{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:LLISXHK35TQL2E2CYTTDV35QFX","short_pith_number":"pith:LLISXHK3","canonical_record":{"source":{"id":"2407.08203","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-11T06:02:27Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"c3c3ae7ecffb972ad30e0751c57cd601a8c47d2404f7d290e083f136aa006cdb","abstract_canon_sha256":"56d0e404890da76fb60ebd612c13b590c305706e8d9035f64f76fb50d29784f7"},"schema_version":"1.0"},"canonical_sha256":"5ad12b9d5bece0bd1342c4e63aefb02dd64eb6d730abed89638af3c9019e8c95","source":{"kind":"arxiv","id":"2407.08203","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.08203","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"arxiv_version","alias_value":"2407.08203v3","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.08203","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"pith_short_12","alias_value":"LLISXHK35TQL","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"pith_short_16","alias_value":"LLISXHK35TQL2E2C","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"pith_short_8","alias_value":"LLISXHK3","created_at":"2026-07-05T10:25:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:LLISXHK35TQL2E2CYTTDV35QFX","target":"record","payload":{"canonical_record":{"source":{"id":"2407.08203","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-11T06:02:27Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"c3c3ae7ecffb972ad30e0751c57cd601a8c47d2404f7d290e083f136aa006cdb","abstract_canon_sha256":"56d0e404890da76fb60ebd612c13b590c305706e8d9035f64f76fb50d29784f7"},"schema_version":"1.0"},"canonical_sha256":"5ad12b9d5bece0bd1342c4e63aefb02dd64eb6d730abed89638af3c9019e8c95","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:25:08.086122Z","signature_b64":"BiulBsqsSotGAVT7SjRQ+NqQf7vfkubkfJRPaPb5PbAGYCt5pmTf96WkB3Sfura1DrJBJYGnlNr+k8kuCqlVDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ad12b9d5bece0bd1342c4e63aefb02dd64eb6d730abed89638af3c9019e8c95","last_reissued_at":"2026-07-05T10:25:08.085564Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:25:08.085564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.08203","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T10:25:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9CYjj4b0ZyNoDquorE1Lp91Ro14v1kIGJcCeYN8mH3w3SfeTMXny8qnYg/YWklpvvjZ2FqmllY6jb4pKY5g6CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T03:20:09.843947Z"},"content_sha256":"e0f336d51b1c30686e74823f2d635e6d4f80a8be0888f5defc3093a2b511da77","schema_version":"1.0","event_id":"sha256:e0f336d51b1c30686e74823f2d635e6d4f80a8be0888f5defc3093a2b511da77"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:LLISXHK35TQL2E2CYTTDV35QFX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"SL(2,Z)-matrixizations of generalized Markov numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.NT","authors_text":"Shuhei Maruyama, Yasuaki Gyoda, Yusuke Sato","submitted_at":"2024-07-11T06:02:27Z","abstract_excerpt":"For $k\\geq 0$, a $k$-generalized Markov number is an integer which appears in some positive integer solution to the $k$-generalized Markov equation $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 + 3k)xyz$. In this paper, we discuss a combinatorial structure of generalized Markov numbers. To investigate this structure in detail, we use two families of matrices: the $k$-generalized Cohn matrices and the $k$-Markov-monodromy matrices, which are elements of $SL(2, \\mathbb{Z})$ whose $(1,2)$-entries are $k$-generalized Markov numbers. We show that these two families of matrices recover the tree structure "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.08203","kind":"arxiv","version":3},"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/2407.08203/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":"2026-07-05T10:25:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4W1vf9+VdK5+SbqXXZ0yxL0HMET1LgONs9hri5/waDqzPCm8tOAyf/+k9yUz3A6OXj0MZTzoJmxiGF9RX7QzAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T03:20:09.844236Z"},"content_sha256":"2c7d26c99876b5e2128f8279fbc88c977f94830069e3ed18e192f164e23ffb42","schema_version":"1.0","event_id":"sha256:2c7d26c99876b5e2128f8279fbc88c977f94830069e3ed18e192f164e23ffb42"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LLISXHK35TQL2E2CYTTDV35QFX/bundle.json","state_url":"https://pith.science