z(20) = 6 — unrefereed candidate certificate-backed computer-assisted proof (Erdős Problem 758 sub-question)

The cochromatic number z(G) of a graph is the least number of classes in a partition of its vertices where every class induces a clique or an independent set; z(n) = max{ z(G) : |V(G)| = n }. The case z(20), previously known only to be 6 or 7, is the sub-question recorded on Erdős Problem #758. This record archives a candidate proof that z(20) = 6, together with the source, proof objects (exact CNFs with DRUP/LRAT unsatisfiability certificates), and dated replay records needed to scrutinise it.

Status: unrefereed candidate computer-assisted proof. A dated replay has been completed on one macOS/arm64 machine. No human mathematician has verified the complete argument; no independent external reproduction or end-to-end proof-assistant formalisation is claimed. The SAT certificates establish the unsatisfiability of the two exact CNFs; the graph-theoretic reduction and encoding equivalence are separate obligations discussed in the paper. Deposit on Zenodo does not upgrade this status.

Provenance (repository-reported): every mathematical step was produced by AI systems; the human participant selected the problem and mediated between systems, and claims none of the mathematics. Phase 1 (unsuccessful): a Nous Hermes agent driving OpenAI's GPT-5.6 Sol. Phase 2: Claude 4.8 (instrumentation, baselines, candidate characterisation). Phase 3 (the reduction): GPT-5.6 Sol (two-core reduction, CNF generator, DPLL solver, DRUP certificates). Phase 4 (verification): Claude Opus 5 (cross-system verification, catalogue and R(4,4)=18 recomputation, z(8) and z(12) re-derivations, the lower bound, and the write-ups). Repository maintenance and publication: Ian Pitchford, CC0 affirmer only to the extent he holds the relevant rights.

Archive: the exact release archive of tag candidate-2026-07-26 (commit 3c7e520fdc0615f5c700761c2b1e5108dcc836e7) of github.com/ipitchford/z20-cochromatic; SHA-256 recorded in the deposit receipt.