The A-group dichotomy for choiceless polynomial time on bounded colour classes
Year: 2026
Type: preprint
Abstract: Let π be a finite family of finite groups closed under isomorphism and subgroups, and let π¦(q,Ο,π) be the full class of finite Ο-structures that are supplied with a linear preorder whose classes have at most q elements and whose induced colour groups Aut(A[C]) all belong to π. We show that choiceless polynomial time with counting (CPT) canonizes π¦(q,Ο,π) for every q and every finite relational vocabulary Ο if and only if every group in π is an A-group, that is, has all its Sylow subgroups abelian. The positive direction covers nonsolvable A-groups such as Aβ
. It decides exact feasibility of bounded coset systems over A-groups without choices, using the abelian-colour canonizer of Abu Zaid, GrΓ€del, Grohe and Pakusa as its only imported logical result. To our knowledge, its new ingredients are the use of the residual series of the class of groups of the form abelian times a product of nonabelian simple groups to locate the top kernels, an exact identity for the top kernels of subdirect constraint groups, the splitting of abelian kernels into whole-variable blocks, the synchronization of nonabelian kernels through whole torsors, and a group-order potential that makes the loop polynomial. For the negative direction, every non-A-group contains a minimal nonabelian p-group M. For each such M we give a vocabulary of maximum arity eight, colour bound |M|, and a total polynomial-time query, feasibility of a Mal'tsev coset system, that CPT does not define on the full class whose colour groups are subgroups of M. The witnesses place regular M-colours and cyclic colours on the periodic Freudenthal triangulation of the three-torus, glued by an exact cup-product cocycle. Their analysis combines a sparse-filling lemma on the cubical torus with the supported hereditarily finite transfer theorem of the recent OpenAI preprint on CPT. So CPT captures polynomial time on these classes for all colour bounds and vocabularies exactly when the family consists of A-groups. If the capture theorem for rank logic on bounded colour classes, deposited earlier from the same program (its proofs, like these, are not yet refereed by humans), is accepted, then rank logic captures polynomial time on every such class, and CPT and rank logic coincide on these classes, for all colour bounds and vocabularies, exactly at the A-group boundary. The negative witnesses have arity eight; whether CPT already fails on ternary structures with Qβ or Dβ colour groups is open. The proofs were produced by an AI model and audited by two independent AI referees, both of which returned the verdict proved; nothing has yet been refereed by humans or formally verified. Provenance and status. The proofs in this deposit were produced by an AI model (an OpenAI GPT model, working in a separate chat thread) in an AI-assisted research program directed by Matthew Liston. They were checked in part by Claude (Anthropic), which also coordinated the program. The proof record was audited by two independent AI referees, OpenAI models each run in a session separate from the one that produced the proof: one audited the positive direction and one the negative direction. Both returned the verdict "proved", with no fatal or major issue; their six minor clarifications are incorporated in the paper (Appendix A), and the proof record itself is unchanged. The referee of the negative direction reran the proving thread's finite checks, which reproduced their recorded outputs byte for byte, and implemented the sparse-filling construction independently; Claude checked the local algebra for nine minimal nonabelian groups with its own code, and reran all of these scripts for this deposit. The negative direction relies on the transfer theorem and computation-encoding lemmas of the OpenAI preprint (Theorem 14 and Section 6), which the referee checked against its text; that preprint is itself new. There has been no human refereeing yet, and there is no machine-checked formal proof. We are confident that the results are correct: the two audits were independent, every repair they requested is incorporated, the referee reran the finite checks, and Claude checked the local algebra independently. Formal verification is planned; the program's formal-verification capacity is currently committed to other results, so we are releasing this version now. Timestamp. The SHA-256 of the proof record [P] (c743054435cfedacc09112eb84dbb6589212946f52b60aa75fa41380984d3a0a) was first published on 6 November 2026 in the description of the CPT deposit, doi:10.5281/zenodo.24024481. Files. a-group-dichotomy-cpt-v1.pdf: the paper (61 pages; Appendix B reproduces the proof record verbatim). SHA-256 e56dc348e86ce447f6bcdf900e4f9cd908d06e2d609bc828a867d930e3bd2824 a-group-dichotomy-cpt-v1.zip: the paper and its LaTeX source, the proof record, the two AI referee reports, the proving thread's verification archive, Claude's independent check, the reruns, a README and SHA256SUMS. SHA-256 5dd6e400d1e7b07e63c1857c1b58910d805b4ffb0e03a14edb591d5ed38e3e5f Records in the zip: [P] proofs/CPT_A_GROUP_DICHOTOMY_PROOF_2026-11-06.txt (the proof record). SHA-256 c743054435cfedacc09112eb84dbb6589212946f52b60aa75fa41380984d3a0a [RA] ai-referee-reports/CPT_A_GROUP_DICHOTOMY_PART_I_REFEREE_REPORT_2026-11-06.txt (AI referee report on the positive direction). SHA-256 3b39cb3a9ee3c06afca571a46dec904c45b5e0fcecac99ad8398b31f11d4c521 [RB] ai-referee-reports/CPT_A_GROUP_NEGATIVE_REFEREE_REPORT_2026-11-06.txt (AI referee report on the negative direction). SHA-256 9369a9a53d04cb325506e8a64b986db1405236f55ec7102f19dc080ef6022cc9 [V] verification/CPT_A_GROUP_DICHOTOMY_VERIFICATION_2026-11-06.zip (the proving thread's finite checks). SHA-256 665ad03ec7430b198c571094b1b4832b7a68b72c1819e444c72819bf637b5213 [CK] checks/claude_b_local_algebra_check.py (Claude's independent check). SHA-256 d7a9da166b9353db6fcc7a4c959b36b04f1804dea1f2964db410960084b80dfa Contact: [email protected]
Author: Liston, Matthew
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225454350