Result 243, Mathematical logic

Separating choiceless counting from polynomial time and witnessed choice

Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽3 defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.

Lean formalization Proof

The bigger picture

Why it matters

Efficient computation can exceed what a logical language expresses when it cannot choose arbitrarily among indistinguishable objects. The manuscripts report two precise limits of a formalism designed to compute on finite structures without a supplied ordering.

What changes?

Choiceless polynomial time with counting permits counting but not arbitrary choices. The first manuscript reports that this formalism cannot express whether linear equations over the three-element field have a simultaneous solution, although this query is decidable in polynomial time. The inputs are unordered finite structures in a fixed binary vocabulary: a fixed collection of relation symbols involving at most two objects. Both reported separations cover the full counting formalism, allowing hereditarily finite sets nested to any finite depth.

What does that help mathematicians do?

Thus counting and unrestricted finite nesting do not suffice to express every polynomial-time query on unordered structures. Separately, the second manuscript reports a fixed sentence using witnessed symmetric choice once that defines a yes-or-no query on every finite input, yet has no equivalent in the original formalism. This identifies choice as a genuine increase in expressive power, not merely a convenient way to write computations.

Are there practical applications?

The immediate value is foundational: these claims clarify which logical resources can describe efficient computation without assuming an input order. They do not provide a faster linear-system solver or show that witnessed symmetric choice captures all polynomial-time queries. Instead, they isolate limits that proposed logical characterizations of polynomial time must address.

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.

2 manuscripts

Choiceless polynomial time with counting does not capture polynomial time

September 23, 2026 34 pages Main result formalized in Lean

We prove that choiceless polynomial time with counting does not capture polynomial time on unordered finite structures, confirming the noncapture conjecture of Blass, Gurevich and Shelah. A linear-consistency query over 𝔽3 in a fixed binary vocabulary is decidable in polynomial time but not in the full counting formalism.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026,
  author = {{OpenAI}},
  title = {{Choiceless polynomial time with counting does not capture polynomial time}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026/paper.pdf}{OAI:Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026}},
  year = {2026}
}

Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting

September 24, 2026 45 pages

We prove that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. A fixed sentence with one witnessed-choice occurrence defines a Boolean query on every finite input that is not definable in the original counting formalism.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026,
  author = {{OpenAI}},
  title = {{Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf}{OAI:Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/243.md.

Separating choiceless counting from polynomial time and witnessed choice

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

Scope

The formalized result separates polynomial time from choiceless polynomial time with counting. It gives an explicit query on finite structures with eight relations that is invariant under isomorphism and decidable in polynomial time, but is not definable in the stated hereditarily finite-set language with cardinality and polynomial bounds on stages and intermediate objects. The query applies to all finite inputs without a promise. The separate witnessed-symmetric-choice result is not included.

The formalization proves that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. It constructs one fixed sentence with exactly one witnessed-choice occurrence that gives a Boolean answer on every finite input, while no sentence of the original counting formalism defines the same class of inputs.

Comparator links

Result Comparator statement
Polynomial-time query outside choiceless polynomial time ChoicelessPolynomialTime.lean
Strict expressive gain from one witnessed symmetric choice WitnessedChoice.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.