{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:U6PCVSFFJQBV7CSHV3XE2XVBKF","short_pith_number":"pith:U6PCVSFF","canonical_record":{"source":{"id":"1501.02027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-09T03:39:38Z","cross_cats_sorted":[],"title_canon_sha256":"9bdb90f5ffaff3e49f680dfb0df01f61483db053951cf39f118d0301a8fd105b","abstract_canon_sha256":"56122556ce20398ab966ac7825e610979443929eced2512ca2802de9ce61a97a"},"schema_version":"1.0"},"canonical_sha256":"a79e2ac8a54c035f8a47aeee4d5ea15159db6df94999ffaf2c9bf92504d91040","source":{"kind":"arxiv","id":"1501.02027","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02027","created_at":"2026-05-18T02:29:45Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02027v1","created_at":"2026-05-18T02:29:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02027","created_at":"2026-05-18T02:29:45Z"},{"alias_kind":"pith_short_12","alias_value":"U6PCVSFFJQBV","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"U6PCVSFFJQBV7CSH","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"U6PCVSFF","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:U6PCVSFFJQBV7CSHV3XE2XVBKF","target":"record","payload":{"canonical_record":{"source":{"id":"1501.02027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-09T03:39:38Z","cross_cats_sorted":[],"title_canon_sha256":"9bdb90f5ffaff3e49f680dfb0df01f61483db053951cf39f118d0301a8fd105b","abstract_canon_sha256":"56122556ce20398ab966ac7825e610979443929eced2512ca2802de9ce61a97a"},"schema_version":"1.0"},"canonical_sha256":"a79e2ac8a54c035f8a47aeee4d5ea15159db6df94999ffaf2c9bf92504d91040","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:45.025997Z","signature_b64":"5w9C1PWbFOL00Kfe5YHEiEjLHH6cP+QuFNLlEFpu79GYO4RsScsbzRNHNwEHVpJwKkJncUv3LJ5EF2puOMikDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a79e2ac8a54c035f8a47aeee4d5ea15159db6df94999ffaf2c9bf92504d91040","last_reissued_at":"2026-05-18T02:29:45.025512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:45.025512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.02027","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":"2026-05-18T02:29:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u6mWEOLUdgoBjZieKqj6uxGsGwgz0uXxji8eB/mK6D5jNctje+QTHTtoSIM5XLzjxco2HXjay7Irb/Or4ZlICQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:04:40.947677Z"},"content_sha256":"e181740eb3e489bd83beceb4f29733ded74dc9e9ece1a23ab2e5f027e69c6c2c","schema_version":"1.0","event_id":"sha256:e181740eb3e489bd83beceb4f29733ded74dc9e9ece1a23ab2e5f027e69c6c2c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:U6PCVSFFJQBV7CSHV3XE2XVBKF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Splines mod m","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Julianna Tymoczko, Nealy Bowden","submitted_at":"2015-01-09T03:39:38Z","abstract_excerpt":"Given a graph whose edges are labeled by ideals in a ring, a generalized spline is a labeling of each vertex by a ring element so that adjacent vertices differ by an element of the ideal associated to the edge. We study splines over the ring Z/mZ. Previous work considered splines over domains, in which very different phenomena occur. For instance when the ring is the integers, the elements of bases for spline modules are indexed by the vertices of the graph. However we prove that over Z/mZ spline modules can essentially have any rank between 1 and n. Using the classification of finite Z-module"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02027","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":""},"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":"2026-05-18T02:29:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GpVO6IYK+VFs0wUPyobRaYKLoNoYLCk1vWrFtK9nbPTHe6tYgp3300nokuJwJmdt7VoKkmFaDzYepcxmn3HvBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:04:40.948066Z"},"content_sha256":"b2172df5be5e5f60e7c9ec7f588ea8faeab2cd0f4ec8c73399e00da7f7bd88c0","schema_version":"1.0","event_id":"sha256:b2172df5be5e5f60e7c9ec7f588ea8faeab2cd0f4ec8c73399e00da7f7bd88c0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U6PCVSFFJQBV7CSHV3XE2XVBKF/bundle.json","state_url":"https