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

The Q10 paper of the Cosmochrony programme reduces the proof of isotropy AH = 2 to the O-series spectral universality hypothesis [U]: maxc |σc(n) - σ_*(n)| ≤ ε(q)σ_*(n) → 0 uniformly over the fitting window n ≤ n_*(q). The present paper establishes [U] with rate ε(q) = O(q-1/2). The proof rests on two inputs: the Lipschitz continuity of the Weil fingerprint energy σc(n) in the reduced character θ = c/q (Lemma 3.2), which follows from a standard operator perturbation estimate on the Weil generators; and the pointwise convergence σc(n) → σ_*(n) for each fixed θ, which follows from the BFS-Carnot-Carathéodory convergence of Q5b at rate O(q-1/2). Lipschitz equicontinuity plus pointwise convergence then yields uniform convergence on compact subsets θ ∈ [θ₁, 1/2] by the Arzelà-Ascoli theorem, while the small-θ regime θ ∈ (0, θ₀(q)) with θ₀(q) = q-1/2 is handled by a Lipschitz triangle argument, requiring no Born-Infeld input. Together these give ε(q) = O(q-1/2), closing the programme.