What was already known
The paper says the previously known unconditional complete list reached class number 100. Broader counts or lists up to class numbers 1000 and 1999 had relied on the generalized Riemann hypothesis. OpenAI's earlier zero-free result supplied an unquantified analytic starting point; Banks independently obtained a different explicit constant without a formal proof.
AI involvement
Kimura says Claude Opus 5.5, under his direction, developed the explicit argument and lower bound, produced the Lean files and numerical certificates, and implemented the sieve and comparison scripts. Kimura chose the problem, the strategy, the class-number target, and the checks, worked through the proofs, and assumes responsibility. An earlier OpenAI model supplied the zero-free theorem and some upstream formalization.
Kimura credits Claude with the explicit zero-gap argument, lower bound, Lean files, certificates and sieve program. Kimura chose the class-number target and strategy, checked the proof and comparisons, and used a separate earlier OpenAI zero-free result as an input.
- Supporting source (arxiv.org) — Introduction and Section 8, Contributions of AI
Importance
If the computational components are correct, this extends unconditional complete classification from class number 100 to 2000 and removes a conditional assumption from several previously published ranges.
Limitations
The reported Lean and independent-kernel checks cover the explicit zero-gap and discriminant-bound theorems, not the final 16,453,493-field classification. That list also relies on an unformalized sieve program, earlier class-group tables, and extraction code. The author reports independent comparisons and spot checks, but the complete computation has not been reproduced in this review. The analytic starting point derives from an earlier OpenAI manuscript, and the author selected the target and method.
Technical verification
The source reports Lean-kernel and independent nanoda checks of the explicit zero-gap and discriminant-bound theorems. These are prerequisites for the classification; the final sieve program and use of prior class-group tables are not formally verified. The author reports spot checks and comparisons, not a full independent reproduction of the final list.
Formal checking
The author reports Lean-kernel and independent nanoda checks of Theorems 1 and 10 only. These analytic bounds are prerequisites, not a formal check of the final list.
- Checker
- Lean kernel and nanoda, as reported by Iwao Kimura
- Scope
- Theorem 1 explicit zero-gap bound and Theorem 10 class-number discriminant bound; excludes sieve and table enumeration
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Kimura explains the explicit-bound argument, the sieve's mathematical basis and how earlier tables enter the classification, and states that he worked through the proofs. This supports understanding of the scoped classification method, without certifying its full computation.
Correction and publication history
Primary Theorem 2, prior unconditional list and GRH-conditional comparison, detailed Section 8 attribution, and distinction between formal analytic checks and unformalized final sieve reviewed; non-notifying local inclusion.
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