{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:UVIYQ7WGS2SOCPN2Z7XISM4V72","short_pith_number":"pith:UVIYQ7WG","canonical_record":{"source":{"id":"2606.31328","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-06-30T08:30:40Z","cross_cats_sorted":[],"title_canon_sha256":"726beb1573241f983d3078c0e5de04c918db3ef7d8064f7812ed173e5b5f42b7","abstract_canon_sha256":"876a562e58a23d7f4d08cef1c207d467fc986fada32486e2e096fb9e6ad051da"},"schema_version":"1.0"},"canonical_sha256":"a551887ec696a4e13dbacfee893395febe52dbb7594210ed0f8e0be3cb6d0805","source":{"kind":"arxiv","id":"2606.31328","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.31328","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"arxiv_version","alias_value":"2606.31328v1","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.31328","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"pith_short_12","alias_value":"UVIYQ7WGS2SO","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"pith_short_16","alias_value":"UVIYQ7WGS2SOCPN2","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"pith_short_8","alias_value":"UVIYQ7WG","created_at":"2026-07-01T01:17:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:UVIYQ7WGS2SOCPN2Z7XISM4V72","target":"record","payload":{"canonical_record":{"source":{"id":"2606.31328","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-06-30T08:30:40Z","cross_cats_sorted":[],"title_canon_sha256":"726beb1573241f983d3078c0e5de04c918db3ef7d8064f7812ed173e5b5f42b7","abstract_canon_sha256":"876a562e58a23d7f4d08cef1c207d467fc986fada32486e2e096fb9e6ad051da"},"schema_version":"1.0"},"canonical_sha256":"a551887ec696a4e13dbacfee893395febe52dbb7594210ed0f8e0be3cb6d0805","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-01T01:17:59.295775Z","signature_b64":"I3kDQGzgFqb6Ao+CdVFuffJ6P8VyPoF1TvYqUHpM0l17CLULYZ3VE3/TDX4Yh21Sz7nWwgzKJVft1uVx5m+WDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a551887ec696a4e13dbacfee893395febe52dbb7594210ed0f8e0be3cb6d0805","last_reissued_at":"2026-07-01T01:17:59.295330Z","signature_status":"signed_v1","first_computed_at":"2026-07-01T01:17:59.295330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.31328","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":"2026-07-01T01:17:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k07s++saxfXIO3yYG/yOUkGUcpkj2uWxMZvaYQpi6liv9wMLEg8NM2215qwDPNpvYTouvSfK8trXvKv5qv0+Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-26T01:56:14.280965Z"},"content_sha256":"67f0bd2f8973b05096be03d1ce9a0a390a7304a14608cb447fabd919c73399d0","schema_version":"1.0","event_id":"sha256:67f0bd2f8973b05096be03d1ce9a0a390a7304a14608cb447fabd919c73399d0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:UVIYQ7WGS2SOCPN2Z7XISM4V72","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $S$-resolvent estimates for the Spinor Dirac operator on manifolds with boundary conditions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Fabrizio Colombo, Irene Sabadini, Ivan Beschastnyi, Simao Andrade Lucas","submitted_at":"2026-06-30T08:30:40Z","abstract_excerpt":"The aim of this paper is to show that the spectral theory based on the S-spectrum is particularly well suited for the Dirac operator on manifolds, even in cases where the operator is not self adjoint. Traditionally, for non-self adjoint operators in the Clifford setting, the literature has often referred to the right spectrum. However, a more comprehensive approach is provided by the theory of the $S$-spectrum, which is the appropriate notion for general operators on Clifford modules. In this work, we show that this theory is particularly well suited for bisectorial Clifford operators. By usin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31328","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/2606.31328/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":"2026-07-01T01:17:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ePEQiHiOeh/CmzAxVVO3dqPFByMUzLh+S8b38sW6O9FqLYKRVJO0rTY2qPWB00opEQybwvuDABnjL8ZjiuHZAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-26T01:56:14.281332Z"},"content_sha256":"2553af4b4d5a4dc4b2a84c4a121c162188e915e58e5c0f7548e074b6696fe233","schema_version":"1.0","event_id":"sha256:2553af4b4d5a4dc4b2a84c4a121c162188e915e58e5c0f7548e074b6696fe233"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UVIYQ7WGS2SOCPN2Z7XISM4V72/bundle.json","state_url":"https://pith.