# Verification report

Date: 2026-09-01

## Mathematical result

`PASS`: the source display and its modulus correction were checked against
final primary text. The analytic audit covers automatic Riesz mass, potential
normalization, complex scaling including the possible atom at zero, Bochner
duality for the bilinear pairing, equality of every projected law, the
countable-dense support argument, global recovery and uniqueness, logarithmic
poles, and realization in every fixed II_1 factor.

## Claim boundary

The theorem resolves bounded commuting tuples and compact positive probability
measures for the real log-modulus kernel. It excludes from its claim the
literal complex-log typo, signed or complex measures, unbounded affiliated
operators, and arbitrary branch domains. No computational claim is used.

## Build and PDF QA

`PASS`: Tectonic 0.17.0 and BibTeX produced a six-page letter-size PDF with
zero LaTeX errors, undefined references/citations, overfull boxes, or
underfull boxes. All six pages were rendered at 144 DPI and directly
inspected; no clipping, margin, formula, bibliography, or legibility defect
was found.

## Public release and license

The author approved public release on 2026-09-02. This paper, its source files, and this verification report are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/