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CS-159 · computer science · mathematics · Computational algebra and number theory

One rational matrix hitting point for noncommutative formulas

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each $n,s\geq 1$, can one deterministically construct in polynomial bit time a single n-tuple of rational matrices of dimension at most $2ns^{2}$ that makes every nonzero size-s noncommutative polynomial formula over every characteristic-zero field nonzero, with polynomially bounded entry bit lengths?

Short English statement

For each $n,s\geq 1$, can one deterministically construct in polynomial bit time a single n-tuple of rational matrices of dimension at most $2ns^{2}$ that makes every nonzero size-s noncommutative polynomial formula over every characteristic-zero field nonzero, with polynomially bounded entry bit lengths?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-159-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-159

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 116; 2026-09-24).

One rational matrix hitting point for noncommutative formulas

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 116; 2026-09-24), the asserted result is that one rational matrix tuple of dimension $O(ns^2)$ detects every nonzero size-$s$ division-free noncommutative formula and is constructible in polynomial bit time.

Mathematical statement of the claimed result

Manuscript 1.
For integers $n,s\ge1$, the claim gives a deterministic algorithm constructing one rational matrix tuple $\mathbf T_{n,s}\in M_d(\mathbb Q)^n$ in polynomial bit time, with $d\le2ns^2$, such that \[ \forall F\text{ of characteristic }0\ \forall0\ne f\in\mathsf{Form}_{n,s}(F):\quad f(\mathbf T_{n,s})\ne0. \] Here $\mathsf{Form}_{n,s}$ denotes noncommutative arithmetic formulas in $n$ variables of size at most $s$, and matrices are viewed in $F$ via $\mathbb Q\hookrightarrow F$. Entry numerators and denominators have bit length $O(\log(ns+1)+ns^2\log(n+1))$. The stated dimensions are $2ns^2$ if $ns\ge2$ and one if $n=s=1$.

Outline of the argument

For manuscript 1 (family 116; 2026-09-24), iterated integrals encode free words faithfully inside differential systems. A vanishing-order bound limits how many Taylor coefficients are needed; those coefficients form the finite matrix space used for the universal substitution.

Formal verification

For manuscript 1 (family 116; 2026-09-24), the documented Lean scope is the universal matrix hitting property over characteristic-zero fields; the construction-time and $O(ns^2)$ dimension estimates are not separately encoded in the selected theorem.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are FormulaHitting.lean, RationalHitting.lean.

Their named declarations include OAI.NCHitting.universal_hitting, OAI.RationalHitting.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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