{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:LMLERR45NPFR3HBQUY726EZD3B","short_pith_number":"pith:LMLERR45","canonical_record":{"source":{"id":"2606.10521","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-09T07:51:08Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"8ef45a3fbe986753764b29f9f741e52d20a2b833a1897a13f7839f1880fb3096","abstract_canon_sha256":"03689a346f0cabdca8269108ce662f53041b59104d776ff3d8ed3e33a63b515f"},"schema_version":"1.0"},"canonical_sha256":"5b1648c79d6bcb1d9c30a63faf1323d851cec86e779f9864020650b4e276f831","source":{"kind":"arxiv","id":"2606.10521","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.10521","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"arxiv_version","alias_value":"2606.10521v1","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.10521","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"pith_short_12","alias_value":"LMLERR45NPFR","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"pith_short_16","alias_value":"LMLERR45NPFR3HBQ","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"pith_short_8","alias_value":"LMLERR45","created_at":"2026-06-10T01:10:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:LMLERR45NPFR3HBQUY726EZD3B","target":"record","payload":{"canonical_record":{"source":{"id":"2606.10521","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-09T07:51:08Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"8ef45a3fbe986753764b29f9f741e52d20a2b833a1897a13f7839f1880fb3096","abstract_canon_sha256":"03689a346f0cabdca8269108ce662f53041b59104d776ff3d8ed3e33a63b515f"},"schema_version":"1.0"},"canonical_sha256":"5b1648c79d6bcb1d9c30a63faf1323d851cec86e779f9864020650b4e276f831","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:10:24.103215Z","signature_b64":"2mDW8LgTOMa5zXQDXGSFzkU/QGSOXTkYHbb2slofWc1wloIBonCzEhfwz94blM8rRjwBtP4QtdIhKvSkZRQDDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b1648c79d6bcb1d9c30a63faf1323d851cec86e779f9864020650b4e276f831","last_reissued_at":"2026-06-10T01:10:24.102335Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:10:24.102335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.10521","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":"2026-06-10T01:10:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kpGAckAMj6WI3wiXn0CA6cOU1yebwaJhxE4UtUO0i2JDMm+UA5eflxxVp3fISSZys7zmp8waBM2HGWVQpsRCAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T14:56:13.893601Z"},"content_sha256":"dc51327d283dd1c5543e47acf004a0700136013e08e84d46435453b5b35d617f","schema_version":"1.0","event_id":"sha256:dc51327d283dd1c5543e47acf004a0700136013e08e84d46435453b5b35d617f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:LMLERR45NPFR3HBQUY726EZD3B","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Unique decomposition of orders","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.NT","authors_text":"Gaurav Digambar Patil","submitted_at":"2026-06-09T07:51:08Z","abstract_excerpt":"We establish a Fundamental Theorem of Orders (FTO), which allows us to express any order (in a number field) uniquely as an intersection of \\textit{irreducible orders}. Along this decomposition, the index (in the ring of integers) distributes multiplicatively, and the conductor factors into pairwise co-prime ideals.\n  We use it to show a more general version of Furtwangler criterion about the structure of conductors of orders over $\\Z$, as this answers a wide variations of such questions. In a future work, we will also give applications to weighted enumeration of number fields."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10521","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/2606.10521/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":"2026-06-10T01:10:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Dql7RHtJReuulLuQ/S7xnRztDvbXYMvGqlL8txWe8pN4qeB8bsgRSlyQBSuHeO6e2ATuO33EFilc3uDoCSllAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T14:56:13.893978Z"},"content_sha256":"f7ea0936adae62701802c72a68da6e678753b511ba1fa6115bbf1048522c0ce6","schema_version":"1.0","event_id":"sha256:f7ea0936adae62701802c72a68da6e678753b511ba1fa6115bbf1048522c0ce6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LMLERR45NPFR3HBQUY726EZD3B/bundle.json","state_url":"https://pith.science/pith/LMLERR45NPFR3HBQUY726EZD3B/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LMLERR45NPFR3HBQUY726EZD3B/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":"2026-06-29T14:56:13Z","links":{"resolver":"https://pith.science/pith/LMLERR45NPFR3HBQUY726EZD3B","bundle":"https://pith.science/pith/LMLERR45NPFR3HBQUY726EZD3B/bundle.json","state":"https://pith.science/pith/LMLERR45NPFR3HBQUY726EZD3B/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LMLERR45NPFR3HBQUY726EZD3B/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LMLERR45NPFR3HBQUY726EZD3B","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":"03689a346f0cabdca8269108ce662f53041b59104d776ff3d8ed3e33a63b515f","cross_cats_sorted":["math.AC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-09T07:51:08Z","title_canon_sha256":"8ef45a3fbe986753764b29f9f741e52d20a2b833a1897a13f7839f1880fb3096"},"schema_version":"1.0","source":{"id":"2606.10521","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.10521","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"arxiv_version","alias_value":"2606.10521v1","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.10521","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"pith_short_12","alias_value":"LMLERR45NPFR","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"pith_short_16","alias_value":"LMLERR45NPFR3HBQ","created_at":"2026-06-10T01:10:24Z"},{"alias_kind":"pith_short_8","alias_value":"LMLERR45","created_at":"2026-06-10T01:10:24Z"}],"graph_snapshots":[{"event_id":"sha256:f7ea0936adae62701802c72a68da6e678753b511ba1fa6115bbf1048522c0ce6","target":"graph","created_at":"2026-06-10T01:10:24Z","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/2606.10521/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish a Fundamental Theorem of Orders (FTO), which allows us to express any order (in a number field) uniquely as an intersection of \\textit{irreducible orders}. Along this decomposition, the index (in the ring of integers) distributes multiplicatively, and the conductor factors into pairwise co-prime ideals.\n  We use it to show a more general version of Furtwangler criterion about the structure of conductors of orders over $\\Z$, as this answers a wide variations of such questions. In a future work, we will also give applications to weighted enumeration of number fields.","authors_text":"Gaurav Digambar Patil","cross_cats":["math.AC"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-09T07:51:08Z","title":"Unique decomposition of orders"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10521","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:dc51327d283dd1c5543e47acf004a0700136013e08e84d46435453b5b35d617f","target":"record","created_at":"2026-06-10T01:10:24Z","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":"03689a346f0cabdca8269108ce662f53041b59104d776ff3d8ed3e33a63b515f","cross_cats_sorted":["math.AC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-09T07:51:08Z","title_canon_sha256":"8ef45a3fbe986753764b29f9f741e52d20a2b833a1897a13f7839f1880fb3096"},"schema_version":"1.0","source":{"id":"2606.10521","kind":"arxiv","version":1}},"canonical_sha256":"5b1648c79d6bcb1d9c30a63faf1323d851cec86e779f9864020650b4e276f831","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b1648c79d6bcb1d9c30a63faf1323d851cec86e779f9864020650b4e276f831","first_computed_at":"2026-06-10T01:10:24.102335Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:10:24.102335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2mDW8LgTOMa5zXQDXGSFzkU/QGSOXTkYHbb2slofWc1wloIBonCzEhfwz94blM8rRjwBtP4QtdIhKvSkZRQDDA==","signature_status":"signed_v1","signed_at":"2026-06-10T01:10:24.103215Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.10521","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc51327d283dd1c5543e47acf004a0700136013e08e84d46435453b5b35d617f","sha256:f7ea0936adae62701802c72a68da6e678753b511ba1fa6115bbf1048522c0ce6"],"state_sha256":"5f00ed5394d9e913169d2b7e1ac2e8c10ceb6cbdaf08b716ab1306331ba5e0dd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NQz2rOwtwla+3igXdDgHP3BTKpQxb7GrC9WBUaI6iIyH/KxgZ8ySdQn2NKpuUO7L5aRi7vFU9kiPsxouUoYwBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T14:56:13.895996Z","bundle_sha256":"d37635e1813379d3855516bae2a2243439b5b990c0680f01aff640a87e3792f9"}}