://pith.science/pith/U6PCVSFFJQBV7CSHV3XE2XVBKF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U6PCVSFFJQBV7CSHV3XE2XVBKF/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":"2026-06-07T23:04:40Z","links":{"resolver":"https://pith.science/pith/U6PCVSFFJQBV7CSHV3XE2XVBKF","bundle":"https://pith.science/pith/U6PCVSFFJQBV7CSHV3XE2XVBKF/bundle.json","state":"https://pith.science/pith/U6PCVSFFJQBV7CSHV3XE2XVBKF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U6PCVSFFJQBV7CSHV3XE2XVBKF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:U6PCVSFFJQBV7CSHV3XE2XVBKF","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":"56122556ce20398ab966ac7825e610979443929eced2512ca2802de9ce61a97a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-09T03:39:38Z","title_canon_sha256":"9bdb90f5ffaff3e49f680dfb0df01f61483db053951cf39f118d0301a8fd105b"},"schema_version":"1.0","source":{"id":"1501.02027","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02027","created_at":"2026-05-18T02:29:45Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02027v1","created_at":"2026-05-18T02:29:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02027","created_at":"2026-05-18T02:29:45Z"},{"alias_kind":"pith_short_12","alias_value":"U6PCVSFFJQBV","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"U6PCVSFFJQBV7CSH","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"U6PCVSFF","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:b2172df5be5e5f60e7c9ec7f588ea8faeab2cd0f4ec8c73399e00da7f7bd88c0","target":"graph","created_at":"2026-05-18T02:29:45Z","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"},"paper":{"abstract_excerpt":"Given a graph whose edges are labeled by ideals in a ring, a generalized spline is a labeling of each vertex by a ring element so that adjacent vertices differ by an element of the ideal associated to the edge. We study splines over the ring Z/mZ. Previous work considered splines over domains, in which very different phenomena occur. For instance when the ring is the integers, the elements of bases for spline modules are indexed by the vertices of the graph. However we prove that over Z/mZ spline modules can essentially have any rank between 1 and n. Using the classification of finite Z-module","authors_text":"Julianna Tymoczko, Nealy Bowden","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-09T03:39:38Z","title":"Splines mod m"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02027","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:e181740eb3e489bd83beceb4f29733ded74dc9e9ece1a23ab2e5f027e69c6c2c","target":"record","created_at":"2026-05-18T02:29:45Z","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":"56122556ce20398ab966ac7825e610979443929eced2512ca2802de9ce61a97a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-09T03:39:38Z","title_canon_sha256":"9bdb90f5ffaff3e49f680dfb0df01f61483db053951cf39f118d0301a8fd105b"},"schema_version":"1.0","source":{"id":"1501.02027","kind":"arxiv","version":1}},"canonical_sha256":"a79e2ac8a54c035f8a47aeee4d5ea15159db6df94999ffaf2c9bf92504d91040","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a79e2ac8a54c035f8a47aeee4d5ea15159db6df94999ffaf2c9bf92504d91040","first_computed_at":"2026-05-18T02:29:45.025512Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:45.025512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5w9C1PWbFOL00Kfe5YHEiEjLHH6cP+QuFNLlEFpu79GYO4RsScsbzRNHNwEHVpJwKkJncUv3LJ5EF2puOMikDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:45.025997Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.02027","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e181740eb3e489bd83beceb4f29733ded74dc9e9ece1a23ab2e5f027e69c6c2c","sha256:b2172df5be5e5f60e7c9ec7f588ea8faeab2cd0f4ec8c73399e00da7f7bd88c0"],"state_sha256":"1aa4efdbed303ede755ebb9a3c206be5415623ada4c52cdde9fab3de5b7d79fc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nwko2VZ+SgfTcJsCiCfhnvI3n02bGG/utN8gxQM+QnP/wSNWfdmTlabbYG1l/0cXAtjyNdherrb1dp0ev5OLAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T23:04:40.951225Z","bundle_sha256":"0980e602ebf8c7d996c5c27ec286eabb8accb5b19cac36c83a6ee8a6740f7ff5"}}