Result 116, Theoretical computer science

Uniform black-box noncommutative identity testing across characteristics

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 formalization Algorithm or complexity result

The bigger picture

Why it matters

When multiplication depends on order, deciding whether a formula always equals zero can be difficult. These unreviewed manuscripts propose universal tests using matrices, square arrays of numbers, that detect every nonzero division-free formula within a chosen size bound.

What changes?

The manuscripts report one matrix substitution detecting all nonzero formulas without division, for n variables and size bound s. Dimension is O(n s squared) in characteristic zero and O(n cubed s to the sixth) in positive characteristic p. Each tuple works over every field of its characteristic, including extension-field coefficients and characteristic two. Construction takes deterministic polynomial bit time; in positive characteristic, the input is a promised prime p in binary, with n and s in unary.

What does that help mathematicians do?

For rational formulas over the rational numbers, allowing inverses, a separate manuscript reports polynomial-size hitting lists constructible in deterministic polynomial bit time. For bounded tree size, every nonzero formula having some defined rational-matrix evaluation has a defined, invertible value on the list. Matrix dimensions, entry bit lengths and total output length are polynomially bounded, without separate bounds on inverse nesting or constant heights. This supplies nonvanishing witnesses while respecting the possibility of undefined inverses.

Are there practical applications?

The immediate value is foundational: these claims give explicit test inputs for black-box identity testing, where a formula is probed through evaluations rather than inspected internally. For division-free formulas within the stated bounds, a zero matrix output would certify the zero polynomial. The abstracts do not establish practical speedups.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Uniform Matrix Hitting Points in Every Positive Characteristic

October 4, 2026 14 pages

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 𝔽_p of dimension $O(n^3s^6)$ 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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026,
  author = {{OpenAI}},
  title = {{Uniform Matrix Hitting Points in Every Positive Characteristic}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026/uniform-matrix-hitting-points-positive-characteristic.pdf}{OAI:Uniform-Matrix-Hitting-Points-in-Every-Positive-Characteristic-October-4-2026}},
  year = {2026}
}

One Rational Matrix Hitting Point for Noncommutative Formulas

September 24, 2026 13 pages

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 $O(ns^2)$ for n variables and formula size s, and the same tuple works over every field of characteristic zero.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026,
  author = {{OpenAI}},
  title = {{One Rational Matrix Hitting Point for Noncommutative Formulas}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026.pdf}{OAI:One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026}},
  year = {2026}
}

Polynomial Hitting Lists for Noncommutative Rational Formulas

September 24, 2026 25 pages Main result formalized in Lean

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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026,
  author = {{OpenAI}},
  title = {{Polynomial Hitting Lists for Noncommutative Rational Formulas}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026.pdf}{OAI:Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/116.md.

Uniform black-box noncommutative identity testing across characteristics

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization gives one explicit tuple of rational matrices that simultaneously detects every nonzero division-free noncommutative formula with at most $s$ gates in $n$ variables, for $n,s\ge1$ . Evaluation at that tuple is a nonzero matrix over every characteristic-zero field.

The selected theorem states this universal hitting property. The paper's deterministic polynomial bit-construction bound and matrix-dimension bound $O(ns^2)$ are not separately asserted in it.

The formalized result constructs polynomial-size hitting lists for noncommutative rational formulas with rational constants, addition, multiplication, and inverse gates. Given the number of variables and a formula-size bound, one deterministic polynomial-time machine outputs rational matrix tuples of a common positive dimension. Every admissible nonzero formula within the size bound evaluates to an invertible matrix on some listed tuple. The matrix dimension, complete binary output length, and running time are all polynomially bounded.

Comparator links

Result	Comparator statement
One rational matrix tuple hitting all bounded-size noncommutative formulas	FormulaHitting.lean
Polynomial hitting lists for rational formulas	RationalHitting.lean