Problem detail · source-aware
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Classify the finite simple graphs of minimum degree at least five whose cycle lengths fail to represent every residue class modulo five. Is residue two the only possible missing residue, and can all such graphs be characterized through an explicit family of exceptional end-blocks together with a condition on the remaining blocks?
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
Large language models contributed substantially to developing and auditing the proof, repairing intermediate arguments, and implementing computational checks. The work included checking rooted-path and cycle constructions, identifying gaps in structural reductions, developing replacement lemmas and corrected attachment arguments, and revising finite residue calculations and literal-cycle geometry checks.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
Unreviewed. The 82-page preprint (Zenodo 10.5281/zenodo.22311785, version 1.0.0 of 4 September 2026) was text-extracted and its introduction, main theorem and disclosure read here; the computational supplement (10.5281/zenodo.22311412) reports 23 passing operations and was not replayed on this site, unlike the Dean-5 supplement, which was. Two dependencies a reader should hold in mind: the proof imports the author's own modulus-five Dean theorem, version 1.0.1, which is itself Candidate here pending independent review; and the earlier Zenodo version of this preprint was withdrawn by the author for errors before this one, which is stated in the submission and matches the deleted record. Tier changed from AI-discovered to AI co-developed: the paper's own disclosure says the results were obtained "with substantial assistance from large language models" and that the author reviewed the arguments and takes responsibility, which is the co-developed pattern on this site, not discovery.
Correctness: unknown · statement fidelity: unaudited · peer review: none
The preprint claims an exact classification. Write $C_5(G)$ for the residues modulo five represented by cycle lengths in $G$. Let $\mathcal E_5={K_6,K_{5,5}}\cup{H_{5,n;t}:2\le t\le5<n}$, where $H_{5,n;t}$ is obtained from $K_{5,n}$ by deleting $5-t$ edges incident with one vertex in the part of size $n$. For every finite simple graph $G$ with minimum degree at least five, exactly one alternative holds: $C_5(G)=\mathbb Z_5$; or every end-block belongs to $\mathcal E_5$ and every non-end-block contains no cycle of length congruent to two modulo five. Every member of $\mathcal E_5$ has cycle-residue spectrum ${0,1,3,4}$. The proof combines structural arguments with finite computational checks. It uses the separately established Dean–5 theorem and its weak-graph strengthening as inputs. The contribution is the stronger stability classification; independent expert review remains pending.
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.