{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:V5PQ6DHXOWFEZIN5ORW6IKSXMH","short_pith_number":"pith:V5PQ6DHX","canonical_record":{"source":{"id":"2604.10124","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-04-11T09:38:43Z","cross_cats_sorted":["cs.FL","math.PR"],"title_canon_sha256":"bb4b5f2ae4b36e26bfc919fcb976156300465a3e9371b81151eb035c76c8284a","abstract_canon_sha256":"c613cf2f4264ed54e2e93e6cbc680b73eab1905cbcb7e3842aac6b51e1cfdc0b"},"schema_version":"1.0"},"canonical_sha256":"af5f0f0cf7758a4ca1bd746de42a5761c3fb55198a0420f85b6ae981af5a23f2","source":{"kind":"arxiv","id":"2604.10124","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.10124","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"arxiv_version","alias_value":"2604.10124v4","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.10124","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"pith_short_12","alias_value":"V5PQ6DHXOWFE","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"pith_short_16","alias_value":"V5PQ6DHXOWFEZIN5","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"pith_short_8","alias_value":"V5PQ6DHX","created_at":"2026-05-27T01:04:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:V5PQ6DHXOWFEZIN5ORW6IKSXMH","target":"record","payload":{"canonical_record":{"source":{"id":"2604.10124","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-04-11T09:38:43Z","cross_cats_sorted":["cs.FL","math.PR"],"title_canon_sha256":"bb4b5f2ae4b36e26bfc919fcb976156300465a3e9371b81151eb035c76c8284a","abstract_canon_sha256":"c613cf2f4264ed54e2e93e6cbc680b73eab1905cbcb7e3842aac6b51e1cfdc0b"},"schema_version":"1.0"},"canonical_sha256":"af5f0f0cf7758a4ca1bd746de42a5761c3fb55198a0420f85b6ae981af5a23f2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T01:04:58.035489Z","signature_b64":"GPZ+HQj+D68rhlERN8x8eZTteGS412Ujavlu8zg/+DS2fDpFDnkB+Hs3WThxC9pn/WQlosUQ+8DVOuzBvUSmCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af5f0f0cf7758a4ca1bd746de42a5761c3fb55198a0420f85b6ae981af5a23f2","last_reissued_at":"2026-05-27T01:04:58.034888Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T01:04:58.034888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.10124","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-27T01:04:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LIe5Bl1j6AJwcZlb4pUX76g+BJGyfqfKwXYU7ovVDRd3O1QU4AR4YYRZCzBGc/7NGGneXLoSIWlJnHOiayiZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T12:53:13.005955Z"},"content_sha256":"c88f6e9df63427316f2661fd4fda82e870a0bb8dae9400eebd7341625b3347ca","schema_version":"1.0","event_id":"sha256:c88f6e9df63427316f2661fd4fda82e870a0bb8dae9400eebd7341625b3347ca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:V5PQ6DHXOWFEZIN5ORW6IKSXMH","target":"graph","payload":{"graph_snapshot":{"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"},"verdict_id":"83ba2a70-6439-47b1-b54e-309463b853e1"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-27T01:04:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PUJysKarZZBB8w+n33DlhOauHJ9T7KPqZ3CnumX5X0OvnXPsYlzPyq0tPOk/KukFzYfqQxe9yTeK2X1RJl9YBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T12:53:13.006928Z"},"content_sha256":"9b1cbe4b423656cf220d7af43982796c40f121104f1dac6c103c0433ddd84476","schema_version":"1.0","event_id":"sha256:9b1cbe4b423656cf220d7af43982796c40f121104f1dac6c103c0433ddd84476"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH/bundle.json","state_url":"https://pith.science/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T12:53:13Z","links":{"resolver":"https://pith.science/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH","bundle":"https://pith.science/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH/bundle.json","state":"https://pith.science/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V5PQ6DHXOWFEZIN5ORW6IKSXMH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