Uniformity Requirements for Weil Fingerprint Energies: Obstructions to the Lipschitz Route to [U]

Q10 proposes a conditional route from the O-series spectral universality hypothesis [U] toward the isotropy identification AH = 2; its coefficient and geometric steps require a separate audit: maxc |σc(n) - σ_*(n)| ≤ ε(q)\,σ_*(n) uniformly over the fitting window n ≤ n_*(q). Version 1 of this paper claimed a proof of [U] with rate O(q-1/2), resting on a Lipschitz continuity of the Weil fingerprint energy in the reduced character θ = c/q, itself derived from an operator perturbation estimate on the Heisenberg Schrödinger generators. That derivation is false, and this version withdraws the claim. We prove instead an exact obstruction: for an odd prime q, the multiplication generators of the finite Heisenberg Schrödinger representations at any two distinct central characters are equidistant, at operator distance exactly 2cos(π/2q), independently of how close the characters are. This equidistance rules out a modulus inferred from individual-generator distances. A separate adjacent-character calculation shows that their summed operator also has no uniform modulus in θ; hence the equicontinuity input of the former proof is unavailable. Two further gaps are recorded and retyped rather than repaired: the fingerprint energy of O25 is an integer Gram Schmidt rank increment on a three-component block, which version 1 silently replaced by a compressed operator norm; and the O(q-1/2) Gromov Hausdorff rate attributed to Q5b appears in neither cited theorem, both of which are rate-free. Hypothesis [U], its rate, and the identification AH = 2 are open on the inputs available here; nothing below shows [U] to be false. Interpretive outlook. The obstruction is informative rather than merely negative: it locates universality, if it holds, in the rank increment of the fingerprint filtration rather than in any norm continuity of the representations, and it states the two estimates a future proof must supply.