{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2001:NFH4IWVCP6H2ISTADQVYEV6OHP","short_pith_number":"pith:NFH4IWVC","canonical_record":{"source":{"id":"math/0106030","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2001-06-05T12:07:49Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"4abe114926a02a606fdb2c9451a4d6880d4886e115f6d74a84c444b0503054e5","abstract_canon_sha256":"43995c847272c81f80da4d6c55410bcbadffc4aaf33df76a264957e03a396dc9"},"schema_version":"1.0"},"canonical_sha256":"694fc45aa27f8fa44a601c2b8257ce3be8cb1553177666b9f15bff39a734bf6f","source":{"kind":"arxiv","id":"math/0106030","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0106030","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"arxiv_version","alias_value":"math/0106030v1","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0106030","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"pith_short_12","alias_value":"NFH4IWVCP6H2","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"pith_short_16","alias_value":"NFH4IWVCP6H2ISTA","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"pith_short_8","alias_value":"NFH4IWVC","created_at":"2026-07-04T14:35:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2001:NFH4IWVCP6H2ISTADQVYEV6OHP","target":"record","payload":{"canonical_record":{"source":{"id":"math/0106030","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2001-06-05T12:07:49Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"4abe114926a02a606fdb2c9451a4d6880d4886e115f6d74a84c444b0503054e5","abstract_canon_sha256":"43995c847272c81f80da4d6c55410bcbadffc4aaf33df76a264957e03a396dc9"},"schema_version":"1.0"},"canonical_sha256":"694fc45aa27f8fa44a601c2b8257ce3be8cb1553177666b9f15bff39a734bf6f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:35:21.834327Z","signature_b64":"sYC0H2EsQLVw/bNifSMqMbB0k7OsEwedZuyqlKJxI0xWXIF1UVgxQauHOCSF1dTRDlQpBHSDnFfQXLoZ0JoKBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"694fc45aa27f8fa44a601c2b8257ce3be8cb1553177666b9f15bff39a734bf6f","last_reissued_at":"2026-07-04T14:35:21.833916Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:35:21.833916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0106030","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":"2026-07-04T14:35:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m6ioK7oRLSuKQvfEe/IwXM10KIQI0v2DxC6NNahodzAf6hyj7ijuCBfr3zbBswwje6KPJIDSy4osJ5niZhn3CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T21:56:31.466950Z"},"content_sha256":"88be4f84050d4a79d7233c153518778b6cd41a55aa73cc4047dccf70f40c1376","schema_version":"1.0","event_id":"sha256:88be4f84050d4a79d7233c153518778b6cd41a55aa73cc4047dccf70f40c1376"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2001:NFH4IWVCP6H2ISTADQVYEV6OHP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Trace Expansions and the Noncommutative Residue for Manifolds with Boundary","license":"","headline":"","cross_cats":["math.SP"],"primary_cat":"math.AP","authors_text":"Elmar Schrohe, Gerd Grubb","submitted_at":"2001-06-05T12:07:49Z","abstract_excerpt":"For a pseudodifferential boundary operator A of integer order \\nu and class zero (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Trace(AB^{-s}) where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary.\n  We prove that Trace(AB^{-s}) has a meromorphic extension to the complex plane with poles at the half-integers s = (n+\\nu-j)/2, j = 0,1,... (possibly double for s<0), and we prove that its residue at zero equals the noncommut"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0106030","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/math/0106030/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":"2026-07-04T14:35:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pSTx+pKHcGWR+sSOMLhQpSaAeK9F4AjHXuO7+vH3Aitt2q+T6EAWxaEHFeygMXvv7LPU0i65ex+a2NyPOTA1AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T21:56:31.467626Z"},"content_sha256":"82212b4174d272c393216d149d80075807bc1a8fdb1c1107ea221bb8f79674e8","schema_version":"1.0","event_id":"sha256:82212b4174d272c393216d149d80075807bc1a8fdb1c1107ea221bb8f79674e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NFH4IWVCP6H2ISTADQVYEV6OHP/bundle.json","state_url":"https://pith.science/pith/NFH4IWVCP6H2ISTADQVYEV6OHP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NFH4IWVCP6H2ISTADQVYEV6OHP/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":"2026-08-03T21:56:31Z","links":{"resolver":"https://pith.science/pith/NFH4IWVCP6H2ISTADQVYEV6OHP","bundle":"https://pith.science/pith/NFH4IWVCP6H2ISTADQVYEV6OHP/bundle.json","state":"https://pith.science/pith/NFH4IWVCP6H2ISTADQVYEV6OHP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NFH4IWVCP6H2ISTADQVYEV6OHP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2001:NFH4IWVCP6H2ISTADQVYEV6OHP","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":"43995c847272c81f80da4d6c55410bcbadffc4aaf33df76a264957e03a396dc9","cross_cats_sorted":["math.SP"],"license":"","primary_cat":"math.AP","submitted_at":"2001-06-05T12:07:49Z","title_canon_sha256":"4abe114926a02a606fdb2c9451a4d6880d4886e115f6d74a84c444b0503054e5"},"schema_version":"1.0","source":{"id":"math/0106030","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0106030","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"arxiv_version","alias_value":"math/0106030v1","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0106030","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"pith_short_12","alias_value":"NFH4IWVCP6H2","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"pith_short_16","alias_value":"NFH4IWVCP6H2ISTA","created_at":"2026-07-04T14:35:21Z"},{"alias_kind":"pith_short_8","alias_value":"NFH4IWVC","created_at":"2026-07-04T14:35:21Z"}],"graph_snapshots":[{"event_id":"sha256:82212b4174d272c393216d149d80075807bc1a8fdb1c1107ea221bb8f79674e8","target":"graph","created_at":"2026-07-04T14:35:21Z","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/math/0106030/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a pseudodifferential boundary operator A of integer order \\nu and class zero (in the Boutet de Monvel calculus) on a compact n-dimensional manifold with boundary, we consider the function Trace(AB^{-s}) where B is an auxiliary system formed of the Dirichlet realization of a second order strongly elliptic differential operator and an elliptic operator on the boundary.\n  We prove that Trace(AB^{-s}) has a meromorphic extension to the complex plane with poles at the half-integers s = (n+\\nu-j)/2, j = 0,1,... (possibly double for s<0), and we prove that its residue at zero equals the noncommut","authors_text":"Elmar Schrohe, Gerd Grubb","cross_cats":["math.SP"],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2001-06-05T12:07:49Z","title":"Trace Expansions and the Noncommutative Residue for Manifolds with Boundary"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0106030","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":"sha256:88be4f84050d4a79d7233c153518778b6cd41a55aa73cc4047dccf70f40c1376","target":"record","created_at":"2026-07-04T14:35:21Z","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":"43995c847272c81f80da4d6c55410bcbadffc4aaf33df76a264957e03a396dc9","cross_cats_sorted":["math.SP"],"license":"","primary_cat":"math.AP","submitted_at":"2001-06-05T12:07:49Z","title_canon_sha256":"4abe114926a02a606fdb2c9451a4d6880d4886e115f6d74a84c444b0503054e5"},"schema_version":"1.0","source":{"id":"math/0106030","kind":"arxiv","version":1}},"canonical_sha256":"694fc45aa27f8fa44a601c2b8257ce3be8cb1553177666b9f15bff39a734bf6f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"694fc45aa27f8fa44a601c2b8257ce3be8cb1553177666b9f15bff39a734bf6f","first_computed_at":"2026-07-04T14:35:21.833916Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:35:21.833916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sYC0H2EsQLVw/bNifSMqMbB0k7OsEwedZuyqlKJxI0xWXIF1UVgxQauHOCSF1dTRDlQpBHSDnFfQXLoZ0JoKBw==","signature_status":"signed_v1","signed_at":"2026-07-04T14:35:21.834327Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0106030","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88be4f84050d4a79d7233c153518778b6cd41a55aa73cc4047dccf70f40c1376","sha256:82212b4174d272c393216d149d80075807bc1a8fdb1c1107ea221bb8f79674e8"],"state_sha256":"35218b4e9ffca4184c9bcedd6d7dac71af9260431209ae1b7d73806215814778"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wOZCZkYnIra4XE7IjQSzVLbM6tt9Y7N/6JhCFwOBynS/PaFOvLMZilz9z62YbKjrDXCiA9mUAYcpoMbv+9tDAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T21:56:31.473702Z","bundle_sha256":"61adb62180e42f7b45a2af3ea3b532bf44d1d24d190908f8ff054d27ee508745"}}