{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:4AIVPJVKVYPLKZO2MV3TU5IAVS","short_pith_number":"pith:4AIVPJVK","canonical_record":{"source":{"id":"2507.13215","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-17T15:21:59Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"86f725534fdb7259e2f6d3b217307cf6381bfa832e0a1355fb8c211689dcb854","abstract_canon_sha256":"b015ed9547eb18d3ba489ae7424f9006ff23f1fe97e4dfccc1239afa03f19bee"},"schema_version":"1.0"},"canonical_sha256":"e01157a6aaae1eb565da65773a7500ac82352b18055e57d1f330f21a8e003e5c","source":{"kind":"arxiv","id":"2507.13215","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.13215","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"arxiv_version","alias_value":"2507.13215v2","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.13215","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"pith_short_12","alias_value":"4AIVPJVKVYPL","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"pith_short_16","alias_value":"4AIVPJVKVYPLKZO2","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"pith_short_8","alias_value":"4AIVPJVK","created_at":"2026-06-12T01:09:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:4AIVPJVKVYPLKZO2MV3TU5IAVS","target":"record","payload":{"canonical_record":{"source":{"id":"2507.13215","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-17T15:21:59Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"86f725534fdb7259e2f6d3b217307cf6381bfa832e0a1355fb8c211689dcb854","abstract_canon_sha256":"b015ed9547eb18d3ba489ae7424f9006ff23f1fe97e4dfccc1239afa03f19bee"},"schema_version":"1.0"},"canonical_sha256":"e01157a6aaae1eb565da65773a7500ac82352b18055e57d1f330f21a8e003e5c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:11.025675Z","signature_b64":"pD0dV3RO5UkcXQNnLaRBW1b5G8c9a7YakFgUiTHx7sx9v0ED5+++7YTAJ36tKKgEa6sBOg1GoYyZgrbup3v7Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e01157a6aaae1eb565da65773a7500ac82352b18055e57d1f330f21a8e003e5c","last_reissued_at":"2026-06-12T01:09:11.024754Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:11.024754Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.13215","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":"2026-06-12T01:09:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UFinmwzFcHKlH82g9IvgzPAWdLOs/nlSTFmWGVje6BSkuTH+LKcZJegtn9IAUXPEgNn1v7wQBvYVzGB79MagDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:30:38.950062Z"},"content_sha256":"686df6e110b53730c1a0671ab3713253e3ff79558e8e7e9c1e79d745dbded70a","schema_version":"1.0","event_id":"sha256:686df6e110b53730c1a0671ab3713253e3ff79558e8e7e9c1e79d745dbded70a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:4AIVPJVKVYPLKZO2MV3TU5IAVS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"From Barcode Entropy to Metric Entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.SG","authors_text":"Basak Z. Gurel, Erman Cineli, Viktor L. Ginzburg","submitted_at":"2025-07-17T15:21:59Z","abstract_excerpt":"We establish a connection between barcode entropy and metric entropy. Namely, we show that the barcode entropy bounds the metric entropy from below for a measure from a specific class of invariant measures associated with a pair of Lagrangian or Legendrian submanifolds, which we call pseudo-chord measures. This inequality refines, via the variational principle, the previously known upper bound of barcode entropy by topological entropy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.13215","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/2507.13215/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":"2026-06-12T01:09:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DGMQyrUSdl2itrAtGp9ftQFo6qHO6UgPKxyhuy2r2M5LVy5jgRW6nmBDnGMibSS+Slno4iAIfgEme6vvr8WQAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T20:30:38.950464Z"},"content_sha256":"980d908ec2a21e4e0438b3d84bc6c3788b57cbc43473a39e5d44a807cb7832e4","schema_version":"1.0","event_id":"sha256:980d908ec2a21e4e0438b3d84bc6c3788b57cbc43473a39e5d44a807cb7832e4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4AIVPJVKVYPLKZO2MV3TU5IAVS/bundle.json","state_url":"https://pith