OpenAI · 116-02 · 2026-09-24 · Theoretical computer science
One Rational Matrix Hitting Point for Noncommutative Formulas
preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026
Family 116. Uniform identity testing for noncommutative formulas.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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 also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.
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 for n variables and formula size s, and the same tuple works over every field of characteristic zero.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 116: Uniform black-box noncommutative identity testing across characteristics. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.