{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KK4FMBUU5P5RXN67OMKVDQXFYX","short_pith_number":"pith:KK4FMBUU","canonical_record":{"source":{"id":"2605.21636","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.CG","submitted_at":"2026-05-20T18:47:17Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"a77dc985e0c7d57b148fec1fe01a5040a6e9d20a88d149b594807daf16d86524","abstract_canon_sha256":"aca42acac0ead11892f3be18da0524434af2922d3768bb578a4e635634baf2c2"},"schema_version":"1.0"},"canonical_sha256":"52b8560694ebfb1bb7df731551c2e5c5cfa9a5cb8f0b2c1ae10d7598fea2fffb","source":{"kind":"arxiv","id":"2605.21636","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21636","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21636v1","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21636","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"pith_short_12","alias_value":"KK4FMBUU5P5R","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"pith_short_16","alias_value":"KK4FMBUU5P5RXN67","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"pith_short_8","alias_value":"KK4FMBUU","created_at":"2026-05-22T01:03:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KK4FMBUU5P5RXN67OMKVDQXFYX","target":"record","payload":{"canonical_record":{"source":{"id":"2605.21636","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.CG","submitted_at":"2026-05-20T18:47:17Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"a77dc985e0c7d57b148fec1fe01a5040a6e9d20a88d149b594807daf16d86524","abstract_canon_sha256":"aca42acac0ead11892f3be18da0524434af2922d3768bb578a4e635634baf2c2"},"schema_version":"1.0"},"canonical_sha256":"52b8560694ebfb1bb7df731551c2e5c5cfa9a5cb8f0b2c1ae10d7598fea2fffb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:03:26.381336Z","signature_b64":"tJmNJt8pvIllYAW3erE1R6HHbI4ZDNvXDB5WA+KsFXvl3JTpuPQWYSSccxg6xzZpIci/CyI+UE9aDUzgwVeKCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52b8560694ebfb1bb7df731551c2e5c5cfa9a5cb8f0b2c1ae10d7598fea2fffb","last_reissued_at":"2026-05-22T01:03:26.380878Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:03:26.380878Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.21636","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":"2026-05-22T01:03:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AFtdscfR0ik1gyQS8Xk/WYkY337sTNONlAPh5I/7TSpN68XuzdrSG+XaNwuPTFp0aJ1nd4pgJmkLSsTaeIwyDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T10:43:50.070204Z"},"content_sha256":"5d76e76b1c0d30fe40cbea65f38b7ab78298a4a68511ce7477162fd5ed391835","schema_version":"1.0","event_id":"sha256:5d76e76b1c0d30fe40cbea65f38b7ab78298a4a68511ce7477162fd5ed391835"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KK4FMBUU5P5RXN67OMKVDQXFYX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bifunction and Interlevel Delaunay Trifiltrations","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"cs.CG","authors_text":"Abhishek Rathod, \\'Angel Javier Alonso, Michael Kerber, Michael Lesnick, Tung Lam","submitted_at":"2026-05-20T18:47:17Z","abstract_excerpt":"A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \\subset \\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21636","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/2605.21636/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":"2026-05-22T01:03:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tl/cIkNPKcoendl9NxJhsJ+E2BZSDjolwjTxbfpvp2odYyv0RFzIFoeS1ccLbqMjenyX7bZ9r1pINd3by+wGBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T10:43:50.070989Z"},"content_sha256":"993b63be71bb1332bbfeecbe715b85136f5b1b1f1f9b8f9460240d01851bc40f","schema_version":"1.0","event_id":"sha256:993b63be71bb1332bbfeecbe715b85136f5b1b1f1f9b8f9460240d01851bc40f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KK4FMBUU5P5RXN67OMKVDQXFYX/bundle.json","state_url":"https://pith.science/pith/KK4FMBUU5P5RXN67OMKVDQXFYX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KK4FMBUU5P5RXN67OMKVDQXFYX/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":"2026-05-26T10:43:50Z","links":{"resolver":"https://pith.science/pith/KK4FMBUU5P5RXN67OMKVDQXFYX","bundle":"https://pith.science