.science/pith/4AIVPJVKVYPLKZO2MV3TU5IAVS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4AIVPJVKVYPLKZO2MV3TU5IAVS/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":"2026-06-24T20:30:38Z","links":{"resolver":"https://pith.science/pith/4AIVPJVKVYPLKZO2MV3TU5IAVS","bundle":"https://pith.science/pith/4AIVPJVKVYPLKZO2MV3TU5IAVS/bundle.json","state":"https://pith.science/pith/4AIVPJVKVYPLKZO2MV3TU5IAVS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4AIVPJVKVYPLKZO2MV3TU5IAVS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:4AIVPJVKVYPLKZO2MV3TU5IAVS","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":"b015ed9547eb18d3ba489ae7424f9006ff23f1fe97e4dfccc1239afa03f19bee","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-17T15:21:59Z","title_canon_sha256":"86f725534fdb7259e2f6d3b217307cf6381bfa832e0a1355fb8c211689dcb854"},"schema_version":"1.0","source":{"id":"2507.13215","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.13215","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"arxiv_version","alias_value":"2507.13215v2","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.13215","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"pith_short_12","alias_value":"4AIVPJVKVYPL","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"pith_short_16","alias_value":"4AIVPJVKVYPLKZO2","created_at":"2026-06-12T01:09:11Z"},{"alias_kind":"pith_short_8","alias_value":"4AIVPJVK","created_at":"2026-06-12T01:09:11Z"}],"graph_snapshots":[{"event_id":"sha256:980d908ec2a21e4e0438b3d84bc6c3788b57cbc43473a39e5d44a807cb7832e4","target":"graph","created_at":"2026-06-12T01:09:11Z","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/2507.13215/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish a connection between barcode entropy and metric entropy. Namely, we show that the barcode entropy bounds the metric entropy from below for a measure from a specific class of invariant measures associated with a pair of Lagrangian or Legendrian submanifolds, which we call pseudo-chord measures. This inequality refines, via the variational principle, the previously known upper bound of barcode entropy by topological entropy.","authors_text":"Basak Z. Gurel, Erman Cineli, Viktor L. Ginzburg","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-17T15:21:59Z","title":"From Barcode Entropy to Metric Entropy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.13215","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":"sha256:686df6e110b53730c1a0671ab3713253e3ff79558e8e7e9c1e79d745dbded70a","target":"record","created_at":"2026-06-12T01:09:11Z","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":"b015ed9547eb18d3ba489ae7424f9006ff23f1fe97e4dfccc1239afa03f19bee","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2025-07-17T15:21:59Z","title_canon_sha256":"86f725534fdb7259e2f6d3b217307cf6381bfa832e0a1355fb8c211689dcb854"},"schema_version":"1.0","source":{"id":"2507.13215","kind":"arxiv","version":2}},"canonical_sha256":"e01157a6aaae1eb565da65773a7500ac82352b18055e57d1f330f21a8e003e5c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e01157a6aaae1eb565da65773a7500ac82352b18055e57d1f330f21a8e003e5c","first_computed_at":"2026-06-12T01:09:11.024754Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-12T01:09:11.024754Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pD0dV3RO5UkcXQNnLaRBW1b5G8c9a7YakFgUiTHx7sx9v0ED5+++7YTAJ36tKKgEa6sBOg1GoYyZgrbup3v7Cw==","signature_status":"signed_v1","signed_at":"2026-06-12T01:09:11.025675Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.13215","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:686df6e110b53730c1a0671ab3713253e3ff79558e8e7e9c1e79d745dbded70a","sha256:980d908ec2a21e4e0438b3d84bc6c3788b57cbc43473a39e5d44a807cb7832e4"],"state_sha256":"036bec0db094f754c748465ec6db363385a54c0ed5052b968d61c31b2ae35452"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3kzUYchaGLQSaadobejA55FTketUvhQO25QNewLHAOiujmc3LSbUgL2+d/WoFyafDU7vFeAlhD3G0iBQCnlgCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T20:30:38.952570Z","bundle_sha256":"43cf86e6ef8b77690b710e707c7e088509a066a6746206f859e791795991c45f"}}