{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:IRPEEA7NDIKHY2NRHT6LE6EWYK","short_pith_number":"pith:IRPEEA7N","canonical_record":{"source":{"id":"2510.24060","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2025-10-28T04:44:18Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"31ecbfa84ee053c7cc034dcd90ad14017e9cd46bdfaa21867f0079c1f5c8b0e7","abstract_canon_sha256":"5a9eda2c0b0f292446ca217714c4b28fa3f2c3d994b0adbfab27fa5284bf0ea5"},"schema_version":"1.0"},"canonical_sha256":"445e4203ed1a147c69b13cfcb27896c2ae5f7e3e9a47c31d2aab6ebfd51a5067","source":{"kind":"arxiv","id":"2510.24060","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.24060","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"arxiv_version","alias_value":"2510.24060v2","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.24060","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"pith_short_12","alias_value":"IRPEEA7NDIKH","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"pith_short_16","alias_value":"IRPEEA7NDIKHY2NR","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"pith_short_8","alias_value":"IRPEEA7N","created_at":"2026-06-10T01:09:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:IRPEEA7NDIKHY2NRHT6LE6EWYK","target":"record","payload":{"canonical_record":{"source":{"id":"2510.24060","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2025-10-28T04:44:18Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"31ecbfa84ee053c7cc034dcd90ad14017e9cd46bdfaa21867f0079c1f5c8b0e7","abstract_canon_sha256":"5a9eda2c0b0f292446ca217714c4b28fa3f2c3d994b0adbfab27fa5284bf0ea5"},"schema_version":"1.0"},"canonical_sha256":"445e4203ed1a147c69b13cfcb27896c2ae5f7e3e9a47c31d2aab6ebfd51a5067","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:09:45.833994Z","signature_b64":"ELhhYvBuYPIoDdhrU54R08v2hGEyGYHQUt0vXmfESBiMYcpOm350wJTIsDN5xjLI4fRY8+xO6NWldRxxlwk3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"445e4203ed1a147c69b13cfcb27896c2ae5f7e3e9a47c31d2aab6ebfd51a5067","last_reissued_at":"2026-06-10T01:09:45.832935Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:09:45.832935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2510.24060","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":"2026-06-10T01:09:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MNIH0GNiAL1jiRgTYoeKtmyYS8nwD+eEZuaeF6Nm8P1kHGdHLD+GaiWb39jb0XpvNYpdk4uHdvrYL4+It/KzBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T13:09:10.754603Z"},"content_sha256":"19409a8130bb05ab74646e299db29b01e2e62a2307484256dd50fc58988fd462","schema_version":"1.0","event_id":"sha256:19409a8130bb05ab74646e299db29b01e2e62a2307484256dd50fc58988fd462"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:IRPEEA7NDIKHY2NRHT6LE6EWYK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Formalizing Schwartz functions and tempered distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"cs.LO","authors_text":"Moritz Doll","submitted_at":"2025-10-28T04:44:18Z","abstract_excerpt":"Distribution theory is a cornerstone of the theory of partial differential equations. We report on the progress of formalizing the theory of tempered distributions in the interactive proof assistant Lean, which is the first formalization in any proof assistant. We give an overview of the mathematical theory and highlight key aspects of the formalization that differ from the classical presentation. As an application, we prove that the Fourier transform extends to a linear isometry on $L^2$ and we define Sobolev spaces via the Fourier transform on tempered distributions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.24060","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/2510.24060/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":"2026-06-10T01:09:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yQlg6+gpAQE5DSCsFuwtUP9/KUr7cFqnJAQVfrF1AfO4rFSWU5q03klgkp3KSYROg8A72Uac84Pq99fU1gq1CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T13:09:10.755426Z"},"content_sha256":"e6f1fe051e745c1a04953ce2e13a011518b68f32a4b38ce03c9e3f8a636767d6","schema_version":"1.0","event_id":"sha256:e6f1fe051e745c1a04953ce2e13a011518b68f32a4b38ce03c9e3f8a636767d6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IRPEEA7NDIKHY2NRHT6LE6EWYK/bundle.json","state_url":"https://pith.