{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VFDABBNKNQOXYRJ4FMH6CSD67Z","short_pith_number":"pith:VFDABBNK","canonical_record":{"source":{"id":"2607.24844","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-24T17:47:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0beeb9de8e364ff361365bd4895c7a3a2439c4a36048e17138a53568e80b47e7","abstract_canon_sha256":"e7457d4f51b3d819c406d0ebde2f6cd99ed34660d8831a32d443b3dd32b11ac1"},"schema_version":"1.0"},"canonical_sha256":"a9460085aa6c1d7c453c2b0fe1487efe6f9ee6283c01c1f2d4871c8084e46921","source":{"kind":"arxiv","id":"2607.24844","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.24844","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.24844v1","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.24844","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"pith_short_12","alias_value":"VFDABBNKNQOX","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"pith_short_16","alias_value":"VFDABBNKNQOXYRJ4","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"pith_short_8","alias_value":"VFDABBNK","created_at":"2026-07-29T00:24:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VFDABBNKNQOXYRJ4FMH6CSD67Z","target":"record","payload":{"canonical_record":{"source":{"id":"2607.24844","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-24T17:47:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"0beeb9de8e364ff361365bd4895c7a3a2439c4a36048e17138a53568e80b47e7","abstract_canon_sha256":"e7457d4f51b3d819c406d0ebde2f6cd99ed34660d8831a32d443b3dd32b11ac1"},"schema_version":"1.0"},"canonical_sha256":"a9460085aa6c1d7c453c2b0fe1487efe6f9ee6283c01c1f2d4871c8084e46921","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-29T00:24:47.377820Z","signature_b64":"BlW5rd0rNyvsLm5hEOQHQNWBk5aUXhqAprvpQkY8LmAkb2qEOSn3CjLtAfAvfcMB3Gk3VMHIxEGXCYuCMHPnCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9460085aa6c1d7c453c2b0fe1487efe6f9ee6283c01c1f2d4871c8084e46921","last_reissued_at":"2026-07-29T00:24:47.377007Z","signature_status":"signed_v1","first_computed_at":"2026-07-29T00:24:47.377007Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.24844","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":"2026-07-29T00:24:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JUr7zJI2Hj73/aLeS79GAx4Vjh0KBjbjc5XSRa/jxjEZW+u8Gx0QzSgOTOvOepRPYGdHOpJc6upPfPBytiYWAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T07:35:21.172995Z"},"content_sha256":"21b92271c5b4949ad115611da1bb03c4b6f8a8c9f2fcf772aa544614ab3cadd8","schema_version":"1.0","event_id":"sha256:21b92271c5b4949ad115611da1bb03c4b6f8a8c9f2fcf772aa544614ab3cadd8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VFDABBNKNQOXYRJ4FMH6CSD67Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Christoffel words as extremal structures in Collatz dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"Carlos Fern\\'andez, Santiago Ib\\'a\\~nez","submitted_at":"2026-07-24T17:47:57Z","abstract_excerpt":"We study the combinatorial structure of parity sequences associated with the accelerated Collatz map with the goal of identifying extremal configurations and relating them to the existence of periodic orbits. To each finite sequence of an orbit, we associate a binary word whose ones encode the odd iterates, and we introduce a functional $C(d)$ on such words which provides an explicit expression for the iterates and characterizes possible periodic cycles. We define a natural rotation action on binary words, compatible with the cyclic structure of periodic orbits, and consider the functional $C_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24844","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.24844/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":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-29T00:24:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w+6cp7tiwtk8ERe4N9tiL95F6LO1J9zE2OqSqIqjRAd+YDLgmQsa0FIFu7RWfNdb7DPRA0nIOjL1kFx8yA1KBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T07:35:21.173593Z"},"content_sha256":"47d023df41569d5d56c52bbed8a9a44cadbe8b027cfeaa36bc9d7d6423473fa3","schema_version":"1.0","event_id":"sha256:47d023df41569d5d56c52bbed8a9a44cadbe8b027cfeaa36bc9d7d6423473fa3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VFDABBNKNQOXYRJ4FMH6CSD67Z/bundle.json","state_url":"https://pith.science/pith/VFDABBNKNQOXYRJ4FMH6CSD67Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VFDABBNKNQOXYRJ4FMH6CSD67Z/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":"2026-08-12T07:35:21Z","links":{"resolver":"https://pith.science/pith/VFDABBNKNQOXYRJ4FMH6CSD67Z","bundle":"https://pith.science/pith/VFDABBNKNQOXYRJ4FMH6CSD67Z/bundle.json","state":"https://pith.science/pith/VFDABBNKNQOXYRJ4FMH6CSD67Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VFDABBNKNQOXYRJ4FMH6CSD67Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VFDABBNKNQOXYRJ4FMH6CSD67Z","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":"e7457d4f51b3d819c406d0ebde2f6cd99ed34660d8831a32d443b3dd32b11ac1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-24T17:47:57Z","title_canon_sha256":"0beeb9de8e364ff361365bd4895c7a3a2439c4a36048e17138a53568e80b47e7"},"schema_version":"1.0","source":{"id":"2607.24844","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.24844","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.24844v1","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.24844","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"pith_short_12","alias_value":"VFDABBNKNQOX","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"pith_short_16","alias_value":"VFDABBNKNQOXYRJ4","created_at":"2026-07-29T00:24:47Z"},{"alias_kind":"pith_short_8","alias_value":"VFDABBNK","created_at":"2026-07-29T00:24:47Z"}],"graph_snapshots":[{"event_id":"sha256:47d023df41569d5d56c52bbed8a9a44cadbe8b027cfeaa36bc9d7d6423473fa3","target":"graph","created_at":"2026-07-29T00:24:47Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.24844/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the combinatorial structure of parity sequences associated with the accelerated Collatz map with the goal of identifying extremal configurations and relating them to the existence of periodic orbits. To each finite sequence of an orbit, we associate a binary word whose ones encode the odd iterates, and we introduce a functional $C(d)$ on such words which provides an explicit expression for the iterates and characterizes possible periodic cycles. We define a natural rotation action on binary words, compatible with the cyclic structure of periodic orbits, and consider the functional $C_","authors_text":"Carlos Fern\\'andez, Santiago Ib\\'a\\~nez","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-24T17:47:57Z","title":"Christoffel words as extremal structures in Collatz dynamics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.24844","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:21b92271c5b4949ad115611da1bb03c4b6f8a8c9f2fcf772aa544614ab3cadd8","target":"record","created_at":"2026-07-29T00:24:47Z","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":"e7457d4f51b3d819c406d0ebde2f6cd99ed34660d8831a32d443b3dd32b11ac1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-07-24T17:47:57Z","title_canon_sha256":"0beeb9de8e364ff361365bd4895c7a3a2439c4a36048e17138a53568e80b47e7"},"schema_version":"1.0","source":{"id":"2607.24844","kind":"arxiv","version":1}},"canonical_sha256":"a9460085aa6c1d7c453c2b0fe1487efe6f9ee6283c01c1f2d4871c8084e46921","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9460085aa6c1d7c453c2b0fe1487efe6f9ee6283c01c1f2d4871c8084e46921","first_computed_at":"2026-07-29T00:24:47.377007Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-29T00:24:47.377007Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BlW5rd0rNyvsLm5hEOQHQNWBk5aUXhqAprvpQkY8LmAkb2qEOSn3CjLtAfAvfcMB3Gk3VMHIxEGXCYuCMHPnCQ==","signature_status":"signed_v1","signed_at":"2026-07-29T00:24:47.377820Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.24844","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21b92271c5b4949ad115611da1bb03c4b6f8a8c9f2fcf772aa544614ab3cadd8","sha256:47d023df41569d5d56c52bbed8a9a44cadbe8b027cfeaa36bc9d7d6423473fa3"],"state_sha256":"2725279f6b4dcafe8749447917740971fc2c97f7dcd45f0c70652142ab514687"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1X+my/d0lMmKCnKw6Q+kifUKchjDBoMjePEeLzRmgmfumdLbiqBTPIUQx7uZ8DqG/Gh4pjnT/JQdf94sYkQBCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T07:35:21.178621Z","bundle_sha256":"8fc44e4d40854a6d2301a009ef2ab14e31b04e66cf8d67bda2f5b50c26eac843"}}