Choiceless polynomial time with counting and finite-field linear algebra does not capture polynomial time
Year: 2026
Type: preprint
Abstract: We give a fixed polynomial-time query that is not definable in choiceless polynomial time with counting (CPT), even when CPT is extended by every finite-field linear-algebraic operator: any Boolean property of tuples of square matrices over a prime field that is invariant under simultaneous similarity, with the prime given by an input-dependent term, operators nested arbitrarily, and even uncomputable oracle operators allowed. This is the CPT analogue of the theorem of Dawar, Grädel and Lichter for fixed-point logic. The witnesses are Heisenberg grids over ℤ/2^{2e}, obtained by a fixed-width interpretation from cyclic Cai–Fürer–Immerman structures over ℤ/2^e, and the query is consistency of a configuration–edge incidence marginal system over ℤ/q. The proof combines the all-primes cyclic-CFI obstruction of Lichter and of Dawar, Grädel and Lichter with sparse source support compression and the supported hereditarily finite comparison of the recent OpenAI preprint on CPT. An explicit incidence-map descent through an auxiliary equality matrix upgrades the earlier rank transfer to simultaneous similarity for every finite tuple of gate matrices, even when representation fibres have size divisible by the characteristic. The underlying type-matrix and congruence-matrix principles belong to the framework of Dawar, Grädel and Pakusa, which builds on the invertible-map game of Dawar and Holm; we claim no priority for them. Two companion results are included: the earlier special case, for rank over a prime field computed by a term; and, for one fixed binary vocabulary, a query definable with rank over 𝔽₃ that is not definable in CPT on structures with colour classes of at most 3^29 elements whose full colour groups are class-two groups of exponent 3. With the capture theorem for bounded colour classes deposited earlier from the same program, rank logic is therefore strictly more expressive than CPT on bounded colour classes, for every colour bound of at least 3^29. The proofs were produced by AI models and audited by independent AI referees; nothing has yet been refereed by humans or formally verified. Provenance and status. The proofs in this deposit were produced by AI models (OpenAI GPT models, working in separate chat threads) in an AI-assisted research program directed by Matthew Liston. They were checked in part by Claude (Anthropic), which also coordinated the program. Each proof record was audited by an independent AI referee (an OpenAI model, run in a session separate from the one that produced the proof). The audit of the main proof returned "proved, with minor formalization and attribution revisions", and those revisions are incorporated in the paper; the audit of the earlier special case returned "proved with fixable gaps", and its nine repairs are incorporated in the main proof record; the audit of the companion separation found no fatal or major mathematical error, and noted that the class-wide consequence depends on a separate rank-canonization note that it did not audit; the paper derives that consequence instead from the capture theorem of the earlier deposit 10.5281/zenodo.23996260. The fixes the referees requested are included in this deposit (Appendix A of the paper). There has been no human refereeing yet, and there is no machine-checked formal proof. We are confident that the results are correct: the audits were independent, every repair they requested is incorporated, and the referees reran the finite checks. Formal verification is planned; the program's formal-verification capacity is currently committed to other results, so we are releasing this version now. Files.cpt-finite-field-linear-algebra-v1.pdf: the paper (57 pages; Appendix B reproduces the main proof record verbatim). SHA-256 09feb2994eb88c01a9f751c5ef72863843f96c32cd86d690d777617cf400ad7bcpt-finite-field-linear-algebra-v1.zip: the paper and its LaTeX source, the three proof records, the three AI referee reports, the reproducibility check with its rerun, a README and SHA256SUMS. SHA-256 dcd7418632a5f35ed0d96802e7cf1e61ad8f0594af4f189dbf94e23662008c80 Records in the zip:[N3] proofs/CPT_ALL_FINITE_FIELD_LINEAR_ALGEBRA_SEPARATION_PROOF_2026-11-05.txt (proof of Theorem 1 (main result)). SHA-256 7ca2a352ea44cf6c207070462d8a7c6fb43b7e2df43a2ab3155a5a1ee963aa6a[UR] proofs/CPT_UNIFORM_RANK_SPARSE_EMBEDDING_PROOF_2026-11-05.txt (proof of Corollary 1.1 (earlier special case)). SHA-256 a9dd50e68c9d24fbebc3c834aa078e14644eb54b3b3c32c47af09ce5c072828a[CS] proofs/CPT_CLASS_TWO_COLOURS_SEPARATION_PROOF_2026-11-05.txt (proof of Theorem 2). SHA-256 14de9622c9e7cf315b87350e669f235bf7bae3591fce3311c61e517ddf5fdbfc[R5] ai-referee-reports/CPT_ALL_FINITE_FIELD_LINEAR_ALGEBRA_REFEREE_REPORT_2026-11-05.txt (AI referee report on [N3]). SHA-256 5d2fd4d1f6557f5bd99a143b26dfa919b1d0b526037f6cb8db06ddf25b4625b6[R4] ai-referee-reports/CPT_UNIFORM_RANK_SPARSE_EMBEDDING_REFEREE_REPORT_2026-11-05.txt (AI referee report on [UR]). SHA-256 5a103275cb8f3066f025e89e1403e5f5121efa8a484993129bba96c96b01a772[RCS] ai-referee-reports/CPT_Class_Two_Colours_Referee_Report_2026-11-05.txt (AI referee report on [CS]). SHA-256 43340b191f6d5dd454b4596a35a127eabd17258f6a63c04e089a97da766aa27eThe digests of [UR], [CS], [R4] and [RCS] were also published in the earlier deposit 10.5281/zenodo.23996260. Further records, timestamped here for priority; they are not part of this deposit's claims.CPT_A_GROUP_DICHOTOMY_PROOF_2026-11-06.txt: for subgroup-closed families of colour groups, CPT canonizes bounded colour classes over all vocabularies exactly when every group has abelian Sylow subgroups. Refereed by two independent AI referees (verdict: proved); a paper is in preparation. SHA-256 c743054435cfedacc09112eb84dbb6589212946f52b60aa75fa41380984d3a0aCPT_WSC_ROUND2_RESULTS_2026-11-06.txt: CPT with witnessed symmetric choice is not contained in CPT with finite-field linear algebra, using the indistinguishability theorem behind Theorem 1 of this paper. Awaiting independent review. SHA-256 052352f29f2a515acf0375f940f2b7d683413bd130ab39139bad12eef218e55a Contact: [email protected]
Author: Liston, Matthew
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225428520