Weil Fingerprint Orthogonality and the Flat Schmidt Spectrum of Conjugate Pairs

The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies wⱼ=1 for every admitted Weil fingerprint direction, across 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=3 Weil fingerprint of Heis₃ (Z/qZ) for any block (c₁, c₂, c₃) is, up to a global phase, the discrete Fourier mode uA = (e^2 iAt/q) ₓ=₀^q-1 at frequency A = c₁b₁+c₂b₂+c₃b₃ q, where (b₁, b₂, b₃) are the b-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 = \|w^GSⱼ\|/\|vⱼ\| = 1 exactly. For matched single-character blocks, the conjugate pair produces sign-reflected frequency sets, so the paired Fourier supports are canonically identified and the purity hypothesis of the companion paper is derived. 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 the Schmidt spectrum of the closed conjugate pair is flat on the residual support, and the reduced entropy is S₄₍ₓ (n) =₏₀₈ₑ (n), where r₏₀₈ₑ (n) =R_-R (n) is the residual rank.