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

Correction notice. This is a major revision (v2.0) of a paper originally titled From Admissibility to Quantum Structure: Phase Coherence and Correlations from Non-Injective Projection (v1.1.2). Independent review found that version's central theorem's published proof unsound, and separately found its Born-rule derivation independently defective. Both are withdrawn. What survives is narrower and unconditional: an exact rigidity theorem and its no-go corollaries, described below.

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.

Quick facts

Author

Jérôme Beau

Work type

Preprint (major correction, v2.0)

Focus

Weil representation, Heisenberg Cayley graph, Fourier support, conjugate character sectors, no-go theorem, frame theory

Key result

$\widehat V_n^{(c)} = \widehat V_n^{(q-c)}$ exactly, for every complete shell

Status

Proved; verified across 1,548 exact checks, $q \in \{13,\ldots,61\}$

Withdrawn (v1.1.2)

Central theorem, Gram–Schmidt rank corollary, Born rule

Open

Genuine multi-basis emergence; the abstract-object bridge

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.
What is withdrawn: v1.1.2's central theorem (its published proof was found unsound), its Gram–Schmidt rank corollary (dimensionally impossible as stated), and its Born-rule derivation (independent proof defects). The singlet correlator and Tsirelson bound survive only as a standard, explicitly conditional textbook remark — not a result of this paper.

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 — that is exactly what v1.1.2 attempted to construct, reading a numerical near-zero residual as an "admissibility-preserved phase coherence" signature. 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 no longer claims to derive quantum-mechanical structure and no longer relies on O18 (parity-fibre structure), O19, O21, or O22 as established pillars — an independent 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.

v1.1.2 had opened the Q-series thematic track on emergent quantum structure, with Q2 and Q3 stated as building on its central theorem. Since that theorem is withdrawn here, whether and how Q2 and Q3 are affected is a separate, not-yet-completed audit — flagged for follow-up, not resolved by this revision.

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