{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:OCXE2HAELSLILHF44E6Q6DTTRH","short_pith_number":"pith:OCXE2HAE","canonical_record":{"source":{"id":"1010.4071","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-10-19T23:08:34Z","cross_cats_sorted":[],"title_canon_sha256":"f7bcd9de1f02aff2489d1012e7dc707c61cf20b67394c9bdf6b7d3b5fbbfc17c","abstract_canon_sha256":"fb2bada7063bac06d49a139856de5e77afed325531b3151cf0a342066ac504b0"},"schema_version":"1.0"},"canonical_sha256":"70ae4d1c045c96859cbce13d0f0e7389e78bb7fff6a8a5a369f272e2ddfe6c33","source":{"kind":"arxiv","id":"1010.4071","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4071","created_at":"2026-05-18T04:24:36Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4071v2","created_at":"2026-05-18T04:24:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4071","created_at":"2026-05-18T04:24:36Z"},{"alias_kind":"pith_short_12","alias_value":"OCXE2HAELSLI","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"OCXE2HAELSLILHF4","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"OCXE2HAE","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:OCXE2HAELSLILHF44E6Q6DTTRH","target":"record","payload":{"canonical_record":{"source":{"id":"1010.4071","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-10-19T23:08:34Z","cross_cats_sorted":[],"title_canon_sha256":"f7bcd9de1f02aff2489d1012e7dc707c61cf20b67394c9bdf6b7d3b5fbbfc17c","abstract_canon_sha256":"fb2bada7063bac06d49a139856de5e77afed325531b3151cf0a342066ac504b0"},"schema_version":"1.0"},"canonical_sha256":"70ae4d1c045c96859cbce13d0f0e7389e78bb7fff6a8a5a369f272e2ddfe6c33","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:36.185832Z","signature_b64":"8HXZstuL5UOj3yMIgg9cXXOdS/S2zGPPlEZaHfSJ+50cg9MIUP7iwDEX7wzJfXHAgNIkUZRb7Tm+FTSEe/mPDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70ae4d1c045c96859cbce13d0f0e7389e78bb7fff6a8a5a369f272e2ddfe6c33","last_reissued_at":"2026-05-18T04:24:36.185283Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:36.185283Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.4071","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":"2026-05-18T04:24:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NcZz6XJX2xJaCs8GL/e3CPy9xt26ayeTwXhx/u3WL8yij/KnOPfjUugxvqfvOTNbYy+lob8TYNj2YO9jaz0pDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T06:43:14.504656Z"},"content_sha256":"6f2b0e2fc5bf03710161bcb7226dcf6537d59b3c4078dc10301b04d1f0dd2650","schema_version":"1.0","event_id":"sha256:6f2b0e2fc5bf03710161bcb7226dcf6537d59b3c4078dc10301b04d1f0dd2650"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:OCXE2HAELSLILHF44E6Q6DTTRH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Morse theory and toric vector bundles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"David Treumann","submitted_at":"2010-10-19T23:08:34Z","abstract_excerpt":"Morelli's computation of the K-theory of a toric variety X associates a polyhedrally constructible function on a real vector space to every equivariant vector bundle E on X. The coherent-constructible correspondence lifts Morelli's constructible function to a complex of constructible sheaves kappa(E). We show that certain filtrations of the cohomology of kappa(E) coming from Morse theory coincide with the Klyachko filtrations of the generic stalk of E. We give Morse-theoretic (i.e. microlocal) conditions for a complex of constructible sheaves to correspond to a vector bundle, and to a nef vect"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4071","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":""},"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":"2026-05-18T04:24:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uNnMVf8jVARbryXNcjvM+Yjrq//SKX2X5TldFtgnkSbQbn78NwlZdr2nfuYMdfgexDRoGzJQjoQx7scPTgJ4CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T06:43:14.504999Z"},"content_sha256":"571bc4ef0195cb4d19e963e6af11be9e0dc3263bcc71b15d9ec59b50a13a4876","schema_version":"1.0","event_id":"sha256:571bc4ef0195cb4d19e963e6af11be9e0dc3263bcc71b15d9ec59b50a13a4876"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OCXE2HAELSLILHF44E6Q6DTTRH/bundle.json","state_url":"https