Fourier-Support Rigidity in Conjugate Weil Sectors: A No-Go Result for Rank-Based Quantum Signatures

Abstract

We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, $q$ an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}), S_q)$, the orbit vectors generated by conjugate Weil sectors $c$ and $q-c$ span identical subspaces of $\mathbb{C}^q$, for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by $1{,}548$ exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms $\rho_{q-c}=\overline{\rho_c}$ is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This is a narrow, structural result: it says nothing about phase coherence, the singlet correlator, the Tsirelson bound, or the Born rule under admissibility. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.