# Paper visual and editorial audit

Date: 2026-08-16 UTC

## Scope

The manuscript contains three explanatory, source-native TikZ figures:

1. the proof architecture, separating the cycle-to-path chain from the three
   alternative residual-cap regimes;
2. the sharpness construction `K_k \vee ((k+1)K_t)`;
3. the DFS representative geometry for the `I_1`, `I_2`, and `I_3` cases.

The proof-architecture figure replaces the manuscript's textual dependency
summary without changing its mathematical content. The other two figures are
schematic and are explicitly described as such. The certification table is
breakable so its heading and first rows stay together.

The publication-facing editorial pass replaced eponymous and implementation
names in the exposition with descriptive mathematical terminology.  In
particular, the infographic no longer uses `RawCore`, `BetaSharp`, or the
internal `I_2` case label; the high-residual section no longer uses
development-history names; and exact implementation identifiers are confined
to the explicitly labeled Lean-declaration correspondence table.  Four
cycle-extension case identifiers were moved from the mathematical proof into
that table.

The pass also designated
`https://www.proofatlas.ai/research/bondy-longest-cycle-proof/` only as the
public information page, removed the private branch label and operational
build-command block, and simplified workflow language in the AI disclosure.
The manuscript contains no GitHub publication or release-tree link.  Its
remaining arXiv link occurs only in the bibliography entry for the
independently cited Ma--Ning--Zhao paper.  No theorem statement, proof step,
inequality, citation, authorship attribution, Lean declaration, or
certification claim changed.

## Independent repair loop

The first visual render exposed four presentation defects: an automatically
hyphenated flow-box label, a sharpness annotation crossing its join lines,
overlapping DFS labels, and a DFS caption that could be read as assigning an
unintended order to the sample indices. All four were repaired. The final
color and grayscale renders of the figure pages were then inspected again.

That second, adversarial pass found two substantive visual ambiguities.  The
first architecture draft incorrectly made the three residual regimes look
sequential, and two outer-clique ovals in the sharpness diagram touched even
though those cliques are pairwise nonadjacent.  The architecture was redrawn
with three separate inputs to a common residual cap, and the outer cliques
were separated.  A final whole-page pass then found a certification-table
page-break problem; a standard breakable table with a repeated header removed
the stranded heading, stretched vertical spacing, and half-empty page.

The DFS caption now distinguishes tree edges from the additional graph edge,
identifies the thick tail segments, and states that only `j < ell_j` is a
structural order relation. The sharpness caption explicitly says that the
displayed join is schematic.

## Clean render and structural checks

The latest source was rendered from an empty output directory with the pinned,
checksum-verified Tectonic 0.16.9 binary and ready isolated cache.  The three
figure crops had already been inspected at 300 dpi in color and grayscale.
The final publication pass rerendered every page and inspected the changed
title/disclosure, infographic, high-residual, certification-table,
ProofAtlas, and appendix pages at full two-times resolution.  The resulting
PDF has:

- 22 pages;
- 17 outline entries;
- 177 links, of which exactly two are external;
- no blank page;
- no text block or vector drawing outside a page boundary;
- no overfull box, underfull box, undefined reference, multiply-defined
  label, or TeX warning in the final log.

The external links are the ProofAtlas information page and the cited
Ma--Ning--Zhao arXiv record.  There is no GitHub link or private branch name.
Static source checks found 131 unique labels, 150 resolved references, 10
matched citation and bibliography keys, balanced environments and braces, and
20 distinct named `Bondy.*` declarations.  Exact code-shaped names appear
only in the formal correspondence table.  PDF title, author, subject, and
keywords are unchanged.

## Final artifact hashes

```text
c06a5e25cc24ca6a37bc4e92fcd0427b423fe411f30047d53db6a0be3c6be77a  Bondy_Longest_Cycle_Formalized_Proof_2026-08-16.tex
f265525ea05387cf556df1f1516985e19c301bf65fc3d9a80f8755ee349d2846  Bondy_Longest_Cycle_Formalized_Proof_2026-08-16.pdf
```