What was already known
OpenAI supplied the 9/4 numerical bound, complex-field construction and original arithmetic-program specification. Classical field descent already gives the characteristic-zero consequence. Extending the construction through positive characteristic requires different separation periods and interpolation nodes. Zuiddam published a separate all-fields Lean development; no priority between these extensions is established here.
AI involvement
The all-fields proof was developed and formalized with GPT-6 Astra and GPT-6.1 Sol under Sela Navot’s direction. The ring repository also discloses AI-generated proof development and manuscript through Codex under human direction. OpenAI’s original construction and proof architecture remain separately credited; Zuiddam’s separate development is related mathematical context, with no AI role inferred for it.
New positive-characteristic separation/interpolation and ring-scope arguments go beyond formal migration of OpenAI’s complex-field proof. Retain Navot’s direction and OpenAI’s original bound.
Importance
Broadens the domain of a reported asymptotic bound from complex arithmetic to arbitrary fields and then arbitrary associative unital rings. The positive-characteristic modifications, integral coefficient identities and ring descent provide substantive formalization and generalization value beyond rebuilding the original complex theorem.
Limitations
This contribution extends scope without improving OpenAI’s numerical bound or determining the exact exponent. It gives arithmetic-operation existence bounds with positive exponent slack, not an endpoint O(n^(9/4)) implementation, practical speedup, bit-complexity bound or efficient uniform generator. The trivial ring has a zero-cost program; its real-valued omega uses an infimum convention, while the lower bound 2 and boundedness checks require a nontrivial ring. No universal semiring extension is claimed. Palomar certifies the recorded formal statements at pinned commits; it does not establish manuscript equivalence, historical priority or human peer review. AIProofWatch reviewed statements, attribution and checking records but did not compile either development or audit the complete proofs. The 7 October report date is the documented all-fields release, not an established earliest discovery date.
Technical verification
A completed rigorous check is documented for the scoped statement. Palomar registry (reported): All-fields formal statements at commit 8375e73ed454e3128500d10d87480c5dd597ff03; declared standard axiom set. Palomar registry (reported): All-rings formal statements at commit 9d10116c89f9bdcdaccd58aacea353d4b7ae6e90; declared standard axiom set.
Formal checking
Registered version 1. Six Comparator claims cover the upper bound, nonvacuity, lower bound, epsilon cost and exact coefficients; no AIProofWatch rerun.
- Checker
- Palomar registry (reported)
- Scope
- All-fields formal statements at commit 8375e73ed454e3128500d10d87480c5dd597ff03; declared standard axiom set.
- Basis
- Reported by the linked checker
- Check date
- 2026-10-08
Formal checking
Registered version 1. Exact integer coefficients and the direct ring cost theorem are included; lower-bound and boundedness claims retain their nontriviality hypotheses.
- Checker
- Palomar registry (reported)
- Scope
- All-rings formal statements at commit 9d10116c89f9bdcdaccd58aacea353d4b7ae6e90; declared standard axiom set.
- Basis
- Reported by the linked checker
- Check date
- 2026-10-11
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. No sufficient account of a named human explaining the scoped central reasoning is established in this assessment. Missing disclosure is not evidence of little understanding.
Correction and publication history
Reviewed historical addition on 14 October 2026. Corrects an omission caused by grouping the broader field/ring developments under the original complex claim. One contribution covers both stages; no duplicate numerical discovery or subscriber alert.
Read this published revision