{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3VY7AUU5XUWFGJZXUA3APNT34C","short_pith_number":"pith:3VY7AUU5","canonical_record":{"source":{"id":"2604.25653","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-28T13:50:06Z","cross_cats_sorted":[],"title_canon_sha256":"331e237d56d5d36e3f1fa1f9c8b366cc0f2303e028d8e8c004f211fbb268c750","abstract_canon_sha256":"9dd76d2a19a6993b35afe09a02729b826bce37089b1e56d2b4a96be20c5a8fd4"},"schema_version":"1.0"},"canonical_sha256":"dd71f0529dbd2c532737a03607b67be092d80008f6a313294d056a1b114712c9","source":{"kind":"arxiv","id":"2604.25653","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.25653","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"2604.25653v2","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.25653","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"3VY7AUU5XUWF","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"pith_short_16","alias_value":"3VY7AUU5XUWFGJZX","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"pith_short_8","alias_value":"3VY7AUU5","created_at":"2026-05-27T01:05:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3VY7AUU5XUWFGJZXUA3APNT34C","target":"record","payload":{"canonical_record":{"source":{"id":"2604.25653","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-28T13:50:06Z","cross_cats_sorted":[],"title_canon_sha256":"331e237d56d5d36e3f1fa1f9c8b366cc0f2303e028d8e8c004f211fbb268c750","abstract_canon_sha256":"9dd76d2a19a6993b35afe09a02729b826bce37089b1e56d2b4a96be20c5a8fd4"},"schema_version":"1.0"},"canonical_sha256":"dd71f0529dbd2c532737a03607b67be092d80008f6a313294d056a1b114712c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T01:05:55.025648Z","signature_b64":"MXcrR3mhSY/0LbOVCDmhNnIFIS5+fb1Yj2M94iGj5M2TL58h7ogN7kxmAye/4jU43WbuJWAPlCRkSAz9GKcfDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd71f0529dbd2c532737a03607b67be092d80008f6a313294d056a1b114712c9","last_reissued_at":"2026-05-27T01:05:55.024866Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T01:05:55.024866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.25653","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-27T01:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SgGpOCTwYbSLCmxCok8E3B8rlmoOHyucqXmQtXhaF3ynK6AJj8fyFEpJLhN2aMNPeFOY7A3Mp2wPUW9qpI5FBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:16:05.518698Z"},"content_sha256":"332badda05b23d850d5c0f1bc0ce01c58609d544423f581c787438adac7e381c","schema_version":"1.0","event_id":"sha256:332badda05b23d850d5c0f1bc0ce01c58609d544423f581c787438adac7e381c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3VY7AUU5XUWFGJZXUA3APNT34C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On numerical semigroups with embedding dimension four","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A geometric procedure determines the Apéry set for any numerical semigroup with embedding dimension four.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kazimierz Chomicz","submitted_at":"2026-04-28T13:50:06Z","abstract_excerpt":"We develop a geometric procedure for finding the Ap\\'ery set of any numerical semigroup with embedding dimension four. Previous methods of comparable strength worked only for embedding dimension three or under very specific conditions. We illustrate our method by finding the Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees of numerical semigroups generated by four consecutive squares and by four consecutive triangular numbers."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We develop a geometric procedure for finding the Apéry set of any numerical semigroup with embedding dimension four.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The geometric procedure correctly identifies the Apéry set for every numerical semigroup with exactly four minimal generators, without hidden restrictions on the generators or the semigroup.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A geometric procedure computes Apéry sets for numerical semigroups with embedding dimension four, yielding Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees for semigroups generated by four consecutive squares and four consecutive triangular numbers.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A geometric procedure determines the Apéry set for any numerical semigroup with embedding dimension four.