The 4-Color Rado Number of $x+y+c=z$: $R(c)=40c+41$ Whenever $c+1$ Is Divisible by 3, 4, 5 or 7

What the AI did

An autonomous clean-room Claude session chose the target problem (4-color Rado numbers), surveyed three mutually-unaware literatures (Malo 2000, Myers 2015, ABEMRS16 2016), re-derived the scaling lemma, proved the synthesis theorem and prime-reduction corollary, built the SAT pipeline and independent verifier, solved all nineteen prime cases, and ran the full two-tier certification (DRAT + independent second encoder). Two independent AI referee agents verified the proof (both CONFIRMED). Human direction limited to run design, operational supervision, and posting.

Verification

Reproduced in substance by this site on 14 August 2026, independently of the repo's code. All 28 coloring certificates were re-checked by an own-code scanner over every monochromatic triple: 28/28 valid, so every lower bound holds outright. Five base cells were fully re-solved with an independently written encoder (own variable layout, own symmetry breaking): satisfiable at $n-1$ and unsatisfiable at $n$ for $c = 0, 2, 3, 4, 5$, matching $R(0)=45$ and the $40c+41$ line exactly. The scaling lemma, its sharpness against the universal lower bound, the synthesis theorem and the prime-reduction corollary were verified by hand; the algebra is elementary and correct. The literature was verified independently: ABEMRS16 is Math. Comp. 85 (2016) 2047-2064 with exactly the claimed authors; Myers' Conjecture 4.9 appears verbatim in the Rutgers thesis; Malo's 2000 thesis is real (Open Prairie, South Dakota State) with $R(1..3)$ in its public abstract, and its full text is bot-gated - so the submitter's hedge about the scaling lemma possibly being Malo's is accurate and could not be resolved from here either. The 2026 papers on this equation were spot-checked and are two-color, as claimed. Not reproduced: the nineteen prime-case UNSAT certificates ($n$ up to 3321), which rest on the bundle's kissat DRAT proofs, drat-trim VERIFIED, with a second independent encoder agreeing on every instance both ran; and no human peer review exists - produced and refereed by AI agents in one pipeline.

Sources

• CodeGitHub repo (synthesis theorem + SAT certificates + dual-encoder verification + independent checker)
• Problem recordABEMRS16, On the n-color Rado number for x_1+...+x_k+c = x_{k+1} (Math. Comp. 85, section 5.5 poses the conjecture) · Myers, Computational Advances in Rado Numbers (Rutgers Ph.D. thesis, 2015) - Conjecture 4.9 · Malo, Four Color Rado Numbers for x_1+x_2+c=x_3 (South Dakota State M.S. thesis, 2000)