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

Abstract

The Q10 paper of the Cosmochrony programme reduces the proof of isotropy $A_H = 2$ to the O-series 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 $\varepsilon(q) = O(q^{-1/2})$. The proof rests on two inputs: the Lipschitz continuity of the Weil fingerprint energy $\sigma_c(n)$ in the reduced character $\theta = c/q$ (Lemma lem:lipschitz), which follows from a standard operator perturbation estimate on the Weil generators; and the pointwise convergence $\sigma_c(n) \to \sigma_*(n)$ for each fixed $\theta$, which follows from the BFS Carnot Carath\'eodory convergence of Q5b at rate $O(q^{-1/2})$. Lipschitz equicontinuity plus pointwise convergence then yields uniform convergence on compact subsets $\theta \in [\theta_1, 1/2]$ by the Arzel\`a Ascoli theorem, while the small-$\theta$ regime $\theta \in (0, \theta_0(q))$ with $\theta_0(q) = q^{-1/2}$ is handled by a Lipschitz triangle argument, requiring no Born Infeld input. Together these give $\varepsilon(q) = O(q^{-1/2})$, closing the programme.