/pith/KK4FMBUU5P5RXN67OMKVDQXFYX/bundle.json","state":"https://pith.science/pith/KK4FMBUU5P5RXN67OMKVDQXFYX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KK4FMBUU5P5RXN67OMKVDQXFYX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KK4FMBUU5P5RXN67OMKVDQXFYX","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":"aca42acac0ead11892f3be18da0524434af2922d3768bb578a4e635634baf2c2","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.CG","submitted_at":"2026-05-20T18:47:17Z","title_canon_sha256":"a77dc985e0c7d57b148fec1fe01a5040a6e9d20a88d149b594807daf16d86524"},"schema_version":"1.0","source":{"id":"2605.21636","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21636","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21636v1","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21636","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"pith_short_12","alias_value":"KK4FMBUU5P5R","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"pith_short_16","alias_value":"KK4FMBUU5P5RXN67","created_at":"2026-05-22T01:03:26Z"},{"alias_kind":"pith_short_8","alias_value":"KK4FMBUU","created_at":"2026-05-22T01:03:26Z"}],"graph_snapshots":[{"event_id":"sha256:993b63be71bb1332bbfeecbe715b85136f5b1b1f1f9b8f9460240d01851bc40f","target":"graph","created_at":"2026-05-22T01:03:26Z","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/2605.21636/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A key property of the Delaunay filtration is that it is topologically (i.e., weakly) equivalent to the offset (union-of-balls) filtration. Recently, this filtration has been extended to point clouds equipped with an $\\mathbb{R}$-valued function, yielding a computable 2-parameter filtration that satisfies an analogous weak equivalence. Motivated in part by the study of time-varying data, we introduce a 3-parameter extension of the Delaunay filtration for point clouds equipped with an $\\mathbb{R}^2$-valued function, also satisfying an analogous weak equivalence. For a point cloud $X \\subset \\mat","authors_text":"Abhishek Rathod, \\'Angel Javier Alonso, Michael Kerber, Michael Lesnick, Tung Lam","cross_cats":["math.AT"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.CG","submitted_at":"2026-05-20T18:47:17Z","title":"Bifunction and Interlevel Delaunay Trifiltrations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21636","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":"sha256:5d76e76b1c0d30fe40cbea65f38b7ab78298a4a68511ce7477162fd5ed391835","target":"record","created_at":"2026-05-22T01:03:26Z","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":"aca42acac0ead11892f3be18da0524434af2922d3768bb578a4e635634baf2c2","cross_cats_sorted":["math.AT"],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"cs.CG","submitted_at":"2026-05-20T18:47:17Z","title_canon_sha256":"a77dc985e0c7d57b148fec1fe01a5040a6e9d20a88d149b594807daf16d86524"},"schema_version":"1.0","source":{"id":"2605.21636","kind":"arxiv","version":1}},"canonical_sha256":"52b8560694ebfb1bb7df731551c2e5c5cfa9a5cb8f0b2c1ae10d7598fea2fffb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"52b8560694ebfb1bb7df731551c2e5c5cfa9a5cb8f0b2c1ae10d7598fea2fffb","first_computed_at":"2026-05-22T01:03:26.380878Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:03:26.380878Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tJmNJt8pvIllYAW3erE1R6HHbI4ZDNvXDB5WA+KsFXvl3JTpuPQWYSSccxg6xzZpIci/CyI+UE9aDUzgwVeKCg==","signature_status":"signed_v1","signed_at":"2026-05-22T01:03:26.381336Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.21636","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d76e76b1c0d30fe40cbea65f38b7ab78298a4a68511ce7477162fd5ed391835","sha256:993b63be71bb1332bbfeecbe715b85136f5b1b1f1f9b8f9460240d01851bc40f"],"state_sha256":"c01318cab38dfcc4057991d2cb077537238ac4ee54d364ab8c8ee9c4706e007a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4ZpT2FETDJIAnsZlIglgQC4VaSC4EX11QiZhaJNsiwhLp5GSuoqRnp/68WyuWI7fDttsxCHpN/zo97gQZXGVDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T10:43:50.074817Z","bundle_sha256":"36f7274483b1b4982f37d08f5dcf7871129eb3687a06f3482051b65d7f0022cd"}}