Overview
The companion entanglement-note 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.
This paper closes it by explicit computation. The $k=3$ Weil fingerprint of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ for any block $(c_1, c_2, c_3)$ factors as a global phase times a single discrete Fourier mode at frequency $$ A = c_1 b_1 + c_2 b_2 + c_3 b_3 \bmod q. $$ Since distinct Fourier modes are orthogonal, any newly admitted fingerprint is already orthogonal to the current span, and $w_j = 1$ is an exact algebraic identity — not a numerical accident.
Core results
- Fingerprint as Fourier mode (Lemma). $\mathbf{f}_t = \frac{e^{2\pi i B/q}}{q} \cdot e^{2\pi i A t/q}$, with $A = c_1 b_1 + c_2 b_2 + c_3 b_3 \bmod q$ and a $t$-independent phase $B$. The frequency $A$ carries all $t$-dependence; the prefactor $1/q$ and phase $B$ are scalar multiples.
- Universal admission orthogonality (Theorem 4). For any fingerprint $\mathbf{f}$ at depth $n$ linearly independent of the prior span $\mathcal{V}_{n-1}$, $$ w_j = \frac{\|w^{\mathrm{GS}}_j\|}{\|\mathbf{f}\|} = 1 \quad \text{exactly}. $$ The Gram–Schmidt process produces an orthonormal subset of the Fourier basis of $\mathbb{C}^q$ and saturates at rank $q$.
- Conjugate frequency symmetry. For matched single-character blocks $(c, 0, 0)$ and $(q-c, 0, 0)$, $A^{(q-c)} = -A^{(c)} \bmod q$ and $V^{(q-c)}_n = \overline{V^{(c)}_n}$.
- Flat Schmidt spectrum (Theorem 7). Under a supplied purity of the joint state, which E1 does not derive, the residual bipartite fibre carries the maximally entangled state $$ |\Omega^{\mathrm{res}}_n\rangle = \frac{1}{\sqrt{r_{\mathrm{pair}}(n)}} \sum_{A \in S_c^{\mathrm{res}}(n)} |\hat e_A\rangle \otimes |\hat e_{-A}\rangle, $$ and the reduced entropy is $S_{\mathrm{ent}}(n) = \log r_{\mathrm{pair}}(n)$ — derived analytically, not merely from the observed vanishing of an admissibility deficit.
Epistemic status update for the companion note
Two items previously listed as open in the entanglement-note are now resolved:
- Analytic proof of $w_j = 1$. Previously numerical ($q \in \{29, 61, 101, 151\}$); now proved (Theorem 4).
- Schmidt-partner pairing of the conjugate pair. Purity of the joint state remains a supplied composition structure, not derived here. What Theorem 7 derives, conditional on that supplied purity, is the diagonal Schmidt pairing $\hat e_A \leftrightarrow \hat e_{-A}$ for matched blocks, from the Fourier-mode structure and conjugation $\rho_{q-c} = \overline{\rho_c}$.
Remaining open item. The matched-block purity argument does not extend automatically to the independently sampled blocks of the canonical O25 pair observable. The O17 equal-count identity ensures equal rank at every shell, but the literal Schmidt-partner pairing $\hat e_A \leftrightarrow \hat e_{-A}$ requires either a canonical block-parameter choice or an additional structural argument.
Mechanism
The algebraic source of admission orthogonality is the linear $t$-dependence of the Weil phase in the Schrödinger representation formula: $(\rho_c(a, b, \gamma) v_0)_t \propto e^{2\pi i c (\gamma + b(t-a))/q}$. Linearity in $t$ yields a pure Fourier mode after the elementwise product of three single-step Weil actions; the orthogonality of distinct Fourier modes over $\mathbb{Z}/q\mathbb{Z}$ then forces $w_j = 1$.
The result is a property of the Weil representation itself, not of the spectral pipeline.
Position in the programme
Constituent of the entanglement sub-programme. Depends on the Weil-block projective capacity machinery of O11 and O12, and uses the equal-count identity of O17 to delimit the open Schmidt-pairing question for general blocks.
References
Beau, J. Weil Fingerprint Orthogonality and the Flat Schmidt Spectrum of Conjugate Pairs. Working paper, 2026. https://doi.org/10.5281/zenodo.20499309