{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:KC73ABE7HLXZIWZ6UDVVHSU26W","short_pith_number":"pith:KC73ABE7","canonical_record":{"source":{"id":"2007.04170","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-07-08T14:59:49Z","cross_cats_sorted":["cs.NA","math.AP"],"title_canon_sha256":"eec987ec8a8f67449ec29cf5c3238ccd70a7c258247b31521492b2dcfb7a0786","abstract_canon_sha256":"8a973375de2c624874205ccffa730f40bdcd386d6b1c121f486bb0321683af53"},"schema_version":"1.0"},"canonical_sha256":"50bfb0049f3aef945b3ea0eb53ca9af591c2ee5b5ae76d5320d96347f0f4fb67","source":{"kind":"arxiv","id":"2007.04170","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.04170","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"arxiv_version","alias_value":"2007.04170v2","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.04170","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"pith_short_12","alias_value":"KC73ABE7HLXZ","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"pith_short_16","alias_value":"KC73ABE7HLXZIWZ6","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"pith_short_8","alias_value":"KC73ABE7","created_at":"2026-07-05T01:25:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:KC73ABE7HLXZIWZ6UDVVHSU26W","target":"record","payload":{"canonical_record":{"source":{"id":"2007.04170","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-07-08T14:59:49Z","cross_cats_sorted":["cs.NA","math.AP"],"title_canon_sha256":"eec987ec8a8f67449ec29cf5c3238ccd70a7c258247b31521492b2dcfb7a0786","abstract_canon_sha256":"8a973375de2c624874205ccffa730f40bdcd386d6b1c121f486bb0321683af53"},"schema_version":"1.0"},"canonical_sha256":"50bfb0049f3aef945b3ea0eb53ca9af591c2ee5b5ae76d5320d96347f0f4fb67","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:25:13.423641Z","signature_b64":"sihbQyij6LJlXr1gjZNGw1ykKz07BdeKqRfX9872s92KwYtSwJvvnS66ZniBCEPJJuCTUSMIGFW4UgQKhKLJBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50bfb0049f3aef945b3ea0eb53ca9af591c2ee5b5ae76d5320d96347f0f4fb67","last_reissued_at":"2026-07-05T01:25:13.423152Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:25:13.423152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2007.04170","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":"2026-07-05T01:25:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O3N0zhrcdjzLO5VwNEvAOGY8lpeAI1pjVAbrZ9ZpejA9wJZ0PlBue5AeC5pxlOd+qEOrpeC5QaxRd7D2MthhAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:56:06.582178Z"},"content_sha256":"53eabb92b3961456ed8dfa2dcfe85dbc519b5ef6a943f73a6f802803542928dc","schema_version":"1.0","event_id":"sha256:53eabb92b3961456ed8dfa2dcfe85dbc519b5ef6a943f73a6f802803542928dc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:KC73ABE7HLXZIWZ6UDVVHSU26W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.AP"],"primary_cat":"math.NA","authors_text":"Carl Leake, Daniele Mortari, Hunter Johnston","submitted_at":"2020-07-08T14:59:49Z","abstract_excerpt":"This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals--called constrained expressions--and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expression"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.04170","kind":"arxiv","version":2},"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/2007.04170/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":"2026-07-05T01:25:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Y+vZa+mPs8yqBVAppkjFXSl8uovm+VIfh0JL8gbRnHbaKZqBmgMMzJiacZsIJdakyDXbOzr6JTiecIK95mrCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:56:06.582600Z"},"content_sha256":"17a2da0e19a08a628c69ff69e94493c0f6de1e1b52e414cede038cdc0726be9f","schema_version":"1.0","event_id":"sha256:17a2da0e19a08a628c69ff69e94493c0f6de1e1b52e414cede038cdc0726be9f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KC73ABE7HLXZIWZ6UDVVHSU26W/bundle.json","state_url":"https://pith.science/pith/KC