The Goldbach exceptional set: an exponent 7/12 and conditional refinement

János Pintz, Ingo Althöfer

Pintz and Althöfer report E(X)≪ε X^(7/12+ε) under a common fixed zero-free half-plane Re(s)>θ₀, θ₀<1, for all Dirichlet L-functions. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most θ∈[1/2,2/3], they report the exponent g(θ)=θ(5−4θ)/(2(2−θ)), which is below 7/12 for θ<2/3.

Sources for this record

What was already known

The paper cites published unconditional exponents 0.72 and 0.709 and a separate unpublished AI-assisted numerical refinement. A common zero-free half-plane eliminates generalized exceptional zeros in the major-arc argument; that is a stronger assumption than the classical zero-free regions.

AI involvement AI-centered

Pintz supplied the major-arc method and paper structure; Althöfer initiated and directed the collaboration. The authors credit ChatGPT with proposing and developing g(θ), algebra and parameter checks, argument expansion and source comparison.

ChatGPT proposed and developed the explicit g(theta) refinement; Pintz supplied the major-arc method. The human-origin 7/12 framework is context.

Importance Notable

A scoped circle-method application of a new prime exponential-sum estimate, plus an AI-assisted explicit family of conditional exceptional-set exponents.

Limitations

The 7/12 conclusion depends on a uniform zero-free-half-plane input supplied in the paper by an unrefereed OpenAI claim. This record keeps that input as a hypothesis rather than presenting the bound as an independently established unconditional result. The sharper g(θ) family needs the stronger stated hypothesis and is not supplied by the 7/8 input. It does not prove the binary Goldbach conjecture. The earlier numerical chain is not an input to these proofs. AIProofWatch has not audited the analytic proof or the underlying OpenAI result; no formal verification or outside checking is claimed.

Technical verification Partial

Only the stated computational, numerical, conditional or component checks are documented for this scope; the full claimed endpoint has not received a documented complete rigorous check in the reviewed evidence. The 7/12 conclusion depends on a uniform zero-free-half-plane input supplied in the paper by an unrefereed OpenAI claim. This record keeps that input as a hypothesis rather than presenting the bound as an independently established unconditional result. The sharper g(θ) family needs the stronger stated hypothesis and is not supplied by the 7/8 input. It does not prove the binary Goldbach conjecture. The earlier numerical chain is not an input to these proofs. AIProofWatch has not audited the analytic proof or the underlying OpenAI result; no formal verification or outside checking is claimed.

Human understanding Few

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

2026-10-16 · publication

Current community lead followed to primary paper, exact conditional hypotheses, prior exponents and documented AI role. Distinct Goldbach-exceptional-set statement, not another OpenAI zero-free discovery. No existing equivalent in the complete catalog.

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