Uniform black-box noncommutative identity testing across characteristics

What OpenAI claims

For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation. (Lean)

Papers (3)

1. Uniform Matrix Hitting Points in Every Positive Characteristic

   4 October 2026 · Preprint, not peer reviewed · PDF

   We construct a single matrix substitution that detects every nonzero size-s division-free noncommutative formula in n variables over every field of a given positive characteristic. One deterministic machine, given a promised prime p in binary and n, s in unary, outputs matrices over 𝔽p of dimension O(n³s⁶) in polynomial bit time. The same tuple works with arbitrary extension-field coefficients, including in characteristic two. The construction also applies to the stated acyclic algebraic path programs.

2. One Rational Matrix Hitting Point for Noncommutative Formulas

   24 September 2026 · Preprint, not peer reviewed · PDF

   We construct, in deterministic polynomial bit time, one tuple of rational matrices that detects every nonzero polynomial computed by a noncommutative division-free formula of a prescribed size. The matrices have dimension O(ns²) for n variables and formula size s, and the same tuple works over every field of characteristic zero.

3. Polynomial Hitting Lists for Noncommutative Rational Formulas

   24 September 2026 · Preprint, not peer reviewed · PDF

   We construct, in deterministic polynomial bit time, a polynomial-size list of rational matrix tuples for noncommutative rational formulas over ℚ of bounded tree size. Every nonzero admissible formula has a defined, invertible value at one tuple, with no separate bounds on inverse nesting or rational constant heights. Matrix dimensions, entry bit lengths, and total output length are polynomially bounded.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.