U1 — Uniform Spectral Universality for Weil Fingerprint Energies

This repository contains the source of the U1 Cosmochrony paper Uniform Spectral Universality for Weil Fingerprint Energies: Proof of [U] with Rate $O(q^{-1/2})$.

This work closes a technical gap in the emergent-geometry sub-programme. The Q10 paper reduces the proof of spatial isotropy ($A_H = 2$) to a single spectral universality hypothesis:

[U]   $\max_c |\sigma_c(n) - \sigma_(n)| \leq \varepsilon(q),\sigma_(n) \to 0$, uniformly over the fitting window $n \leq n_*(q)$.

The present paper establishes [U] with rate $\varepsilon(q) = O(q^{-1/2})$, turning the isotropy result of Q10 into an unconditional statement.

Conceptual Overview

The proof rests on two independent inputs about the Weil fingerprint energy $\sigma_c(n)$, seen as a function of the reduced character $\theta = c/q$:

1. Lipschitz continuity in $\theta$ (equicontinuity). $\sigma_c(n)$ is Lipschitz in $\theta = c/q$, uniformly in $n$, as a consequence of a standard operator-perturbation estimate on the Weil generators.

2. Pointwise convergence for each fixed $\theta$. $\sigma_c(n) \to \sigma_*(n)$ pointwise, inherited from the BFS–Carnot–Carathéodory convergence of Q5b at rate $O(q^{-1/2})$.

Lipschitz equicontinuity together with pointwise convergence yields uniform convergence on compact subsets $\theta \in [\theta_1, 1/2]$ by the Arzelà–Ascoli theorem. The complementary small-$\theta$ regime $\theta \in (0, \theta_0(q))$ with $\theta_0(q) = q^{-1/2}$ is handled by a Lipschitz triangle argument and requires no Born–Infeld input. Combining the two regimes gives the uniform rate $\varepsilon(q) = O(q^{-1/2})$.

Core Results

1. Uniform universality [U] holds, with explicit rate $\varepsilon(q) = O(q^{-1/2})$.
2. The rate is structural: it is inherited from the $O(q^{-1/2})$ convergence of the underlying sub-Riemannian (Q5b) geometry, not fitted.
3. Isotropy becomes unconditional: the $A_H = 2$ result of Q10 no longer depends on an assumed universality hypothesis.
4. The argument is elementary given its two inputs: perturbative Lipschitz continuity plus Arzelà–Ascoli, with no appeal to the Born–Infeld dynamics.

What This Paper Does Not Assume

• no Born–Infeld input in the small-$\theta$ regime,
• no fitted convergence rate (the $O(q^{-1/2})$ rate is derived from Q5b),
• no uniformity beyond the compact fitting window $n \leq n_*(q)$,
• no dynamical or background-geometry assumptions.

Keywords

Weil representation, Heisenberg group, BFS convergence, Lipschitz continuity, Arzelà–Ascoli, spectral universality, Gromov–Hausdorff convergence.

Repository Contents

u1/
├── tex/         # LaTeX sources (u1.tex, cosmochrony-bibliography.bib, references.bib)
├── out/         # Compiled paper PDF (u1.pdf)
├── compile.sh   # Build script (pdflatex + bibtex)
├── zenodo.json  # Zenodo deposition metadata
└── README.md

Compilation

bash compile.sh
# or manually:
pdflatex -output-directory=out tex/u1.tex
cd out && bibtex u1 && cd ..
pdflatex -output-directory=out tex/u1.tex
pdflatex -output-directory=out tex/u1.tex

Links

• 🔗 DOI: 10.5281/zenodo.19881146
• 🌐 Website: https://cosmochrony.org/science/emergent-geometry/u1/

Citation

If you reference this work, please cite:

J. Beau, Uniform Spectral Universality for Weil Fingerprint Energies: Proof of [U] with Rate $O(q^{-1/2})$, Zenodo, 2026. DOI: 10.5281/zenodo.19881146.

Acknowledgements

Portions of the editorial refinement benefited from iterative interactions with large language models, used as analytical assistants for exploring alternative formulations, checking internal consistency, and improving clarity. All claims, interpretations, and final formulations remain the sole responsibility of the author.

Contributions

This repository is intended as a research reference. Critical feedback, independent analyses, and formal scrutiny are welcome. Please open an issue to discuss the Lipschitz continuity estimate, the Arzelà–Ascoli argument, the small-$\theta$ regime, or the inherited $O(q^{-1/2})$ rate.