science/pith/UVIYQ7WGS2SOCPN2Z7XISM4V72/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UVIYQ7WGS2SOCPN2Z7XISM4V72/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":"2026-07-26T01:56:14Z","links":{"resolver":"https://pith.science/pith/UVIYQ7WGS2SOCPN2Z7XISM4V72","bundle":"https://pith.science/pith/UVIYQ7WGS2SOCPN2Z7XISM4V72/bundle.json","state":"https://pith.science/pith/UVIYQ7WGS2SOCPN2Z7XISM4V72/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UVIYQ7WGS2SOCPN2Z7XISM4V72/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:UVIYQ7WGS2SOCPN2Z7XISM4V72","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":"876a562e58a23d7f4d08cef1c207d467fc986fada32486e2e096fb9e6ad051da","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-06-30T08:30:40Z","title_canon_sha256":"726beb1573241f983d3078c0e5de04c918db3ef7d8064f7812ed173e5b5f42b7"},"schema_version":"1.0","source":{"id":"2606.31328","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.31328","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"arxiv_version","alias_value":"2606.31328v1","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.31328","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"pith_short_12","alias_value":"UVIYQ7WGS2SO","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"pith_short_16","alias_value":"UVIYQ7WGS2SOCPN2","created_at":"2026-07-01T01:17:59Z"},{"alias_kind":"pith_short_8","alias_value":"UVIYQ7WG","created_at":"2026-07-01T01:17:59Z"}],"graph_snapshots":[{"event_id":"sha256:2553af4b4d5a4dc4b2a84c4a121c162188e915e58e5c0f7548e074b6696fe233","target":"graph","created_at":"2026-07-01T01:17:59Z","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/2606.31328/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The aim of this paper is to show that the spectral theory based on the S-spectrum is particularly well suited for the Dirac operator on manifolds, even in cases where the operator is not self adjoint. Traditionally, for non-self adjoint operators in the Clifford setting, the literature has often referred to the right spectrum. However, a more comprehensive approach is provided by the theory of the $S$-spectrum, which is the appropriate notion for general operators on Clifford modules. In this work, we show that this theory is particularly well suited for bisectorial Clifford operators. By usin","authors_text":"Fabrizio Colombo, Irene Sabadini, Ivan Beschastnyi, Simao Andrade Lucas","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-06-30T08:30:40Z","title":"The $S$-resolvent estimates for the Spinor Dirac operator on manifolds with boundary conditions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31328","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":"sha256:67f0bd2f8973b05096be03d1ce9a0a390a7304a14608cb447fabd919c73399d0","target":"record","created_at":"2026-07-01T01:17:59Z","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":"876a562e58a23d7f4d08cef1c207d467fc986fada32486e2e096fb9e6ad051da","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-06-30T08:30:40Z","title_canon_sha256":"726beb1573241f983d3078c0e5de04c918db3ef7d8064f7812ed173e5b5f42b7"},"schema_version":"1.0","source":{"id":"2606.31328","kind":"arxiv","version":1}},"canonical_sha256":"a551887ec696a4e13dbacfee893395febe52dbb7594210ed0f8e0be3cb6d0805","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a551887ec696a4e13dbacfee893395febe52dbb7594210ed0f8e0be3cb6d0805","first_computed_at":"2026-07-01T01:17:59.295330Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-01T01:17:59.295330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I3kDQGzgFqb6Ao+CdVFuffJ6P8VyPoF1TvYqUHpM0l17CLULYZ3VE3/TDX4Yh21Sz7nWwgzKJVft1uVx5m+WDA==","signature_status":"signed_v1","signed_at":"2026-07-01T01:17:59.295775Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.31328","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67f0bd2f8973b05096be03d1ce9a0a390a7304a14608cb447fabd919c73399d0","sha256:2553af4b4d5a4dc4b2a84c4a121c162188e915e58e5c0f7548e074b6696fe233"],"state_sha256":"e9f1e4bf02790346603b13dc62dc95dfb241b1858bbaec27d8190d11f75be9d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ajOMW1UH4tspNcjyrd/1anoUT0nYT9FPac1foqaxzFTWT/2+HhaJH/roli9U6ksHQVCPIBDPZALJBFwhLjzABA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-26T01:56:14.283794Z","bundle_sha256":"e06b207aff9ffcbbbfc36bb377870f2d78793dde68ff7ceacf105c234d2d94cf"}}