Source authenticated

The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture

Proves the divisible rank-three case: rank exactly three and ground-set size a multiple of three, with no restriction to simple, paving, representable or graphic matroids. The part not previously in the literature is the non-simple sub-case, since McGuinness had settled all paving matroids and a rank-three matroid is paving exactly when it has no parallel pairs. Combined with the coprime-case theorem of van den Heuvel and Thomasse, this covers every finite uniformly dense rank-three matroid. The conjecture remains open in higher rank.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Aug 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-verified. Publication: preprint. AI contribution: ai-discovered. Imported under CC BY 4.0.

Canonical aliases: The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture · Divisible rank-three KUM

Confidence: Not scored

Registry verification: lean verified · preprint · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

GPT-5.6 Sol
model · ai model contributor · OpenAI

Claude Opus 5 (for some Lean formalization and paper write-up)
model · ai model contributor · Anthropic

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture

specialist registry · passed

The new divisible rank-three theorem is formalized end to end in Lean 4. The repository builds successfully with 3,046 jobs, zero errors and zero warnings. The principal theorem contains no sorry, admit, custom axiom declaration, unsafe declaration or use of native_decide; Lean reports only the standard Mathlib axioms propext, Classical.choice and Quot.sound. The conclusion for all rank-three matroids additionally invokes the published coprime-case theorem of van den Heuvel and Thomassé, which is not formalized in this repository. The divisible ingredient is therefore Lean-verified, and the complete rank-three result is established modulo that named literature theorem. This is a complete resolution in rank three but a partial result toward the unrestricted Kajitani–Ueno–Miyano conjecture, which remains open in higher rank. The manuscript is an unrefereed Zenodo preprint.

Source ↗

The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture

preprint · pending

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

Source ↗

Lean repo

code · pending

code link supplied by the VibeMathed registry.

Source ↗

Archived repo

code · pending

code link supplied by the VibeMathed registry.