What was already known
Exact odd-cycle capacities beyond C5 remain unknown. Earlier model-generated independent sets improved four cycles, and Gao’s human product method gave C7 ≥ 3.258789153908.
AI involvement
ChatGPT-5.6 Sol Pro and Claude Opus 5 assisted the finite search; earlier model-generated sets fed part of it. Buys, Polak and Zuiddam supplied the product argument and Lean formalization.
Documented model-generated base independent sets and BPZ model-assisted tuple search supplied central retained finite constructions; human work supplied Gao-style propagation and formalization.
Importance
Adds seven formally checked lower-bound constructions across small odd cycles.
Limitations
The gains are small and no capacity is determined. A later independent-set construction already exceeds the C11 figure, and later heterogeneous recursion exceeds the C7 figure. The origin of each of the seven base tuples is not individually documented.
Technical verification
BPZ report Lean checks of all seven explicit base tuples and formalization of their stated lower bounds. Itty et al. separately report author checks of feasibility for their four earlier model-generated sets; that human check has narrower scope. These are source-reported checks, not AIProofWatch reruns.
Formal checking
Lean formal proof over explicit finite literals
- Checker
- Lean kernel via spectra-research/shannon-capacity-lean
- Scope
- Seven valid base tuples, product derivation and the paper’s seven capacity lower bounds
- Basis
- Reported by the linked checker
Formal checking
author verification of graph independence for the returned sets
- Checker
- Itty, Rosin, Carstensen and Reichman
- Scope
- Feasibility of their four earlier independent-set constructions in C7^10, C11^6, C13^6 and C15^8
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors explain the valid-tuple conditions, Gao product iteration and each derived bound; the model search heuristic itself need not be fully understood to explain the constructions.
Correction and publication history
Adds seven formally checked lower-bound constructions across small odd cycles.
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