Weil Fingerprint Orthogonality and the Flat Schmidt Spectrum of Conjugate Pairs

Abstract

The companion entanglement paper established numerically that the normalized Gram–Schmidt admission residual satisfies $w_j=1$ for every admitted Weil fingerprint direction, across $q\in\{29,61,101,151\}$ 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 ${\mathrm{Heis}}_3(\mathbb{Z}/q\mathbb{Z})$ for any block $(c_1,c_2,c_3)$ is, up to a global phase, the discrete Fourier mode ${\hat{u}}_A = (e^{2\pi iAt/q})_{t=0}^{q-1}$ at frequency $A = c_1b_1+c_2b_2+c_3b_3 \bmod q$, where $(b_1,b_2,b_3)$ 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 $w_j = \|w^{\mathrm{GS}}_j\|/\|v_j\| = 1$ exactly. For matched single-character blocks, the conjugate pair produces sign-reflected frequency sets, so the paired Fourier supports are canonically identified; conditional on the companion paper's supplied purity of the joint state, this fixes the diagonal Schmidt-partner pairing. 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, conditional on the companion paper's supplied purity of the joint state, the Schmidt spectrum of the conjugate pair is flat on the residual support for matched blocks, and the reduced entropy is ${S_{\mathrm{ent}}}(n)=\log{r_{\mathrm{pair}}}(n)$, where ${r_{\mathrm{pair}}}(n)=R_\infty-R(n)$ is the residual rank.