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

Current Zenodo release: version 2.1 (2026-10-26). Official title: Uniformity Requirements for Weil Fingerprint Energies: Obstructions to the Lipschitz Route to [U].

Overview

Hypothesis [U] asks for a relative, uniform character-independence estimate for Weil fingerprint energies on Heisenberg graphs. Version 1 of U1 claimed to prove it with rate $O(q^{-1/2})$. Version 2.0 withdraws that proof and the rate; [U] remains open.

U1 instead proves an exact obstruction for the finite Heisenberg–Schrödinger representations: multiplication generators at any two distinct central characters have operator distance $2\cos(\pi/2q)$, tending to $2$. This concerns individual generators. A separate adjacent-character example shows that their summed operator also lacks the uniform modulus claimed by the former argument.

Status. The generator obstruction is proved without Q5a–Q5b hypotheses; [U] and its proposed rate are open. U1 does not show [U] to be false.

Core contributions

Individual generators: for distinct $c,c'$ at prime $q$, $\|\rho_{q,c}(X)-\rho_{q,c'}(X)\|=2\cos(\pi/2q)$, independently of the pair.
Summed operator: cancellation can occur for $c'=-c$. Separately, for $c=1,c'=2$, its distance is at least $2\cos(\pi/q)+2\cos(2\pi/q)\to4$ while $|c-c'|/q\to0$.
Observable mismatch: O25 measures a Gram–Schmidt rank increment on sampled three-component blocks, not an identified compressed operator norm.
Unsupported rate: the cited Q5b convergence result is qualitative; it supplies no $O(q^{-1/2})$ Gromov–Hausdorff rate or transfer to the O25 count.

Consequences for the geometric emergence cluster

U1 withdraws its claim to close Q10 and the Q7 bridge. Q10 version 2.0.1 keeps [U] as an explicit open input, shows in its Proposition 3.2 that the inputs cited for [U] do not imply it, and derives no coefficient value. W1 version 2.0.1 withdraws the weight-stabilisation claim that imported U1's withdrawn quantitative rate. Any other paper using that rate needs its dependence checked at its own stage.

Interpretive outlook. The obstruction concerns this operator-norm route, not spectral universality itself. A proof of [U] would need a comparison to the actual sampled-block rank increment and a quantitative relative estimate.

Relation to the Cosmochrony programme

The result separates a proved statement about finite Heisenberg generators from the still-open uniformity of Weil fingerprint energies. The associated Weil action is distinct from the Heisenberg multiplication and shift representation used in the obstruction. The programme registry records this distinction and leaves the downstream coefficient and metric identifications pending.

References

Jérôme Beau. Uniformity Requirements for Weil Fingerprint Energies: Obstructions to the Lipschitz Route to [U], version 2.0, 2026. doi:10.5281/zenodo.19881146