{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:U26JOPWFSA7C7DTSMSV4XEC752","short_pith_number":"pith:U26JOPWF","canonical_record":{"source":{"id":"2602.05185","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-02-05T01:26:32Z","cross_cats_sorted":["math.CO","math.OA","math.SP"],"title_canon_sha256":"7e56e49d60566c1976ac023ee669ece880faa776391a8e02e26476af7800888b","abstract_canon_sha256":"9653afd28afee7588f13873545d5940d989500752017462c23d5293ca9c2369d"},"schema_version":"1.0"},"canonical_sha256":"a6bc973ec5903e2f8e7264abcb905fee85687582c22fa3b621621bc8ef14d5d7","source":{"kind":"arxiv","id":"2602.05185","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.05185","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"arxiv_version","alias_value":"2602.05185v2","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.05185","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"pith_short_12","alias_value":"U26JOPWFSA7C","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"pith_short_16","alias_value":"U26JOPWFSA7C7DTS","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"pith_short_8","alias_value":"U26JOPWF","created_at":"2026-06-11T00:08:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:U26JOPWFSA7C7DTSMSV4XEC752","target":"record","payload":{"canonical_record":{"source":{"id":"2602.05185","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-02-05T01:26:32Z","cross_cats_sorted":["math.CO","math.OA","math.SP"],"title_canon_sha256":"7e56e49d60566c1976ac023ee669ece880faa776391a8e02e26476af7800888b","abstract_canon_sha256":"9653afd28afee7588f13873545d5940d989500752017462c23d5293ca9c2369d"},"schema_version":"1.0"},"canonical_sha256":"a6bc973ec5903e2f8e7264abcb905fee85687582c22fa3b621621bc8ef14d5d7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T00:08:12.223998Z","signature_b64":"aysSOyg/WtTGox00JZDyhdCouRcl32KPwl0j1pELuw/kgBMv+fGxB0S6dOdIBl3g487bKxEmvyMVtWNkvWM2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6bc973ec5903e2f8e7264abcb905fee85687582c22fa3b621621bc8ef14d5d7","last_reissued_at":"2026-06-11T00:08:12.222962Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T00:08:12.222962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2602.05185","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":"2026-06-11T00:08:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FH7ent6AluzJXc2epwlpbJmtiysQJauYKUVTydFfOdaqXAHVoZq//blFuPiA/G4RUxGLY0t0hMoiPA9LFsYTDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:36:31.530684Z"},"content_sha256":"5362b6d8707ef36322b21b1d9dc91e6e85a9a0df17eefb6cb007963aad28fee2","schema_version":"1.0","event_id":"sha256:5362b6d8707ef36322b21b1d9dc91e6e85a9a0df17eefb6cb007963aad28fee2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:U26JOPWFSA7C7DTSMSV4XEC752","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral Theory for Borel PMP Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.OA","math.SP"],"primary_cat":"math.LO","authors_text":"Alexander Tenenbaum, Cecelia Higgins, Pieter Spaas","submitted_at":"2026-02-05T01:26:32Z","abstract_excerpt":"We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.05185","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/2602.05185/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":"2026-06-11T00:08:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iVj1GtdElRh/7xQ9xucZCj/eSuvZ+By9NNRiRA8Ly4UJDmtR36FYGYo6Xa2AAzAfcIeqy26nm7X73IZnLGE7Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:36:31.531640Z"},"content_sha256":"c3f18d205be75aa3cbfa08ee7957342b02140c897a41838f677ee493dc498dde","schema_version":"1.0","event_id":"sha256:c3f18d205be75aa3cbfa08ee7957342b02140c897a41838f677ee493dc498dde"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U26JOPWFSA7C7DTSMSV4XEC752/bundle.json","state_url":"https://pith.science/pith/U26JOPWFSA7C7DTSMSV4XEC752/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U26JOPWFSA7C7DTSMSV4XEC752/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":"2026-06-12T02:36:31Z","links":{"resolver":"https://pith.science/pith/U26JOPWFSA7C7DTSMSV4XEC752","bundle":"https://pith.science/pith/U26JOPWFSA7C7DTSMSV4XEC752/bundle.json","state":"https://pith.science/pith/U26JOPWFSA7C7DTSMSV4XEC752/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U26JOPWFSA7C7DTSMSV4XEC752/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:U26JOPWFSA7C7DTSMSV4XEC752","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":"9653afd28afee7588f13873545d5940d989500752017462c23d5293ca9c2369d","cross_cats_sorted":["math.CO","math.OA","math.SP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-02-05T01:26:32Z","title_canon_sha256":"7e56e49d60566c1976ac023ee669ece880faa776391a8e02e26476af7800888b"},"schema_version":"1.0","source":{"id":"2602.05185","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.05185","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"arxiv_version","alias_value":"2602.05185v2","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.05185","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"pith_short_12","alias_value":"U26JOPWFSA7C","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"pith_short_16","alias_value":"U26JOPWFSA7C7DTS","created_at":"2026-06-11T00:08:12Z"},{"alias_kind":"pith_short_8","alias_value":"U26JOPWF","created_at":"2026-06-11T00:08:12Z"}],"graph_snapshots":[{"event_id":"sha256:c3f18d205be75aa3cbfa08ee7957342b02140c897a41838f677ee493dc498dde","target":"graph","created_at":"2026-06-11T00:08:12Z","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/2602.05185/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matc","authors_text":"Alexander Tenenbaum, Cecelia Higgins, Pieter Spaas","cross_cats":["math.CO","math.OA","math.SP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-02-05T01:26:32Z","title":"Spectral Theory for Borel PMP Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.05185","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":"sha256:5362b6d8707ef36322b21b1d9dc91e6e85a9a0df17eefb6cb007963aad28fee2","target":"record","created_at":"2026-06-11T00:08:12Z","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":"9653afd28afee7588f13873545d5940d989500752017462c23d5293ca9c2369d","cross_cats_sorted":["math.CO","math.OA","math.SP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2026-02-05T01:26:32Z","title_canon_sha256":"7e56e49d60566c1976ac023ee669ece880faa776391a8e02e26476af7800888b"},"schema_version":"1.0","source":{"id":"2602.05185","kind":"arxiv","version":2}},"canonical_sha256":"a6bc973ec5903e2f8e7264abcb905fee85687582c22fa3b621621bc8ef14d5d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a6bc973ec5903e2f8e7264abcb905fee85687582c22fa3b621621bc8ef14d5d7","first_computed_at":"2026-06-11T00:08:12.222962Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T00:08:12.222962Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aysSOyg/WtTGox00JZDyhdCouRcl32KPwl0j1pELuw/kgBMv+fGxB0S6dOdIBl3g487bKxEmvyMVtWNkvWM2BQ==","signature_status":"signed_v1","signed_at":"2026-06-11T00:08:12.223998Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.05185","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5362b6d8707ef36322b21b1d9dc91e6e85a9a0df17eefb6cb007963aad28fee2","sha256:c3f18d205be75aa3cbfa08ee7957342b02140c897a41838f677ee493dc498dde"],"state_sha256":"cd58e777dc6f129892ca901628be911025428032086c93b9f017bab839615af2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IQWl1LT8oelrksweb1df8S1+snBaLbRHcUY7jwnErlw6+qpV4Fs3Yy3m4dL3xPGvzRsXmR1f+3BpV77aCksNCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T02:36:31.536103Z","bundle_sha256":"8c9dd67eb4de810ed66ea69416172ee0bfb283a71f0c9ba1944f82225cc89b00"}}