{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:PANEHHS6CTJRVATXHQF55YVOGP","short_pith_number":"pith:PANEHHS6","canonical_record":{"source":{"id":"2605.14234","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-14T00:59:30Z","cross_cats_sorted":[],"title_canon_sha256":"a35cc5342783cc7ccbaf14eee78eff3dc7fe3f7ec5a7d6847f3bb05a6714aade","abstract_canon_sha256":"e7cec9a6e9b300c83f65ec0759b2cf287ddf9832a238f56bf055e041251e4784"},"schema_version":"1.0"},"canonical_sha256":"781a439e5e14d31a82773c0bdee2ae33fe12cfc911b76c92865926e4112c815b","source":{"kind":"arxiv","id":"2605.14234","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14234","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14234v1","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14234","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"pith_short_12","alias_value":"PANEHHS6CTJR","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"PANEHHS6CTJRVATX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"PANEHHS6","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:PANEHHS6CTJRVATXHQF55YVOGP","target":"record","payload":{"canonical_record":{"source":{"id":"2605.14234","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-14T00:59:30Z","cross_cats_sorted":[],"title_canon_sha256":"a35cc5342783cc7ccbaf14eee78eff3dc7fe3f7ec5a7d6847f3bb05a6714aade","abstract_canon_sha256":"e7cec9a6e9b300c83f65ec0759b2cf287ddf9832a238f56bf055e041251e4784"},"schema_version":"1.0"},"canonical_sha256":"781a439e5e14d31a82773c0bdee2ae33fe12cfc911b76c92865926e4112c815b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:10.711809Z","signature_b64":"9/L/wH+gYDDKnNS0wKzbH6RRxW5fDwAvLiTEaOUd7WCyLpEQV9X9k/Vn47BOnWPHT6qNW+eU4NmV06YfpYGtBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"781a439e5e14d31a82773c0bdee2ae33fe12cfc911b76c92865926e4112c815b","last_reissued_at":"2026-05-17T23:39:10.711358Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:10.711358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.14234","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HNtPLyJE7cFw/Xl0sUxiFx9iuJTTinaRG21AelKW3h6N/HOz+jhqfOlhP6VxWDET47m9AZCqADs/KuOWSmiiDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T14:14:56.069547Z"},"content_sha256":"f8524bb75e0044793b8631595c8a99c94a3a9ca0aa6d2ef3a06ecbc992f97d23","schema_version":"1.0","event_id":"sha256:f8524bb75e0044793b8631595c8a99c94a3a9ca0aa6d2ef3a06ecbc992f97d23"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:PANEHHS6CTJRVATXHQF55YVOGP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Group Theory of the Kolakoski Sequence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Noah MacAulay","submitted_at":"2026-05-14T00:59:30Z","abstract_excerpt":"Run-length decoding is an operation on sequences in which a positive integer $a$ is replaced by a run(sequence of repeated elements) of length $a$. Iterated run-length decodings applied to sequences with alphabets consisting of pairs of positive integers $\\{p, q\\}$ have attracted attention from mathematicians, most notably in their role defining the well-known Kolakoski sequence. $n$-th-iterated run-length decodings are controlled by naturally associated permutation automata $A^{p,q}_n$. Here we study the transformation groups $\\mathcal{K}^{p,q}_n$ of these automata. They are subgroups of the "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd n.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the transformation groups K^{p,q}_n are subgroups of (and likely equal to) the recursively defined group J_n^{p,q} whose limit is weakly regular branch.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Transformation groups of run-length decoding automata for Kolakoski-like sequences are subgroups of binary tree automorphisms with recursive structure, allowing exact count of maximal-length orbits when the iteration depth is odd.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"48099d468852d75b6a64ec764a2ae261bbcf0bb7e2090a4ae1b757153d2fd589"},"source":{"id":"2605.14234","kind":"arxiv","version":1},"verdict":{"id":"1a7e53dd-7b6c-44bf-98ea-0b77216a631b","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:42:53.011340Z","strongest_claim":"As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd n.","one_line_summary":"Transformation groups of run-length decoding automata for Kolakoski-like sequences are subgroups of binary tree automorphisms with recursive structure, allowing exact count of maximal-length orbits when the iteration depth is odd.