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

We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group Heis₃(Z/qZ), q an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph Cay(Heis₃(Z/qZ), Sq), the orbit vectors generated by conjugate Weil sectors c and $q-c$ span identical subspaces of Cq, 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 ρq-c=ρc̄ 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.