What was already known
An earlier planar example gave 1.139747…; an unconditional planar ceiling of 2 was known. The E(3) and E(4) numbers were earlier lower-bound examples, not exact envelope theorems.
AI involvement
Nikolenko directed and wrote the project and reports verifying this paper. He credits GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 with most key ideas, proofs and verification tools, without assigning individual steps.
The paper explicitly says models supplied most key ideas in the work, including proofs; the particular idea-by-idea attribution remains unresolved.
Importance
The planar lower bound improves and the local envelope theorems sharpen the restricted theory.
Limitations
The envelope and decomposition bounds are class-specific; none is a universal upper bound. The public flow counterexample and assigned four-terminal planar value concern earlier, weaker witnesses.
Technical verification
Nikolenko reports checking the entire paper. The finite planar witness and envelope certificate ledger were also checked with exact arithmetic and second implementations. These are source-reported checks; the full pipeline was not run here.
Independent checking
Personally verified and edited the complete manuscript
- Checker
- Sergey Nikolenko, author
- Scope
- Paper III’s planar lower bound and local-envelope theorems
- Basis
- Reported by the linked checker
Formal checking
Exact rational certificate checks, including raw-arc reconstruction and second implementations
- Checker
- Project’s separately written certificate implementations
- Scope
- Six-terminal planar witness and 2,015-cell E(4) closure ledger
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Nikolenko explains the exact-mean principle, factorization, bad-state/cover duality and why the earlier lower-bound examples become local envelopes.
Correction and publication history
The planar lower bound improves and the local envelope theorems sharpen the restricted theory.
Read this published revision