/pith/LLISXHK35TQL2E2CYTTDV35QFX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LLISXHK35TQL2E2CYTTDV35QFX/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":"2026-08-15T03:20:09Z","links":{"resolver":"https://pith.science/pith/LLISXHK35TQL2E2CYTTDV35QFX","bundle":"https://pith.science/pith/LLISXHK35TQL2E2CYTTDV35QFX/bundle.json","state":"https://pith.science/pith/LLISXHK35TQL2E2CYTTDV35QFX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LLISXHK35TQL2E2CYTTDV35QFX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:LLISXHK35TQL2E2CYTTDV35QFX","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":"56d0e404890da76fb60ebd612c13b590c305706e8d9035f64f76fb50d29784f7","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-11T06:02:27Z","title_canon_sha256":"c3c3ae7ecffb972ad30e0751c57cd601a8c47d2404f7d290e083f136aa006cdb"},"schema_version":"1.0","source":{"id":"2407.08203","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.08203","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"arxiv_version","alias_value":"2407.08203v3","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.08203","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"pith_short_12","alias_value":"LLISXHK35TQL","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"pith_short_16","alias_value":"LLISXHK35TQL2E2C","created_at":"2026-07-05T10:25:08Z"},{"alias_kind":"pith_short_8","alias_value":"LLISXHK3","created_at":"2026-07-05T10:25:08Z"}],"graph_snapshots":[{"event_id":"sha256:2c7d26c99876b5e2128f8279fbc88c977f94830069e3ed18e192f164e23ffb42","target":"graph","created_at":"2026-07-05T10:25:08Z","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/2407.08203/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For $k\\geq 0$, a $k$-generalized Markov number is an integer which appears in some positive integer solution to the $k$-generalized Markov equation $x^2 + y^2 + z^2 + k(yz + zx + xy) = (3 + 3k)xyz$. In this paper, we discuss a combinatorial structure of generalized Markov numbers. To investigate this structure in detail, we use two families of matrices: the $k$-generalized Cohn matrices and the $k$-Markov-monodromy matrices, which are elements of $SL(2, \\mathbb{Z})$ whose $(1,2)$-entries are $k$-generalized Markov numbers. We show that these two families of matrices recover the tree structure ","authors_text":"Shuhei Maruyama, Yasuaki Gyoda, Yusuke Sato","cross_cats":["math.AG","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-11T06:02:27Z","title":"SL(2,Z)-matrixizations of generalized Markov numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.08203","kind":"arxiv","version":3},"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:e0f336d51b1c30686e74823f2d635e6d4f80a8be0888f5defc3093a2b511da77","target":"record","created_at":"2026-07-05T10:25:08Z","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":"56d0e404890da76fb60ebd612c13b590c305706e8d9035f64f76fb50d29784f7","cross_cats_sorted":["math.AG","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-11T06:02:27Z","title_canon_sha256":"c3c3ae7ecffb972ad30e0751c57cd601a8c47d2404f7d290e083f136aa006cdb"},"schema_version":"1.0","source":{"id":"2407.08203","kind":"arxiv","version":3}},"canonical_sha256":"5ad12b9d5bece0bd1342c4e63aefb02dd64eb6d730abed89638af3c9019e8c95","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ad12b9d5bece0bd1342c4e63aefb02dd64eb6d730abed89638af3c9019e8c95","first_computed_at":"2026-07-05T10:25:08.085564Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:25:08.085564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BiulBsqsSotGAVT7SjRQ+NqQf7vfkubkfJRPaPb5PbAGYCt5pmTf96WkB3Sfura1DrJBJYGnlNr+k8kuCqlVDw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:25:08.086122Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.08203","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e0f336d51b1c30686e74823f2d635e6d4f80a8be0888f5defc3093a2b511da77","sha256:2c7d26c99876b5e2128f8279fbc88c977f94830069e3ed18e192f164e23ffb42"],"state_sha256":"0fcdd84c5d64929913d0dfe4f0ae8fc012f3db6a459567b71e61ee6fdc8b2556"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jj+yDgJn25LQPh8Mt5zqm7stiSCMNR+/aMfp+1a0iRw/Z3loSHfeKjSGybO/UOoWm62JATg8fNAatm+ta6zOAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T03:20:09.847815Z","bundle_sha256":"737383c4b93acde57368e6f7f03a8bfb431953f72ae79bdd92036abfe6702843"}}