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The 4-color Rado number of x+y+c=z: general case

The claim is $R(c) = 40c+41$ for every $c \ge 2$ , reduced to three finite facts: the base value $R(2) = 121$ , and the unsatisfiability of a 321-position and a 521-position spoke template. The reduction is Lean-checked and holds for every $D \ge 1$ ; the two unsatisfiability results carry DRAT proofs. This completes the partial entry for the same conjecture, which proved it for roughly two thirds of integers via a scaling lemma; that lemma is now one of three legs, covering the branch where $d$ is divisible by 3. The supporting results are worth more than the headline for anyone deciding whether to believe it: the paper also shows every band relaxation is satisfiable, which is why previous attempts stalled, and that the affine method alone is exactly sharp and can never finish.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 19, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. Editorial metadata: CC BY 4.0; linked content retains its own rights.

Canonical aliases: The 4-color Rado number of x+y+c=z: general case · 4-color Rado: $R(c)=40c+41$, all $c \ge 2$

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Claude Fable 5
model · ai model contributor · Anthropic

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The 4-color Rado number of x+y+c=z: general case

specialist registry · passed

Unreviewed: AI-produced, no peer review, and no authoritative tracker has accepted it. The artifact is unusually well organised, though, and some of it was checked here. Checked: the repository's CI is green on the verification workflow (CNF regeneration, hash checks, Lean reduction check); lean/Rado.lean is 354 lines with no sorry, no axiom declarations and no native_decide; and the theorem structure matches the prose, in that `upper` takes the three finite facts as explicit hypotheses, so Lean proves the reduction and the SAT work discharges the leaves rather than the Lean claiming the whole theorem. Not checked here: the two DRAT proofs were not re-verified, the SAT solves were not re-run, and the Lean was not rebuilt. By the repository's own logs drat-trim takes 998 s and 1125 s on the two templates, so this is compute rather than judgement, and it is exactly what site-confirmed would require. Worth noting in the submission's favour: a second, independently written encoder reproduces both unsatisfiability results from the definitions, the templates are regenerated from the Lean definitions and hash-checked against pins, and R(88) = 3561 was solved directly as a positive control.

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The 4-color Rado number of x+y+c=z: general case

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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CLAIMS.md - every public claim mapped to the artifact backing it

witness · pending

other link supplied by the VibeMathed registry.

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The Lean 4 reduction (core Lean, no Mathlib)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.