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"09b6d48dcc1adcc0ccd561e37c1bb0d36c8a43ddc29e1a927cd6188f5ce5dcdf"},"source":{"id":"2604.25653","kind":"arxiv","version":2},"verdict":{"id":"4bf5550a-09c2-4b31-8d2e-70aaa1fe39f5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T14:52:12.579019Z","strongest_claim":"We develop a geometric procedure for finding the Apéry set of any numerical semigroup with embedding dimension four.","one_line_summary":"A geometric procedure computes Apéry sets for numerical semigroups with embedding dimension four, yielding Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees for semigroups generated by four consecutive squares and four consecutive triangular numbers.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The geometric procedure correctly identifies the Apéry set for every numerical semigroup with exactly four minimal generators, without hidden restrictions on the generators or the semigroup.","pith_extraction_headline":"A geometric procedure determines the Apéry set for any numerical semigroup with embedding dimension four."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25653/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T04:36:14.093379Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:53:15.567569Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"516737f9bedfc0f334987487fae465634bb4920bb113d8e0fb3e67d8caa0b296"},"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":"4bf5550a-09c2-4b31-8d2e-70aaa1fe39f5"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-27T01:05:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mQAhD3JDlFkpjsvy1LIDacnnS/5bX/jCUEnVTi7Wqd0NVJ1w/x8PlFykaz8SKe0026QZqcWVvmI+I4JeQNy8Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:16:05.519209Z"},"content_sha256":"5f4211b24684b4966afc7dcfacfa4adaed17dda3d76fa01e6f7ddbee754c6e11","schema_version":"1.0","event_id":"sha256:5f4211b24684b4966afc7dcfacfa4adaed17dda3d76fa01e6f7ddbee754c6e11"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3VY7AUU5XUWFGJZXUA3APNT34C/bundle.json","state_url":"https://pith.science/pith/3VY7AUU5XUWFGJZXUA3APNT34C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3VY7AUU5XUWFGJZXUA3APNT34C/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":"2026-06-10T21:16:05Z","links":{"resolver":"https://pith.science/pith/3VY7AUU5XUWFGJZXUA3APNT34C","bundle":"https://pith.science/pith/3VY7AUU5XUWFGJZXUA3APNT34C/bundle.json","state":"https://pith.science/pith/3VY7AUU5XUWFGJZXUA3APNT34C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3VY7AUU5XUWFGJZXUA3APNT34C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3VY7AUU5XUWFGJZXUA3APNT34C","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":"9dd76d2a19a6993b35afe09a02729b826bce37089b1e56d2b4a96be20c5a8fd4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-28T13:50:06Z","title_canon_sha256":"331e237d56d5d36e3f1fa1f9c8b366cc0f2303e028d8e8c004f211fbb268c750"},"schema_version":"1.0","source":{"id":"2604.25653","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.25653","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"arxiv_version","alias_value":"2604.25653v2","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.25653","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"pith_short_12","alias_value":"3VY7AUU5XUWF","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"pith_short_16","alias_value":"3VY7AUU5XUWFGJZX","created_at":"2026-05-27T01:05:55Z"},{"alias_kind":"pith_short_8","alias_value":"3VY7AUU5","created_at":"2026-05-27T01:05:55Z"}],"graph_snapshots":[{"event_id":"sha256:5f4211b24684b4966afc7dcfacfa4adaed17dda3d76fa01e6f7ddbee754c6e11","target":"graph","created_at":"2026-05-27T01:05:55Z","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":"We develop a geometric procedure for finding the Apéry set of any numerical semigroup with embedding dimension four."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The geometric procedure correctly identifies the Apéry set for every numerical semigroup with exactly four minimal generators, without hidden restrictions on the generators or the semigroup."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A geometric procedure computes Apéry sets for numerical semigroups with embedding dimension four, yielding Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees for semigroups generated by four consecutive squares and four consecutive triangular numbers."