Choiceless polynomial time on graphs of maximum degree three: finite-field linear algebra versus canonical towers
Year: 2026
Type: preprint
Abstract: We study logics for polynomial time on finite graphs of maximum degree three. We give one polynomial-time query on connected graphs of maximum degree three 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 operators allowed. So neither CPT, nor rank logic, nor CPT with rank captures polynomial time on these graphs. The witnesses are the growing-ring Heisenberg grids of our earlier CPT paper, encoded into graphs of maximum degree three by origin-free binary coset trees over numerical index-two chains; that paper's support and transfer argument, which rests on the supported hereditarily finite comparison of the recent OpenAI preprint on CPT, is verified again on the new graphs. Polynomial-time canonization of graphs of bounded degree is classical (Babai and Luks), so some logic captures polynomial time on them; we study one choiceless operator that suffices, canonical placement of bounded-fibre towers: layered membership diagrams with ordered roots in which every vertex has at most r parents and at most b vertices of a level share a parent set. CPT with the (D−1,D)-tower operator captures polynomial time on graphs of maximum degree D, and on graphs of maximum degree three the (2,2)-operator suffices; (2,2) is the coordinatewise least pair, since fibre bound one or parent bound one adds nothing to CPT. Unbounded tower height is necessary: every fixed-height tower operator reduces to bounded-colour canonization, which rank logic performs, and so fails by the lower bound. For each fibre bound B and each odd prime p > B, a solvability query over Fₚ on structures in a fixed binary vocabulary escapes CPT with tower operators of fibre bound at most B, and no fixed fibre bound handles every degree bound. A further record, not yet independently refereed, gives a query on binary structures, definable with rank over F₂ alone, that CPT with all tower operators cannot define; so over all finite structures towers and rank are incomparable even at one fixed prime. We also include the program's earlier route to noncapture by CPT on these graphs, through a characteristic-two Heisenberg grid; it also shows that CPT fails on one binary vocabulary with colour classes of at most 2²⁹ elements whose colour groups are 2-groups of class at most two and exponent dividing four, where, by the capture theorem for bounded colour classes deposited earlier from this program, rank logic captures polynomial time. The proofs were produced by AI models; all but the fixed-prime record were audited by independent AI referees, and 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. Three of the four proof records were audited by independent AI referees, OpenAI models each run in a session separate from the one that produced the proof. The audit of the tower record returned the verdict "proved" for its lower bound on graphs of maximum degree three and for its capture, height and prime results, and "proved with fixable gaps" for the construction with parent bound two, which Claude had proposed and which the referee completed; the audit of the two characteristic-two records returned "proved" for both, with no fatal or major issue. Every repair the referees requested is incorporated in the paper (Appendix A), and the proof records themselves are unchanged. The fourth record, the fixed-prime separation (Theorem 5), has not yet been independently refereed, and the paper labels it so. The paper also states three results from a further record of the program on witnessed symmetric choice; they are AI-produced and not yet refereed, and are labelled so. The referees reran the finite checks, and Claude reran all of them again for this deposit, reproducing the recorded outputs. The results depend on the recent OpenAI preprint on CPT and on three earlier deposits of this program (doi:10.5281/zenodo.23996260, doi:10.5281/zenodo.24024481 and doi:10.5281/zenodo.24025833), none of them refereed by humans. There has been no human refereeing yet, and there is no machine-checked formal proof. We are confident that the AI-refereed 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. Further records, timestamped here for priority. CPT_BOUNDED_ARITY_QUANTIFIERS_PROOF_2026-11-07.txt (an AI-produced proof record from the same program, not part of this paper): for every k, a total polynomial-time query, on structures in a fixed vocabulary of maximum arity 8k+2, that is not definable in CPT with counting extended by every isomorphism-invariant Lindström quantifier whose argument relations have arity at most k and are taken over the input universe, as the record specifies. SHA-256 50d99a67f934295df79c5108e1d7597c190e634beba7e1727acd87c679278865. Its AI referee report, CPT_BOUNDED_ARITY_QUANTIFIERS_REFEREE_REPORT_2026-11-07.txt (an OpenAI model in a session separate from the one that produced the proof; not human review), returned the verdict "proved", requesting only minor revisions (no fatal or major issue). SHA-256 7797885360fc197dad2765e922f8b7f8f9bae941fa39352902aa6161416c4005. Files. degree-three-cpt-v1.pdf: the paper (76 pages; Appendix B reproduces the tower record verbatim). SHA-256 5f0fe7b286a2a1454ec61706064601ebd8912db33b216f1a2b71398573f15a70 degree-three-cpt-v1.zip: the paper and its LaTeX source, the four proof records, the two AI referee reports, the finite checks with Claude's reruns, a README and SHA256SUMS. SHA-256 8cceb8d7fcbf2968783bd5965065d9319e9b905dca70a322a1f726498d5d43f6 Records in the zip: [T] proofs/CPT_BOUNDED_FIBRE_TOWER_OPERATOR_2026-11-07.txt (the tower record; reproduced in Appendix B of the paper). SHA-256 9e73b6a5171e3a026477c47bd7eb7476a3902b6e027a58b002adaa7c81484d9f [T2] proofs/CPT_TOWER_FIXED_PRIME_RANK_SEPARATION.txt (the fixed-prime record; not yet independently refereed). SHA-256 2b3f2b6cd000bc0ec14e80aa4fd708adb30a4e2d81d364674e95b1d9e76992d2 [S] proofs/CPT_SUBCUBIC_SEPARATION_PROOF_2026-11-05.txt (CPT fails on connected graphs of maximum degree three). SHA-256 76b5a9e62da43a9cb0e0cba5dcd1558fc2e1e88dad510fa9bec8251db27eb7b9 [D1] proofs/CPT_Fails_on_Bounded_2_Group_Colour_Classes_2026-11-05.txt (the characteristic-two grid on bounded colour classes). SHA-256 a28f8e999ec65d75d014ec74923a56c9d8d8c3ae10a01bebda046cd55d8601f0 [RT] ai-referee-reports/CPT_TOWER_SUBCUBIC_REFEREE_REPORT_2026-11-06.txt (AI referee report on [T] and the construction with parent bound two). SHA-256 a9da8e7ba5b93d8e79add6c29e6613d04d342a489ab234ed9d29c85dec2c2289 [RDS] ai-referee-reports/CPT_Characteristic_2_Referee_Report_2026-11-06.txt (AI referee report on [D1] and [S]). SHA-256 fc3ffdda206dac1cb82636626520212f5fae69ec3be97640570330c61a1d83a9 Cited in the paper and not deposited: DEGREE3_WSC_RESULTS_2026-11-07.txt (AI-produced record on witnessed symmetric choice on these graphs; not yet refereed). SHA-256 b367dc6757900f96ec1e678a2b122cbcdff8e2c3270bbcf3f76beeab69bee39d 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: W7225567882