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

The companion entanglement note and the orthogonality paper E1 identify two distinct entanglement entropies of the conjugate Weil pair ,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 mean-zero lemma is unconditional: the zero mode is always removed by admissibility. Spectral atomicity itself – that the admissible projection Πq equals span₀,eξc,eq-ξc\ – is confirmed numerically to machine precision for the primes tested, but its general analytical proof rests on O23's admissible-rank-exactly-three result, which O23 itself states as conditional on the absence of additional effective symmetries beyond Born–Infeld parity, within the scope of O18 – an explicit hypothesis here labelled (H-rank). Under (H-rank), removing the zero mode leaves the two-dimensional Born–Infeld parity pair ξc,eq-ξc\, identified with the spin-12 representation. 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 Sₑₙₜᵃᵈᵐ=log 2 for general canonical blocks. This admissible pair entropy is therefore conditional on both (H-rank) and (H-inv), not unconditional. The second object is the full residual fibre-level rank rₚₐᵢᵣ(n)=R_∞-R(n) defined on the full Gram–Schmidt basis (rank q at saturation, not the rank-3 admissible projection). E1 proves Sₑₙₜ(n)=log(rₚₐᵢᵣ(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 [Hᵣₑₛ], and on extending O17's equal-residual-dimension result beyond its own toy model to the real admissibility pipeline. The metaplectic dilation φ_ω is excluded as a bridge for [Hᵣₑₛ] because it acts between blocks, not within the residual support of one block.