Q1 — Fourier-Support Rigidity in Conjugate Weil Sectors

This repository contains the source of the Q1 Cosmochrony paper Fourier-Support Rigidity in Conjugate Weil Sectors: A No-Go Result for Rank-Based Quantum Signatures.

For every complete (inversion-symmetric) BFS shell of the Heisenberg Cayley graph $\mathrm{Cay}(\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z}), S_q)$, $q$ an odd prime, 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 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, independent of any coherence measure.

Two no-go corollaries characterise exactly which diagnostics this rigidity forecloses: Fourier support, subspace rank, projectors, dimensions, and principal angles; and, more precisely, any quantity invariant under independent rephasing and reindexing of the underlying frame — a class that includes normalised Bargmann-type invariants (proved trivial by an explicit frame-orbit argument), but explicitly excludes phase-gauge-dependent quantities such as a raw Gram-matrix entry.

Verified independently by 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.

What this paper does not claim: it does not derive Hilbert-space structure, the singlet correlator, the Tsirelson bound, the Born rule, or non-locality. See the paper's Introduction for the full list.

Weil representation, Heisenberg group, Cayley graph, Fourier support, conjugate character sectors, no-go theorem, frame theory, Gram matrix, rank-based diagnostics.

q1/
├── code/ # Numerical validation scripts (+ requirements.txt)
├── tex/ # LaTeX sources (main + cosmochrony-bibliography.bib + external-refs.bib)
├── out/ # Compiled paper PDF (Q1-paper.pdf)
├── zenodo.json # Zenodo deposition metadata
└── README.md

Key scripts: exact_fourier_support.py (the exact integer-arithmetic proof, multi-prime sweep), test_regression.py (full regression suite, including the frame-invariance and negative-control checks), frame_invariance_check.py (frame-orbit / Bargmann-invariant verification), separator.py (numeric SVD cross-check), weil_pipeline_compat.py (reconstructed pre-correction pipeline, used only as the negative control).

cd code
pip install -r requirements.txt
# Full regression suite (all checks below, plus shell/coherence sanity checks
# and the coordinate-dependent-phase negative control)
python3 test_regression.py
# The multi-prime exact sweep reported in the paper (1,548 checks, 0 failures)
python3 exact_fourier_support.py --primes 13,17,29,37,53,61
# Frame-orbit / Bargmann-invariant check on its own (q=29)
python3 frame_invariance_check.py

All checks use fixed, deterministic seeds where randomness is involved (e.g. np.random.default_rng(0) in the negative control) and report exit code 0 on success.

• 🔗 DOI: 10.5281/zenodo.19561060 (concept DOI)
• 🌐 Website: https://cosmochrony.org/science/quantum-structure/q1/

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

Portions of the editorial refinement benefited from iterative interactions with large language models, used as analytical assistants. All claims and final formulations remain the sole responsibility of the author.