{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:6L7R6AX2INMZXTNR4TWV7NSNVH","short_pith_number":"pith:6L7R6AX2","canonical_record":{"source":{"id":"2412.20825","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-12-30T09:37:13Z","cross_cats_sorted":[],"title_canon_sha256":"d7e042c617e544cfdde1334ef8204008a2a65a036a5a4fdd06af7f7074434484","abstract_canon_sha256":"3db5956af9a1daf2f6c429fa7dfea5cfd5a2ba17e39f91d6921d92a7ca6c418d"},"schema_version":"1.0"},"canonical_sha256":"f2ff1f02fa43599bcdb1e4ed5fb64da9c373788d812c4aee660536b9add7c198","source":{"kind":"arxiv","id":"2412.20825","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.20825","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"arxiv_version","alias_value":"2412.20825v3","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.20825","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"pith_short_12","alias_value":"6L7R6AX2INMZ","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"pith_short_16","alias_value":"6L7R6AX2INMZXTNR","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"pith_short_8","alias_value":"6L7R6AX2","created_at":"2026-05-25T02:02:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:6L7R6AX2INMZXTNR4TWV7NSNVH","target":"record","payload":{"canonical_record":{"source":{"id":"2412.20825","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-12-30T09:37:13Z","cross_cats_sorted":[],"title_canon_sha256":"d7e042c617e544cfdde1334ef8204008a2a65a036a5a4fdd06af7f7074434484","abstract_canon_sha256":"3db5956af9a1daf2f6c429fa7dfea5cfd5a2ba17e39f91d6921d92a7ca6c418d"},"schema_version":"1.0"},"canonical_sha256":"f2ff1f02fa43599bcdb1e4ed5fb64da9c373788d812c4aee660536b9add7c198","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:02:06.264225Z","signature_b64":"hHgVNOFCcy5qeopKuc0TmZcnZVdAu2lqaZLp0274ugNtvx04DUWGy1YxrNoBBGCfXjUMAKX0dILx8MEq7+IYAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2ff1f02fa43599bcdb1e4ed5fb64da9c373788d812c4aee660536b9add7c198","last_reissued_at":"2026-05-25T02:02:06.263218Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:02:06.263218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.20825","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-25T02:02:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M/TM8b4jWOv+u51yGIyvAKwsCUoRrzm8uAUS3FNrUcDuFs5j4SYISkUD5l9gDvX4J3Qyoz9QCval49FRvcElBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:11:40.114230Z"},"content_sha256":"8a38d8546333643ed25f425cdc00094a7333aaf7dc3e8a85e16ef1b50434ae3f","schema_version":"1.0","event_id":"sha256:8a38d8546333643ed25f425cdc00094a7333aaf7dc3e8a85e16ef1b50434ae3f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:6L7R6AX2INMZXTNR4TWV7NSNVH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The calculus of Duistermaat's triple index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Graham Cox, Gregory Berkolaiko, Selim Sukhtaiev, Yuri Latushkin","submitted_at":"2024-12-30T09:37:13Z","abstract_excerpt":"In this paper we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago, this index has lately been the subject of renewed attention, due to its central role in eigenvalue interlacing problems on quantum graphs (and more abstractly for self-adjoint extensions of symmetric operators). Here we give an axiomatic characterization of the index that leads to elementary proofs of its fundamental properties. We also relate the index to other quantities often appearing in symplectic geometry, such"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.20825","kind":"arxiv","version":3},"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/2412.20825/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":"2026-05-25T02:02:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yG6b+oKyGPs4IWsZjZq8kALrgnlFGcS8sb+K1CCr3eltUEs+dCaCtqNxyhzRHKqack7Ws5ltLDb8i7lB9Ky/DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:11:40.114999Z"},"content_sha256":"d7b7ab0e22bb9bf773592545cb77ca870fdd6dc54a154f582f2e283a0a30e180","schema_version":"1.0","event_id":"sha256:d7b7ab0e22bb9bf773592545cb77ca870fdd6dc54a154f582f2e283a0a30e180"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6L7R6AX2INMZXTNR4TWV7NSNVH/bundle.json","state