://pith.science/pith/OCXE2HAELSLILHF44E6Q6DTTRH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OCXE2HAELSLILHF44E6Q6DTTRH/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":"2026-06-12T06:43:14Z","links":{"resolver":"https://pith.science/pith/OCXE2HAELSLILHF44E6Q6DTTRH","bundle":"https://pith.science/pith/OCXE2HAELSLILHF44E6Q6DTTRH/bundle.json","state":"https://pith.science/pith/OCXE2HAELSLILHF44E6Q6DTTRH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OCXE2HAELSLILHF44E6Q6DTTRH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:OCXE2HAELSLILHF44E6Q6DTTRH","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":"fb2bada7063bac06d49a139856de5e77afed325531b3151cf0a342066ac504b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-10-19T23:08:34Z","title_canon_sha256":"f7bcd9de1f02aff2489d1012e7dc707c61cf20b67394c9bdf6b7d3b5fbbfc17c"},"schema_version":"1.0","source":{"id":"1010.4071","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4071","created_at":"2026-05-18T04:24:36Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4071v2","created_at":"2026-05-18T04:24:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4071","created_at":"2026-05-18T04:24:36Z"},{"alias_kind":"pith_short_12","alias_value":"OCXE2HAELSLI","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"OCXE2HAELSLILHF4","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"OCXE2HAE","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:571bc4ef0195cb4d19e963e6af11be9e0dc3263bcc71b15d9ec59b50a13a4876","target":"graph","created_at":"2026-05-18T04:24:36Z","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"},"paper":{"abstract_excerpt":"Morelli's computation of the K-theory of a toric variety X associates a polyhedrally constructible function on a real vector space to every equivariant vector bundle E on X. The coherent-constructible correspondence lifts Morelli's constructible function to a complex of constructible sheaves kappa(E). We show that certain filtrations of the cohomology of kappa(E) coming from Morse theory coincide with the Klyachko filtrations of the generic stalk of E. We give Morse-theoretic (i.e. microlocal) conditions for a complex of constructible sheaves to correspond to a vector bundle, and to a nef vect","authors_text":"David Treumann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-10-19T23:08:34Z","title":"Morse theory and toric vector bundles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4071","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":"sha256:6f2b0e2fc5bf03710161bcb7226dcf6537d59b3c4078dc10301b04d1f0dd2650","target":"record","created_at":"2026-05-18T04:24:36Z","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":"fb2bada7063bac06d49a139856de5e77afed325531b3151cf0a342066ac504b0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-10-19T23:08:34Z","title_canon_sha256":"f7bcd9de1f02aff2489d1012e7dc707c61cf20b67394c9bdf6b7d3b5fbbfc17c"},"schema_version":"1.0","source":{"id":"1010.4071","kind":"arxiv","version":2}},"canonical_sha256":"70ae4d1c045c96859cbce13d0f0e7389e78bb7fff6a8a5a369f272e2ddfe6c33","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70ae4d1c045c96859cbce13d0f0e7389e78bb7fff6a8a5a369f272e2ddfe6c33","first_computed_at":"2026-05-18T04:24:36.185283Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:24:36.185283Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8HXZstuL5UOj3yMIgg9cXXOdS/S2zGPPlEZaHfSJ+50cg9MIUP7iwDEX7wzJfXHAgNIkUZRb7Tm+FTSEe/mPDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:24:36.185832Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.4071","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6f2b0e2fc5bf03710161bcb7226dcf6537d59b3c4078dc10301b04d1f0dd2650","sha256:571bc4ef0195cb4d19e963e6af11be9e0dc3263bcc71b15d9ec59b50a13a4876"],"state_sha256":"381b2e6f4d29bf95ba524df94df325673f311c0220fb704c9031de6bc9195da1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AOXpJk8Em8XPpiG4JBYJ0y1rhG+Rbagv+PW6YqDiUGM9PN9iyArWS/upSomqpfGVIhTevQivRYzjhcyora6BBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T06:43:14.506948Z","bundle_sha256":"3722ed9fe20f6d0fd87c2e0901a25b36b36d492d91e65bea1ed86b77bc563b21"}}