{"paper":{"title":"On a Constraint on Invariant Measures of Certain Cellular Automata","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata.","cross_cats":["cs.FL","math.PR"],"primary_cat":"math.DS","authors_text":"Matan Tal","submitted_at":"2026-04-11T09:38:43Z","abstract_excerpt":"In [6], a constraint on invariant measures of bi-permutative cellular automata has been observed: fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index. When the alphabet is a finite group and the automaton is multiplication of two neighbors, that set is in fact a coset of some subgroup. In the present paper, we strengthen the formulations in [6] and investigate further the implications of this constraint. In the finite group case mentioned above, relations between some attrib"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index; in the finite group multiplication case this set is a coset of some subgroup. The constraint induces a factor with respect to the shift, and zero-entropy invariant measures on that factor correspond to positive-entropy measures on the original system.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The conjecture that the class of RLP subshifts is much larger than bi-permutative cellular automata, while only one additional example is proved to belong to it; the paper relies on the prior observation in [6] without re-deriving it from scratch.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A constraint linking fixed positive-index values to uniform conditional probabilities at the zero index is strengthened for bi-permutative cellular automata, with group-structure relations, zero-entropy factors, and a partial generalization to RLP subshifts.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6117f3737fbbf53cb3a14fa1f05e54f7ecb3c93cd417ffb79be71fe3acb13719"},"source":{"id":"2604.10124","kind":"arxiv","version":4},"verdict":{"id":"83ba2a70-6439-47b1-b54e-309463b853e1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:29:40.916764Z","strongest_claim":"Fixed values at the positive indices determine almost-surely a uniform conditional probability on the subset of values of positive conditional probability at the zero index; in the finite group multiplication case this set is a coset of some subgroup. The constraint induces a factor with respect to the shift, and zero-entropy invariant measures on that factor correspond to positive-entropy measures on the original system.","one_line_summary":"A constraint linking fixed positive-index values to uniform conditional probabilities at the zero index is strengthened for bi-permutative cellular automata, with group-structure relations, zero-entropy factors, and a partial generalization to RLP subshifts.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The conjecture that the class of RLP subshifts is much larger than bi-permutative cellular automata, while only one additional example is proved to belong to it; the paper relies on the prior observation in [6] without re-deriving it from scratch.","pith_extraction_headline":"Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.10124/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}