_url":"https://pith.science/pith/6L7R6AX2INMZXTNR4TWV7NSNVH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6L7R6AX2INMZXTNR4TWV7NSNVH/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":"2026-06-08T17:11:40Z","links":{"resolver":"https://pith.science/pith/6L7R6AX2INMZXTNR4TWV7NSNVH","bundle":"https://pith.science/pith/6L7R6AX2INMZXTNR4TWV7NSNVH/bundle.json","state":"https://pith.science/pith/6L7R6AX2INMZXTNR4TWV7NSNVH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6L7R6AX2INMZXTNR4TWV7NSNVH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:6L7R6AX2INMZXTNR4TWV7NSNVH","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":"3db5956af9a1daf2f6c429fa7dfea5cfd5a2ba17e39f91d6921d92a7ca6c418d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-12-30T09:37:13Z","title_canon_sha256":"d7e042c617e544cfdde1334ef8204008a2a65a036a5a4fdd06af7f7074434484"},"schema_version":"1.0","source":{"id":"2412.20825","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.20825","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"arxiv_version","alias_value":"2412.20825v3","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.20825","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"pith_short_12","alias_value":"6L7R6AX2INMZ","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"pith_short_16","alias_value":"6L7R6AX2INMZXTNR","created_at":"2026-05-25T02:02:06Z"},{"alias_kind":"pith_short_8","alias_value":"6L7R6AX2","created_at":"2026-05-25T02:02:06Z"}],"graph_snapshots":[{"event_id":"sha256:d7b7ab0e22bb9bf773592545cb77ca870fdd6dc54a154f582f2e283a0a30e180","target":"graph","created_at":"2026-05-25T02:02:06Z","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/2412.20825/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago, this index has lately been the subject of renewed attention, due to its central role in eigenvalue interlacing problems on quantum graphs (and more abstractly for self-adjoint extensions of symmetric operators). Here we give an axiomatic characterization of the index that leads to elementary proofs of its fundamental properties. We also relate the index to other quantities often appearing in symplectic geometry, such","authors_text":"Graham Cox, Gregory Berkolaiko, Selim Sukhtaiev, Yuri Latushkin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-12-30T09:37:13Z","title":"The calculus of Duistermaat's triple index"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.20825","kind":"arxiv","version":3},"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":"sha256:8a38d8546333643ed25f425cdc00094a7333aaf7dc3e8a85e16ef1b50434ae3f","target":"record","created_at":"2026-05-25T02:02:06Z","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":"3db5956af9a1daf2f6c429fa7dfea5cfd5a2ba17e39f91d6921d92a7ca6c418d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2024-12-30T09:37:13Z","title_canon_sha256":"d7e042c617e544cfdde1334ef8204008a2a65a036a5a4fdd06af7f7074434484"},"schema_version":"1.0","source":{"id":"2412.20825","kind":"arxiv","version":3}},"canonical_sha256":"f2ff1f02fa43599bcdb1e4ed5fb64da9c373788d812c4aee660536b9add7c198","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f2ff1f02fa43599bcdb1e4ed5fb64da9c373788d812c4aee660536b9add7c198","first_computed_at":"2026-05-25T02:02:06.263218Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:02:06.263218Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hHgVNOFCcy5qeopKuc0TmZcnZVdAu2lqaZLp0274ugNtvx04DUWGy1YxrNoBBGCfXjUMAKX0dILx8MEq7+IYAw==","signature_status":"signed_v1","signed_at":"2026-05-25T02:02:06.264225Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.20825","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8a38d8546333643ed25f425cdc00094a7333aaf7dc3e8a85e16ef1b50434ae3f","sha256:d7b7ab0e22bb9bf773592545cb77ca870fdd6dc54a154f582f2e283a0a30e180"],"state_sha256":"6b835dac454ee328ed9db51032aaa16d505beb908ab40ff9b11fd72c110fe694"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J9YDbPEcLzbmyjHmbuEApHmCQXa4QdW+0cE9O3DKPRDFgyJOi9uYqJfTn2tfgJy7RsCS7uGO4xKe2KHcmlnYAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T17:11:40.120234Z","bundle_sha256":"94c494aca6a07788a2ba2a6e1659f1b7512fc46c60fb2e9a9e1aa09964159776"}}