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CS-155 · computer science · mathematics · Computational algebra and number theory

Matrix hitting points in every positive characteristic

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Given a promised prime p in binary and n,s in unary, can one construct in polynomial bit time an n-tuple of d×d matrices over $F_p$, $d=O(n^{3}s^{6})$, on which every nonzero free noncommutative polynomial formula of size $\leq s$ over every field of characteristic p evaluates nonzero?

Short English statement

Given a promised prime p in binary and n,s in unary, can one construct in polynomial bit time an n-tuple of d×d matrices over $F_p$, $d=O(n^{3}s^{6})$, on which every nonzero free noncommutative polynomial formula of size $\leq s$ over every field of characteristic p evaluates nonzero?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-155-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-155

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; 2026-10-04).

Matrix hitting points in every positive characteristic

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; 2026-10-04), the asserted result is that every positive characteristic has one polynomial-size prime-field matrix tuple detecting all nonzero bounded-size division-free noncommutative formulas, including arbitrary extension coefficients.

Mathematical statement of the claimed result

Manuscript 1.
A prime $p$ is supplied in binary, with primality promised, and positive integers $n,s$ in unary. The claim is a uniform deterministic algorithm producing $d=O(n^3s^6)$ and matrices $T_1,\ldots,T_n\in\operatorname{Mat}_d(\mathbb F_p)$ such that \[ \forall F\text{ with }\operatorname{char}F=p\ \forall 0\ne f\in F\langle x_1,\ldots,x_n\rangle: \quad\operatorname{formula\ size}(f)\le s\ \Longrightarrow\ f(T_1,\ldots,T_n)\ne0. \] The matrices are viewed over $F$ through its prime subfield, so one tuple hits formulas with arbitrary extension-field coefficients. With $L=\lceil\log_2p\rceil$, its total bit time and dense binary output length are at most \[ C(n+s+L+1)^k \] for absolute positive integers $C,k$; entries are ordinary residues from $0$ to $p-1$.

Outline of the argument

For manuscript 1 (family 116; 2026-10-04), a multiplicative-shift system faithfully encodes word series. Derivative-row and minor arguments bound the first nonzero scalar coefficient, so truncating the word operators yields a uniform finite matrix construction without differentiation's characteristic obstruction.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 116; 2026-10-04).

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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