73ABE7HLXZIWZ6UDVVHSU26W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KC73ABE7HLXZIWZ6UDVVHSU26W/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":"2026-08-14T14:56:06Z","links":{"resolver":"https://pith.science/pith/KC73ABE7HLXZIWZ6UDVVHSU26W","bundle":"https://pith.science/pith/KC73ABE7HLXZIWZ6UDVVHSU26W/bundle.json","state":"https://pith.science/pith/KC73ABE7HLXZIWZ6UDVVHSU26W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KC73ABE7HLXZIWZ6UDVVHSU26W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:KC73ABE7HLXZIWZ6UDVVHSU26W","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":"8a973375de2c624874205ccffa730f40bdcd386d6b1c121f486bb0321683af53","cross_cats_sorted":["cs.NA","math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-07-08T14:59:49Z","title_canon_sha256":"eec987ec8a8f67449ec29cf5c3238ccd70a7c258247b31521492b2dcfb7a0786"},"schema_version":"1.0","source":{"id":"2007.04170","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.04170","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"arxiv_version","alias_value":"2007.04170v2","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.04170","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"pith_short_12","alias_value":"KC73ABE7HLXZ","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"pith_short_16","alias_value":"KC73ABE7HLXZIWZ6","created_at":"2026-07-05T01:25:13Z"},{"alias_kind":"pith_short_8","alias_value":"KC73ABE7","created_at":"2026-07-05T01:25:13Z"}],"graph_snapshots":[{"event_id":"sha256:17a2da0e19a08a628c69ff69e94493c0f6de1e1b52e414cede038cdc0726be9f","target":"graph","created_at":"2026-07-05T01:25:13Z","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/2007.04170/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals--called constrained expressions--and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expression","authors_text":"Carl Leake, Daniele Mortari, Hunter Johnston","cross_cats":["cs.NA","math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-07-08T14:59:49Z","title":"The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.04170","kind":"arxiv","version":2},"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":"sha256:53eabb92b3961456ed8dfa2dcfe85dbc519b5ef6a943f73a6f802803542928dc","target":"record","created_at":"2026-07-05T01:25:13Z","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":"8a973375de2c624874205ccffa730f40bdcd386d6b1c121f486bb0321683af53","cross_cats_sorted":["cs.NA","math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2020-07-08T14:59:49Z","title_canon_sha256":"eec987ec8a8f67449ec29cf5c3238ccd70a7c258247b31521492b2dcfb7a0786"},"schema_version":"1.0","source":{"id":"2007.04170","kind":"arxiv","version":2}},"canonical_sha256":"50bfb0049f3aef945b3ea0eb53ca9af591c2ee5b5ae76d5320d96347f0f4fb67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"50bfb0049f3aef945b3ea0eb53ca9af591c2ee5b5ae76d5320d96347f0f4fb67","first_computed_at":"2026-07-05T01:25:13.423152Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:25:13.423152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sihbQyij6LJlXr1gjZNGw1ykKz07BdeKqRfX9872s92KwYtSwJvvnS66ZniBCEPJJuCTUSMIGFW4UgQKhKLJBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:25:13.423641Z","signed_message":"canonical_sha256_bytes"},"source_id":"2007.04170","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53eabb92b3961456ed8dfa2dcfe85dbc519b5ef6a943f73a6f802803542928dc","sha256:17a2da0e19a08a628c69ff69e94493c0f6de1e1b52e414cede038cdc0726be9f"],"state_sha256":"da2ce198e0bc3f95ad3ba91280ad704bb661ec85ea9f04510edff4d5e745bf1d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mzXAUNl5/3Eh8YsMUL6LYE6wx6/lc+hSKTiSElg62YsZF6oEBuxRhYbbmZy7uJNOnu5OXXlBxS4G6cKfOXDDCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T14:56:06.585876Z","bundle_sha256":"f793481b8ab6f4ebfa63934262a0e76b397c3854b128cdf4b4c8b616b37ced8d"}}