extending O17's equal-residual-dimension result beyond its own toy model to the real admissibility pipeline. 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.