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 ûA = (e2π iAt/q)ₜ₌₀q-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 = \|wGSⱼ\|/\|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 Sₑₙₜ(n)=ₚₐᵢᵣ(n), where rₚₐᵢᵣ(n)=R_∞-R(n) is the residual rank.