# Verification report

Title: Strong Approximate Innerness for Diagonal Graph Actions without Aperiodicity

Author: Alper Ferudun, Mercury Software GmbH. Manuscript date: 18 October 2026. Problem: OWR-15585-006. Version: 1.0.

The complete result concerns finite abelian edge-scaling actions on finite strongly connected noncycle graph C*-algebras, and consequently diagonal quasi-free actions on simple purely infinite Cuntz--Krieger algebras. It removes the period and outerness assumptions from the applicable Barlak--Szabo theorem by explicit reductions. It does not claim a result for arbitrary actions, nonabelian groups, or approximate representability.

The mathematical self-audit checks fixed-full-corner lifting, restriction of outerness, the return-graph isomorphism, simplicity of the outer fixed algebra, diagonal vertex coboundaries, character extension through nonsplit subgroups, and both polar corrections. All fixedness hypotheses are tracked. The exact source question was checked against PDF page 21 / printed page 2621 of Oberwolfach Report 42/2017. Barlak--Szabo Corollary 2.8 is the essential external analytic input, not a new result of this paper.

The included verify_finite.py uses exact integer arithmetic and standard-library group enumeration. It checks 149 small strongly connected zero-one graphs, 3,194 finite actions, one specified nonsplit C4 example and 16 matrix-unit frame identities. The normal and optimized Python runs must agree. These computations check finite identities only; the general C*-algebra proof is written in the manuscript. No theorem prover or independent human referee was used.

The source package includes the manuscript, this report, the exact script and research scope/audit notes. Third-party PDFs and extracted third-party source text are not redistributed. A clean extracted-source build and direct inspection of all seven PDF pages are recorded in the local build record before release. The bounded literature review did not locate an earlier statement of this full extension; no absolute-priority guarantee follows from that search. Corrections and prior-work pointers are welcome.

Reproduction: compile main.tex with Tectonic or a standard LaTeX installation, then run `python3 reproducibility/verify_finite.py` and `python3 -O reproducibility/verify_finite.py`. Both scripts should return identical PASS reports.