What was already known
Threefold cone results were established. Waldron had a human-origin fourfold outline but retained a gap; existing fourfold tools relied on log-resolution hypotheses.
AI involvement
Waldron supplied the proof outline and essential ingredients. ChatGPT-5.6 Sol reportedly changed the approach around a gap; Waldron and Codex then edited the proof.
The model reportedly supplied an essential way around the gap in Waldron’s otherwise human-origin proof strategy.
Importance
Extends a central cone-theorem step of the minimal-model program to the stated fourfold setting.
Limitations
The theorem is conditional on log resolution and numerical effectiveness. The model’s exact gap repair is not identified, and no completed human or machine check of the final theorem is documented.
Technical verification
A paper proof and AI-only review exist, but no completed claim-specific human check or qualifying machine verification is documented.
Human understanding
Waldron describes the reduction and edited the resulting proof; no central unexplained mechanism is identified.
Correction and publication history
Extends a central cone-theorem step of the minimal-model program to the stated fourfold setting.
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