This repository contains the source of the Q5a Cosmochrony paper Canonical Fourier Filtration and the Obstruction to a Spatial Continuum Limit — What the Admissible Fibre Does and Does Not Converge To.
The paper addresses the Q5 problem posed in the Foundation paper — how the discrete admissible fibre (F_n \simeq V_\rho \subset L^2(\mathbb{Z}/q\mathbb{Z})) relates to continuous structure in the large-(q) limit — and answers a sharper preliminary question: what does the canonical filtration actually converge to, under the published admissibility form and normalisation?
- Exact identification (proved). With the pipeline's initial vector (the uniform state), the canonical filtration stage is exactly the toric Fourier window (\Omega_n = \mathrm{span}{e^{2\pi i b x/q} : |b| \le n}), of dimension (\min(2n+1, q)). Every fixed toric mode is captured once the published saturation depth (n_1(q)) exceeds its index; balanced (line-scale) profiles are rejected at all measured primes. The stage's physical bandwidth is exactly (\sqrt{2\pi},(x_1(q)/C_{\mathrm{Heis}})^{1/4}), governed by the critical coverage.
- Zero-form theorem (proved). With the published prefactor (q^{-2}) and bounded weights, the admissibility form tends to zero uniformly on norm-bounded sets; its Mosco limit is the zero form.
- Normalisation no-go (proved under the measured weight behaviour). If the modulation and translation weights both remain positive and of order one, no common scalar normalisation produces a non-trivial finite toric differential operator: preserving the derivative sector makes the modulation sector diverge; preserving the modulation sector eliminates the derivative. An anisotropic renormalisation is not derived anywhere in the corpus and would be a new input.
- Dual window limit (conditional). Under the frequency blow-up (u = b/n_1(q)), the rescaled form converges to a Dirichlet form on (L^2([-1,1])) in the rescaled frequency variable, with a potential term controlled by the critical coverage. A dual-space statement: it does not produce a spatial continuum.
Q5 therefore remains open. The downstream identification of a flat spatial co-metric from a limit operator (-A\partial_x^2) on (L^2(\mathbb{R})) (companion paper Q5b) rests on an input that is not established here. A balanced-scale route remains on record as a conditional possibility, tied to a future non-vanishing asymptotic bound on the critical coverage (x_1(q)) — never as a current result.
The numerical statements (balanced-profile rejection at the measured primes) are produced by
the deterministic script code/canonical_sector_test.py (published depths only, no
recalibration). Note: the script imports spectral_O12.py from the O25 repository
(admissibility/o25/code/ in the Cosmochrony workspace); reproducing it requires that
repository alongside this one.
J. Beau, Canonical Fourier Filtration and the Obstruction to a Spatial Continuum Limit, Zenodo, 2026. DOI: 10.5281/zenodo.19642369
- 🌐 Programme website: https://cosmochrony.org
- 💻 GitHub organization: https://github.com/Cosmochrony
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.