science/pith/IRPEEA7NDIKHY2NRHT6LE6EWYK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IRPEEA7NDIKHY2NRHT6LE6EWYK/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":"2026-06-10T13:09:10Z","links":{"resolver":"https://pith.science/pith/IRPEEA7NDIKHY2NRHT6LE6EWYK","bundle":"https://pith.science/pith/IRPEEA7NDIKHY2NRHT6LE6EWYK/bundle.json","state":"https://pith.science/pith/IRPEEA7NDIKHY2NRHT6LE6EWYK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IRPEEA7NDIKHY2NRHT6LE6EWYK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:IRPEEA7NDIKHY2NRHT6LE6EWYK","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":"5a9eda2c0b0f292446ca217714c4b28fa3f2c3d994b0adbfab27fa5284bf0ea5","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2025-10-28T04:44:18Z","title_canon_sha256":"31ecbfa84ee053c7cc034dcd90ad14017e9cd46bdfaa21867f0079c1f5c8b0e7"},"schema_version":"1.0","source":{"id":"2510.24060","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2510.24060","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"arxiv_version","alias_value":"2510.24060v2","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.24060","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"pith_short_12","alias_value":"IRPEEA7NDIKH","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"pith_short_16","alias_value":"IRPEEA7NDIKHY2NR","created_at":"2026-06-10T01:09:45Z"},{"alias_kind":"pith_short_8","alias_value":"IRPEEA7N","created_at":"2026-06-10T01:09:45Z"}],"graph_snapshots":[{"event_id":"sha256:e6f1fe051e745c1a04953ce2e13a011518b68f32a4b38ce03c9e3f8a636767d6","target":"graph","created_at":"2026-06-10T01:09:45Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2510.24060/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Distribution theory is a cornerstone of the theory of partial differential equations. We report on the progress of formalizing the theory of tempered distributions in the interactive proof assistant Lean, which is the first formalization in any proof assistant. We give an overview of the mathematical theory and highlight key aspects of the formalization that differ from the classical presentation. As an application, we prove that the Fourier transform extends to a linear isometry on $L^2$ and we define Sobolev spaces via the Fourier transform on tempered distributions.","authors_text":"Moritz Doll","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2025-10-28T04:44:18Z","title":"Formalizing Schwartz functions and tempered distributions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.24060","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":"sha256:19409a8130bb05ab74646e299db29b01e2e62a2307484256dd50fc58988fd462","target":"record","created_at":"2026-06-10T01:09:45Z","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":"5a9eda2c0b0f292446ca217714c4b28fa3f2c3d994b0adbfab27fa5284bf0ea5","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2025-10-28T04:44:18Z","title_canon_sha256":"31ecbfa84ee053c7cc034dcd90ad14017e9cd46bdfaa21867f0079c1f5c8b0e7"},"schema_version":"1.0","source":{"id":"2510.24060","kind":"arxiv","version":2}},"canonical_sha256":"445e4203ed1a147c69b13cfcb27896c2ae5f7e3e9a47c31d2aab6ebfd51a5067","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"445e4203ed1a147c69b13cfcb27896c2ae5f7e3e9a47c31d2aab6ebfd51a5067","first_computed_at":"2026-06-10T01:09:45.832935Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:09:45.832935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ELhhYvBuYPIoDdhrU54R08v2hGEyGYHQUt0vXmfESBiMYcpOm350wJTIsDN5xjLI4fRY8+xO6NWldRxxlwk3Ag==","signature_status":"signed_v1","signed_at":"2026-06-10T01:09:45.833994Z","signed_message":"canonical_sha256_bytes"},"source_id":"2510.24060","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19409a8130bb05ab74646e299db29b01e2e62a2307484256dd50fc58988fd462","sha256:e6f1fe051e745c1a04953ce2e13a011518b68f32a4b38ce03c9e3f8a636767d6"],"state_sha256":"fc3d273b71ea48c4fff8fa612bb8bb9125ebe14fce5a504bb740c7842af59175"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"acd5pADElwMLazE+HqD6mraDQXtuNSka0xNHc7XtxOn+cvNbO+X7TvePDolBDPlO6K4BvIDfoDmrx2VWMjZyAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T13:09:10.759618Z","bundle_sha256":"9506bf14906ba1d793df311d531eb720ccd46654481b773b1a5f02348edb539e"}}