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Ten Advances in Mathematics and Theoretical Computer Science

OpenAI

A direct 3SAT reduction proves NP-hardness of approximating Euclidean closest vector within n^(1/400), under the dimension and promise conventions of Chapter 7. The chapter gives separate consequences for binary nearest codeword and finite p norms.

Sources for this record

What was already known

Earlier Euclidean CVP hardness factors were smaller or used different assumptions. Song later reported stronger deterministic factors; that result does not check this reduction.

AI involvement AI-led

OpenAI attributes the argument and its Lean formalization to Astra. Humans selected the problem and prepared the manuscript with the model.

OpenAI’s original disclosure assigns the argument and subsequent Lean formalization to Astra; human manuscript preparation is separately credited.

Importance Major

Establishes polynomial-factor hardness for a foundational lattice problem.

Limitations

The reported Lean certificate covers the stated formal promise. Human reviewers reconstructed the reduction but left some threshold and degree calculations compressed. Song’s stronger theorem is separate.

Technical verification Formal (Lean 4.32.0 with Mathlib and Comparator + human)

The project reports a complete Lean certificate for the stated formal theorem. Human assessments have narrower scope; semantic matching and any manuscript gaps remain separately noted.

Independent checking

chapter-specific mathematical review of the published argument, with the limited coverage summarized in the audit. Table 1; Chapter 7 discussion

Checker
Chapter 7 human reviewers in the Sienicki et al. review corpus
Scope
reduction architecture, valuation and exponent bookkeeping, with compressed thresholds
Basis
Reported by the linked checker
Evidence source

Formal checking

Lean 4.32.0 formal certificate in GapCVP.lean; comparator target for GapCVP.Comparator.gapCVP400IsNPHard; manifest reports zero sorry and only standard axioms. main_results entry GapCVP.Comparator.gapCVP400IsNPHard; status and axioms

Checker
OpenAI ten-proofs project, Lean kernel and Comparator configuration
Scope
n^(1/400)-factor Euclidean GapCVP NP-hard promise
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

Human Chapter 7 reviews reconstruct the Reed–Solomon encoding, algebraic recovery, reduction soundness and exponent conversion, while noting compressed threshold calculations.

Correction and publication history