# Verification report

Manuscript: A Six-Vertex Counterexample to Vertex-Averaged Ordered Reachability.
Author: Alper Ferudun, Mercury Software GmbH. Date: 28 September 2026.
Corpus identifier: AIM-COMBINATORICS-0177. Unrefereed preprint.

The two layer-neighborhood columns in the manuscript define the entire
witness. Every row in each layer has two distinct neighbors and excludes
its starting vertex. E1 intersect reverse(E2) is empty. With only two labels
this excludes all rainbow cycles. Strictly increasing paths have at most
two edges, so the displayed union formula exhausts all possible endpoints.
The six support sizes are 5,5,5,5,5,4, with sum 29 rather than at least 30.

The supplied standard-library checker recursively enumerates all increasing
walks and separately computes the integer product (I+A1)(I+A2). Each route
agrees with the fixed table. All 720 vertex permutations were checked.
Uniform blow-ups for m=1,...,12 agree with total 6m+23m^2; the manuscript
proves the formula for every positive integer m by lifting base paths.
Two executions, Python 3.13.5 and 3.12.14, passed 1599 assertions each.

The reversed order has total 35. Averaging both orders gives 16/3>5,
explicitly showing that the witness is not a counterexample to that variant.
The upstream five-vertex pointwise witness totals 21 against a threshold of
20, retained as a negative control. Solver optimality was not used in the
proof and no minimum-order claim is made.

The primary-source wording omits its averaging variable. The proved result
addresses only the stated fixed-order, uniform-start interpretation. It is
not a disproof of the Caccetta-Haggkvist conjecture or the unordered rainbow
path conjecture. The source and bounded novelty review are in the ZIP.

The latest source compiled successfully with the desktop editor. Cached
Tectonic produced the three-page PDF without layout warnings. All three
pages were rendered at 1800 pixels and visually reviewed: equations, table,
author footnote, references, page numbering and section breaks are legible.
A first build split the theorem across pages and gave a 0.39pt vertical
overflow warning; it was preserved internally and corrected before release.

This is an originating-researcher self-audit, not independent peer review
or formal proof-assistant verification. The literature search does not
certify priority. AI assistance supported research, calculations, proof
development and writing; the author retains responsibility for the text.