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the transformation groups K^{p,q}_n are subgroups of (and likely equal to) the recursively defined group J_n^{p,q} whose limit is weakly regular branch.","pith_extraction_headline":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths."},"references":{"count":5,"sample":[{"doi":"","year":1939,"title":"Rufus Oldenburger, Exponent trajectories in symbolic dynamics, Trans. Amer. Math. Soc., Vol. 46 (1939), pp. 453-466","work_id":"a1f06b72-18fb-4db4-abd1-725721e76050","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"Kolakoski, Self Generating Runs, Problem 5304, American Math","work_id":"3eed7634-77da-48a4-8e07-b974219bbb35","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"The Kolakoski sequence and related conjectures about orbits","work_id":"41b9176e-3671-42df-90e0-85a25207cc96","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"Notes on the Kolakoski sequence","work_id":"9d6b0f83-56db-4a8d-9832-3f7cae865664","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Branch groups","work_id":"f2549d1b-fc81-4e1b-bbe5-c705324e8e20","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":5,"snapshot_sha256":"2c2c97cfeb31176f2595baaf543ac4ff0ebd272b29e09ba28878b712df8ac6c2","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"088287ea65fe4512288c80f224a5cd7b372f5ec3dcab37f5031272fc48610359"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"1a7e53dd-7b6c-44bf-98ea-0b77216a631b"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:39:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C9slreme8EXWEyvNphfPkdD1qs1gVHTba39DIkc8ApyS6k4AzrAfKVrTY6up/lBFrdU37SsRyxU8hPGoc2ipCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T14:14:56.070048Z"},"content_sha256":"8bf538fb08ac01ac97d2f20aae75fa42751756420c288eb4de66c2a5deb79e9e","schema_version":"1.0","event_id":"sha256:8bf538fb08ac01ac97d2f20aae75fa42751756420c288eb4de66c2a5deb79e9e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PANEHHS6CTJRVATXHQF55YVOGP/bundle.json","state_url":"https://pith.science/pith/PANEHHS6CTJRVATXHQF55YVOGP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PANEHHS6CTJRVATXHQF55YVOGP/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":"2026-06-10T14:14:56Z","links":{"resolver":"https://pith.science/pith/PANEHHS6CTJRVATXHQF55YVOGP","bundle":"https://pith.science/pith/PANEHHS6CTJRVATXHQF55YVOGP/bundle.json","state":"https://pith.science/pith/PANEHHS6CTJRVATXHQF55YVOGP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PANEHHS6CTJRVATXHQF55YVOGP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:PANEHHS6CTJRVATXHQF55YVOGP","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":"e7cec9a6e9b300c83f65ec0759b2cf287ddf9832a238f56bf055e041251e4784","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-14T00:59:30Z","title_canon_sha256":"a35cc5342783cc7ccbaf14eee78eff3dc7fe3f7ec5a7d6847f3bb05a6714aade"},"schema_version":"1.0","source":{"id":"2605.14234","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14234","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14234v1","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14234","created_at":"2026-05-17T23:39:10Z"},{"alias_kind":"pith_short_12","alias_value":"PANEHHS6CTJR","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"PANEHHS6CTJRVATX","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"PANEHHS6","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:8bf538fb08ac01ac97d2f20aae75fa42751756420c288eb4de66c2a5deb79e9e","target":"graph","created_at":"2026-05-17T23:39:10Z","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":"As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd n."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"That the transformation groups K^{p,q}_n are subgroups of (and likely equal to) the recursively defined group J_n^{p,q} whose limit is weakly regular branch."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Transformation groups of run-length decoding automata for Kolakoski-like sequences are subgroups of binary tree automorphisms with recursive structure, allowing exact count of maximal-length orbits when the iteration depth is odd."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths."}],"snapshot_sha256":"48099d468852d75b6a64ec764a2ae261bbcf0bb7e2090a4ae1b757153d2fd589"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"088287ea65fe4512288c80f224a5cd7b372f5ec3dcab37f5031272fc48610359"},"paper":{"abstract_excerpt":"Run-length decoding is an operation on sequences in which a positive integer $a$ is replaced by a run(sequence of repeated elements) of length $a$. Iterated run-length decodings applied to sequences with alphabets consisting of pairs of positive integers $\\{p, q\\}$ have attracted attention from mathematicians, most notably in their role defining the well-known Kolakoski sequence. $n$-th-iterated run-length decodings are controlled by naturally associated permutation automata $A^{p,q}_n$. Here we study the transformation groups $\\mathcal{K}^{p,q}_n$ of these automata. They are subgroups of the ","authors_text":"Noah MacAulay","cross_cats":[],"headline":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-14T00:59:30Z","title":"Group Theory of the Kolakoski Sequence"},"references":{"count":5,"internal_anchors":0,"resolved_work":5,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Rufus Oldenburger, Exponent trajectories in symbolic dynamics, Trans. Amer. Math. Soc., Vol. 46 (1939), pp. 453-466","work_id":"a1f06b72-18fb-4db4-abd1-725721e76050","year":1939},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Kolakoski, Self Generating Runs, Problem 5304, American Math","work_id":"3eed7634-77da-48a4-8e07-b974219bbb35","year":1965},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"The Kolakoski sequence and related conjectures about orbits","work_id":"41b9176e-3671-42df-90e0-85a25207cc96","year":2020},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Notes on the Kolakoski sequence","work_id":"9d6b0f83-56db-4a8d-9832-3f7cae865664","year":1994},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Branch groups","work_id":"f2549d1b-fc81-4e1b-bbe5-c705324e8e20","year":2003}],"snapshot_sha256":"2c2c97cfeb31176f2595baaf543ac4ff0ebd272b29e09ba28878b712df8ac6c2"},"source":{"id":"2605.14234","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T02:42:53.011340Z","id":"1a7e53dd-7b6c-44bf-98ea-0b77216a631b","model_set":{"reader":"grok-4.3"},"one_line_summary":"Transformation groups of run-length decoding automata for Kolakoski-like sequences are subgroups of binary tree automorphisms with recursive structure, allowing exact count of maximal-length orbits when the iteration depth is odd.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Transformation groups of run-length decoding automata for Kolakoski sequences permit explicit counting of maximal orbits for odd iteration depths.","strongest_claim":"As an application we determine the number of maximal-length orbits of the automata given an arbitrary input sequence for odd n.","weakest_assumption":"That the transformation groups K^{p,q}_n are subgroups of (and likely equal to) the recursively defined group J_n^{p,q} whose limit is weakly regular branch."}},"verdict_id":"1a7e53dd-7b6c-44bf-98ea-0b77216a631b"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f8524bb75e0044793b8631595c8a99c94a3a9ca0aa6d2ef3a06ecbc992f97d23","target":"record","created_at":"2026-05-17T23:39:10Z","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":"e7cec9a6e9b300c83f65ec0759b2cf287ddf9832a238f56bf055e041251e4784","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-14T00:59:30Z","title_canon_sha256":"a35cc5342783cc7ccbaf14eee78eff3dc7fe3f7ec5a7d6847f3bb05a6714aade"},"schema_version":"1.0","source":{"id":"2605.14234","kind":"arxiv","version":1}},"canonical_sha256":"781a439e5e14d31a82773c0bdee2ae33fe12cfc911b76c92865926e4112c815b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"781a439e5e14d31a82773c0bdee2ae33fe12cfc911b76c92865926e4112c815b","first_computed_at":"2026-05-17T23:39:10.711358Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:10.711358Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9/L/wH+gYDDKnNS0wKzbH6RRxW5fDwAvLiTEaOUd7WCyLpEQV9X9k/Vn47BOnWPHT6qNW+eU4NmV06YfpYGtBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:10.711809Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14234","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f8524bb75e0044793b8631595c8a99c94a3a9ca0aa6d2ef3a06ecbc992f97d23","sha256:8bf538fb08ac01ac97d2f20aae75fa42751756420c288eb4de66c2a5deb79e9e"],"state_sha256":"19dc7bb39a23ac6417839ed900c6118a819488551ac4bb15d4d446585fffb6bd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QYX1dBJlWjsaWuTEXPGKo05BwIac6rloLQT/FMqBRNnGb3mrX/EWkmmHHi88UYtYUaepvlhruSOT9cIfdcelCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T14:14:56.072370Z","bundle_sha256":"9d928c1df24300829009e63e213c6d42eb7de041f6d39f0b2ffc22f7924ac3cb"}}