# Verification and scope

A Three-Round Peeling Bound for Internal Partitions of 5- and 8-Regular Graphs

The five-page English manuscript gives a complete written proof of the threshold 500000 for simple 5- and 8-regular graphs, refining Lin-Zeng's stated 10000000 bound. It does not solve arbitrary regular degree or claim new existence results for degrees 5 and 8. Whole source: PARTIALLY_SOLVED. New source closures: zero.

Detailed originating-researcher self-audit covered cut signs, unequal bisection sizes, exceptional-set vertices, overlapping predecessor witnesses, all three deletion rounds, integer rounding, strict surplus and the nonempty core extension. No remaining mathematical gap was found. This is not independent review or proof-assistant verification.

The final checker uses integer/rational arithmetic and networkx 3.5. Normal and optimized runs agree on 796323 verified conditions. The 136 eligible atlas graph/parameter combinations yield 13232 accepted local-premise configurations, including 1102 with all three layers nonempty. Eighty-eight seeded regular graphs produce 55656 enumerated bisection cuts and 1473 no-core bisections. The scalar domain contains 100001 consecutive N values from 250000 and 1000 seeded larger values. The written positive-slope inequalities, not the finite samples, prove all N>=250000.

All five final rendered pages were inspected. Built-in and portable LaTeX compilation succeed; the portable PDF has identical page content streams and extracted text. The source ZIP contains the checker, baselines, proof audit, limitations and source hashes. Third-party full texts are excluded. The author line is only Alper Ferudun; Mercury Software GmbH and contacts are in a footnote. AI assistance is disclosed in ordinary prose without a personal human-review claim. License: CC BY 4.0. Bounded novelty review is not absolute priority certification.