What was already known
Guevara and coauthors derived the recursion and a determinant in a special decay region. General tree-dependent cut factors blocked direct use of the ordinary Matrix-Tree theorem.
AI involvement
Kehoe selected and directed the problem. Codex supplied the root-cone idea, cut/flow proof, verification programs and initial manuscript; a separate LLM referee checked selected claims with fresh code.
Codex supplied the central root-cone reformulation and proof under Kehoe's direction; the human role is problem selection and project oversight.
Importance
Provides a root-cone route through a stated obstruction and new formulas in specified chambers.
Limitations
There is no universal determinant formula. Lean checks the abstract cut/flow core, but not the graph-tree bridge, full weighted enumerator, chamber counts or five-point formula. Finite exact-code checks cover selected claims; no independent human audit of the statement map is documented.
Technical verification
The v0.3.0 release reports a passing Lean 4 check for the cut, flow and root-cone core, plus exact-code checks for small chambers and the five-point identity. Lean assumes a tree-cut structure rather than constructing it from a graph tree, and does not encode the whole weighted enumerator or chamber claims. The informal-to-formal statement map has not had an independent audit.
Independent checking
Lean checks the abstract cut, divergence and positive-flow results. The graph-tree bridge and weighted enumerator remain outside it.
- Checker
- Project Lean 4 CI and leanchecker
- Scope
- Spinor cut identity, divergence and positive tree-flow equivalence, and abstract open-root-cone membership
- Basis
- Reported by the linked checker
Independent checking
Exact chamber enumeration and finite identity checks cover the stated four- and five-block examples.
- Checker
- Project exact-code suites and independent LLM referee implementations
- Scope
- Four- and five-block chamber counts, selected determinant cases and the non-decay five-point formula
- Basis
- Reported by the linked checker
Human understanding
The proof and explanation were generated by Codex and checked by another LLM; no human account explaining the central root-cone mechanism was located. Few is the evidence default.
Correction and publication history
Provides a root-cone route through a stated obstruction and new formulas in specified chambers.
Read this published revision