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A geometric procedure determines the Apéry set for any numerical semigroup with embedding dimension four."}],"snapshot_sha256":"09b6d48dcc1adcc0ccd561e37c1bb0d36c8a43ddc29e1a927cd6188f5ce5dcdf"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"name":"ai_meta_artifact","ran_at":"2026-05-21T04:36:14.093379Z","status":"completed","version":"1.0.0"},{"findings_count":0,"name":"doi_compliance","ran_at":"2026-05-19T20:53:15.567569Z","status":"completed","version":"1.0.0"}],"endpoint":"/pith/2604.25653/integrity.json","findings":[],"snapshot_sha256":"516737f9bedfc0f334987487fae465634bb4920bb113d8e0fb3e67d8caa0b296","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We develop a geometric procedure for finding the Ap\\'ery set of any numerical semigroup with embedding dimension four. Previous methods of comparable strength worked only for embedding dimension three or under very specific conditions. We illustrate our method by finding the Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees of numerical semigroups generated by four consecutive squares and by four consecutive triangular numbers.","authors_text":"Kazimierz Chomicz","cross_cats":[],"headline":"A geometric procedure determines the Apéry set for any numerical semigroup with embedding dimension four.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-28T13:50:06Z","title":"On numerical semigroups with embedding dimension four"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.25653","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-07T14:52:12.579019Z","id":"4bf5550a-09c2-4b31-8d2e-70aaa1fe39f5","model_set":{"reader":"grok-4.3"},"one_line_summary":"A geometric procedure computes Apéry sets for numerical semigroups with embedding dimension four, yielding Frobenius numbers, genera, Betti elements, minimal presentations, and catenary degrees for semigroups generated by four consecutive squares and four consecutive triangular numbers.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"A geometric procedure determines the Apéry set for any numerical semigroup with embedding dimension four.","strongest_claim":"We develop a geometric procedure for finding the Apéry set of any numerical semigroup with embedding dimension four.","weakest_assumption":"The geometric procedure correctly identifies the Apéry set for every numerical semigroup with exactly four minimal generators, without hidden restrictions on the generators or the semigroup."}},"verdict_id":"4bf5550a-09c2-4b31-8d2e-70aaa1fe39f5"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:332badda05b23d850d5c0f1bc0ce01c58609d544423f581c787438adac7e381c","target":"record","created_at":"2026-05-27T01:05:55Z","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":"9dd76d2a19a6993b35afe09a02729b826bce37089b1e56d2b4a96be20c5a8fd4","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-28T13:50:06Z","title_canon_sha256":"331e237d56d5d36e3f1fa1f9c8b366cc0f2303e028d8e8c004f211fbb268c750"},"schema_version":"1.0","source":{"id":"2604.25653","kind":"arxiv","version":2}},"canonical_sha256":"dd71f0529dbd2c532737a03607b67be092d80008f6a313294d056a1b114712c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd71f0529dbd2c532737a03607b67be092d80008f6a313294d056a1b114712c9","first_computed_at":"2026-05-27T01:05:55.024866Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-27T01:05:55.024866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MXcrR3mhSY/0LbOVCDmhNnIFIS5+fb1Yj2M94iGj5M2TL58h7ogN7kxmAye/4jU43WbuJWAPlCRkSAz9GKcfDg==","signature_status":"signed_v1","signed_at":"2026-05-27T01:05:55.025648Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.25653","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:332badda05b23d850d5c0f1bc0ce01c58609d544423f581c787438adac7e381c","sha256:5f4211b24684b4966afc7dcfacfa4adaed17dda3d76fa01e6f7ddbee754c6e11"],"state_sha256":"c4cc94650b6a46338d913270c94d33c58ae4b58ae4adda8d5a1b6f93b01fe1ea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GZgV8kDplVWm6UfvqYFnclYX3Bh0G4B1aPH5TMX7+M27X3PvQvGbMNKO+wChoBHXrBJD4zX19+6WLTSkQcaFDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T21:16:05.521982Z","bundle_sha256":"dcfd96555c7f2363b048d3ddb6c08645ca12f2e03ddee494961cc92bc876cb23"}}