Uniform Spectral Universality for Weil Fingerprint Energies: Proof of [U] with Rate O(q^{-1/2})

Overview

Hypothesis [U] — the uniform spectral universality of Weil fingerprint energies — is a foundational assumption for several results in the geometric emergence cluster. It asserts that, as the prime $q$ grows, the fingerprint energies of the Weil representation on Heisenberg graphs converge uniformly, independently of the character.

U1 upgrades [U] from a working hypothesis to a proved theorem. The proof combines two ingredients:

A Weil-generator Lipschitz bound: the Weil generators act with bounded Lipschitz constant on the space of fingerprint energies, controlling how fast energies can vary as the character changes.
BFS–Carnot–Carathéodory convergence: BFS distance on the Heisenberg graph converges to the Carnot–Carathéodory metric, providing the geometric substrate for the uniformity argument.

The result is a quantitative rate $\varepsilon(q) = O(q^{-1/2})$ for the deviation from universality.

Status. Proved.

Core contributions

Proof of [U]: hypothesis [U] is proved rigorously, converting a working assumption into a theorem with explicit quantitative control.
Weil-generator Lipschitz bound: the Weil generators satisfy a Lipschitz condition on fingerprint energy space, bounding character-to-character variation.
BFS–Carnot–Carathéodory convergence: as $q \to \infty$, the BFS graph metric on the Heisenberg group converges to the sub-Riemannian Carnot–Carathéodory distance, providing the geometric backbone of the universality proof.
Explicit rate $O(q^{-1/2})$: the convergence rate is not merely qualitative but quantified, enabling downstream applications (Q10, W1) to bound error terms explicitly.

Foundational role in the geometric emergence cluster

U1 is a hub paper: its proof of [U] with explicit rate unlocks three downstream results simultaneously:

Q10 (su(2)-isotropy): uses [U] to force character-independence of the large-$q$ effective quadratic form, establishing $A_H = 2$.
W1 (Dirichlet form stabilisation): uses [U] to prove hypothesis [H-w], showing that admissibility weights $a_q(s)$ converge to a positive limit.
O30 (equatorial correction): uses the rate $O(q^{-1/2})$ to bound the equatorial correction to the spectral admissibility constraint.

Before U1. [U] was a standing hypothesis. After U1, it is a theorem, and all results conditional on [U] become unconditional (subject only to Q5a hypotheses).

Relation to the Cosmochrony programme

U1 resolves one of the key open hypotheses in the Q5a admissibility framework. Before U1, several steps in the metric emergence chain (Q10, W1) were conditional on [U]. After U1, those steps are fully justified.

The proof method — Lipschitz bounds on Weil generators combined with geometric convergence to a sub-Riemannian limit — is characteristic of the Cosmochrony approach: structural properties of the Heisenberg group provide the analytic control needed to pass from discrete spectral data to continuous geometric structures.

References

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