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

Overview

This paper proves a rigidity theorem in finite representation theory, in the finite Weil (oscillator) representation of the Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ acting on $\mathbb{C}^q$. For every complete (inversion-symmetric) Breadth-First-Search shell of the associated Cayley graph, the orbit vectors generated by conjugate character sectors $c$ and $q-c$ span identical subspaces — for the special character block used by the original numerical pipeline and for 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, and independent of any coherence measure. Two no-go corollaries then characterise exactly which diagnostics this rigidity puts out of reach — not only Fourier support, subspace rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the underlying frame, a class that includes normalised Bargmann-type invariants.

Independent numerical verification: 1,548 exact integer-arithmetic checks across six primes with zero failures, plus a negative control confirming the criterion correctly detects failure on a deliberately truncated shell.

Central message. A natural class of representation-theoretic diagnostics — support, rank, Gram-derived quantities — cannot distinguish conjugate Weil sectors, for a structural reason (collapse to single-frequency Fourier lines), not a numerical coincidence. This paper does not derive quantum-mechanical structure.

Core contributions

Exact conjugate-sector Fourier-support rigidity: for every complete BFS shell, conjugate Weil-sector fingerprint vectors span identical subspaces of $\mathbb{C}^q$, proved by a five-step argument reducing every vector to a single pure Fourier mode.
No-go corollaries: no diagnostic built from Fourier support, rank, projectors, dimensions, or principal angles can separate the sectors; more precisely, no diagnostic invariant under independent rephasing and reindexing of the underlying frame can either — proved by an explicit frame-orbit construction, not asserted from Gram-matrix orbit-equality alone.
Independent numerical verification: 1,548 exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell.
Scope: this paper does not derive quantum-mechanical structure — no result here establishes phase coherence, the singlet correlator, the Tsirelson bound, or the Born rule from admissibility.

Interpretation

Support, rank, and Gram-derived quantities are natural, representation-theoretic diagnostics, and one might plausibly have expected some such diagnostic to separate conjugate Weil sectors. This paper shows that expectation fails for a structural reason, not a numerical coincidence: the pure-Fourier-line reduction that proves the rigidity identity simultaneously forecloses the richer phase geometry that would be needed to separate the sectors by any rephasing/permutation-invariant frame diagnostic.

Interpretive outlook (not a further result). Read as a methodological instance for the wider spectral admissibility programme, this episode is a cautionary one: a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism. This is offered as a single instance, not a general claim about other diagnostics used elsewhere in the O-series or Q-series programme. The genuine open question — whether a single Weil/Heisenberg source can produce multiple, phase-distinguishable quantum-like frames by some other construction — remains exactly as open after this paper as before it.

Relation to the Cosmochrony programme

This paper does not derive quantum-mechanical structure, and does not rely on O18 (parity-fibre structure), O19, O21, or O22 as established pillars — an audit of those papers found the fibre-identification and canonical-observable claims they had been cited for invalid or unestablished. What remains is self-contained: a representation-theoretic rigidity theorem about the pipeline's own uniform-probe construction, and its no-go corollaries.

Q2 and Q3 state results built on a central theorem this construction does not establish. Whether and how they are affected is a separate, not-yet-completed audit, flagged for follow-up.

References

Jérôme Beau. Fourier-Support Rigidity in Conjugate Weil Sectors: A No-Go Result for Rank-Based Quantum Signatures, 2026. doi:10.5281/zenodo.19561060