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Math-2725 · mathematics · Logic

Strict expressive gain from witnessed symmetric choice

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Does adding witnessed symmetric choice to choiceless polynomial time with counting strictly increase the Boolean queries definable on all finite structures, even when one total sentence uses only one witnessed-choice occurrence?

Short English statement

Does adding witnessed symmetric choice to choiceless polynomial time with counting strictly increase the Boolean queries definable on all finite structures, even when one total sentence uses only one witnessed-choice occurrence?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2725-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2725

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 243; 2026-09-24).

Strict expressive gain from witnessed symmetric choice

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 243; 2026-09-24), the asserted result is that one witnessed symmetric-choice occurrence defines a query beyond ordinary choiceless polynomial time with counting, while giving an answer on every finite input.

Mathematical statement of the claimed result

Manuscript 1.
In choiceless polynomial time with counting (CPT), computations use the full hereditarily finite domain without a fixed rank bound. Its witnessed symmetric choice extension (WSC) allows a choice accompanied by automorphism witnesses for the entire input vocabulary, fixing parameters and all previously constructed states. The claim exhibits a fixed vocabulary of eight binary relation symbols and a polynomially bounded CPT+WSC sentence with one occurrence of witnessed symmetric choice such that \[ Q\in\operatorname{CPT+WSC},\qquad Q\notin\operatorname{CPT}, \] where $Q$ is its Boolean query on all finite structures in that vocabulary. The nondefinability assertion ranges over all CPT sentences, not a bounded-rank fragment.

Outline of the argument

For manuscript 1 (family 243; 2026-09-24), edge and configuration atoms encode compatible states on a charged box. Explicit automorphisms witness symmetric choices and preserve earlier selections; a guarded fixed sentence detects zero charge, while the sparse-support counting argument prevents the original formalism from doing so.

Formal verification

For manuscript 1 (family 243; 2026-09-24), OpenAI lists Lean statements for this assertion: one witnessed symmetric-choice occurrence defines a query beyond ordinary choiceless polynomial time with counting, while giving an answer on every finite input.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are ChoicelessPolynomialTime.lean, WitnessedChoice.lean.

Their named declarations include OAI.CPTSeparation.main, OAI.WitnessedChoice.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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