:V5PQ6DHXOWFEZIN5ORW6IKSXMH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c613cf2f4264ed54e2e93e6cbc680b73eab1905cbcb7e3842aac6b51e1cfdc0b","cross_cats_sorted":["cs.FL","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-04-11T09:38:43Z","title_canon_sha256":"bb4b5f2ae4b36e26bfc919fcb976156300465a3e9371b81151eb035c76c8284a"},"schema_version":"1.0","source":{"id":"2604.10124","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.10124","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"arxiv_version","alias_value":"2604.10124v4","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.10124","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"pith_short_12","alias_value":"V5PQ6DHXOWFE","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"pith_short_16","alias_value":"V5PQ6DHXOWFEZIN5","created_at":"2026-05-27T01:04:58Z"},{"alias_kind":"pith_short_8","alias_value":"V5PQ6DHX","created_at":"2026-05-27T01:04:58Z"}],"graph_snapshots":[{"event_id":"sha256:9b1cbe4b423656cf220d7af43982796c40f121104f1dac6c103c0433ddd84476","target":"graph","created_at":"2026-05-27T01:04:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata."}],"snapshot_sha256":"6117f3737fbbf53cb3a14fa1f05e54f7ecb3c93cd417ffb79be71fe3acb13719"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.10124/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Matan Tal","cross_cats":["cs.FL","math.PR"],"headline":"Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-04-11T09:38:43Z","title":"On a Constraint on Invariant Measures of Certain Cellular Automata"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.10124","kind":"arxiv","version":4},"verdict":{"created_at":"2026-05-10T16:29:40.916764Z","id":"83ba2a70-6439-47b1-b54e-309463b853e1","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"Fixed positive indices determine uniform conditional probabilities on a coset at zero for invariant measures of bi-permutative cellular automata.","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.","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."}},"verdict_id":"83ba2a70-6439-47b1-b54e-309463b853e1"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c88f6e9df63427316f2661fd4fda82e870a0bb8dae9400eebd7341625b3347ca","target":"record","created_at":"2026-05-27T01:04:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c613cf2f4264ed54e2e93e6cbc680b73eab1905cbcb7e3842aac6b51e1cfdc0b","cross_cats_sorted":["cs.FL","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-04-11T09:38:43Z","title_canon_sha256":"bb4b5f2ae4b36e26bfc919fcb976156300465a3e9371b81151eb035c76c8284a"},"schema_version":"1.0","source":{"id":"2604.10124","kind":"arxiv","version":4}},"canonical_sha256":"af5f0f0cf7758a4ca1bd746de42a5761c3fb55198a0420f85b6ae981af5a23f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af5f0f0cf7758a4ca1bd746de42a5761c3fb55198a0420f85b6ae981af5a23f2","first_computed_at":"2026-05-27T01:04:58.034888Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-27T01:04:58.034888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GPZ+HQj+D68rhlERN8x8eZTteGS412Ujavlu8zg/+DS2fDpFDnkB+Hs3WThxC9pn/WQlosUQ+8DVOuzBvUSmCg==","signature_status":"signed_v1","signed_at":"2026-05-27T01:04:58.035489Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.10124","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c88f6e9df63427316f2661fd4fda82e870a0bb8dae9400eebd7341625b3347ca","sha256:9b1cbe4b423656cf220d7af43982796c40f121104f1dac6c103c0433ddd84476"],"state_sha256":"342478a4cd5d4116e77b24b27c2f616699906e6e83d7f919449e58395d8f9162"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BmL0tK4Tpp/G0qcv9cBmoW6i/yDbRQNfjBXttpmblPPu12hpUp8OCBtJ246Iq+T79jTN0YQiEKPlSM70qdgiDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T12:53:13.011807Z","bundle_sha256":"8513368bc8cfb2e1310ffecf6bfdcf3ccfe02107ce18134bf8822216547172a0"}}