OpenAI reports one matrix hitting point in every positive characteristic

Given a prime p in binary and positive n,s in unary, one deterministic polynomial-bit-time machine outputs a tuple of matrices over F_p of dimension O(n³s⁶) that detects every nonzero division-free noncommutative formula of size at most s in n variables over every field of characteristic p.

Sources for this record

What was already known

An earlier companion gave one rational matrix tuple in characteristic zero. Earlier deterministic black-box hitting sets for noncommutative branching programs were quasipolynomial under a field-size condition; internal-description identity tests do not give this uniform black-box tuple.

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Notable

Supplies a polynomial-size universal black-box matrix test across every positive characteristic, including two.

Limitations

The formula has no division gates, and p is promised prime. The linked Lean scope checks characteristic-zero hitting points and rational-formula hitting lists, not this positive-characteristic construction.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The OpenAI-authored manuscript reports and argues the result, but the reviewed sources do not document a completed check of the full assessed claim.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, prior-work, and precise partial or mismatched Lean-scope review for non-notifying local inclusion.

Read this published revision