{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TPNQSK5ICWQUODUQVBJOIINPID","short_pith_number":"pith:TPNQSK5I","canonical_record":{"source":{"id":"2605.20586","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CT","submitted_at":"2026-05-20T00:48:45Z","cross_cats_sorted":[],"title_canon_sha256":"e222c584d71edfff7d97a7ae560200c1e764528c84134efa10bebdc58616758c","abstract_canon_sha256":"4383a20c0c786c4767b9d32e7b163d509e8ddeed5bc1dd4dba5b6a077407bc57"},"schema_version":"1.0"},"canonical_sha256":"9bdb092ba815a1470e90a852e421af40c562d904ec5d3909ded72c102bd83f04","source":{"kind":"arxiv","id":"2605.20586","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20586","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20586v1","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20586","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"pith_short_12","alias_value":"TPNQSK5ICWQU","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"pith_short_16","alias_value":"TPNQSK5ICWQUODUQ","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"pith_short_8","alias_value":"TPNQSK5I","created_at":"2026-05-21T01:04:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TPNQSK5ICWQUODUQVBJOIINPID","target":"record","payload":{"canonical_record":{"source":{"id":"2605.20586","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CT","submitted_at":"2026-05-20T00:48:45Z","cross_cats_sorted":[],"title_canon_sha256":"e222c584d71edfff7d97a7ae560200c1e764528c84134efa10bebdc58616758c","abstract_canon_sha256":"4383a20c0c786c4767b9d32e7b163d509e8ddeed5bc1dd4dba5b6a077407bc57"},"schema_version":"1.0"},"canonical_sha256":"9bdb092ba815a1470e90a852e421af40c562d904ec5d3909ded72c102bd83f04","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-21T01:04:43.543462Z","signature_b64":"OAgogMjyqUHJaSFZZKIVE7ZL4oivgwcFG1NL4bTTLhnMNwhoc25SUD8NpwEqXfhhAo7RD8f1d225RNWw/GpoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bdb092ba815a1470e90a852e421af40c562d904ec5d3909ded72c102bd83f04","last_reissued_at":"2026-05-21T01:04:43.543010Z","signature_status":"signed_v1","first_computed_at":"2026-05-21T01:04:43.543010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.20586","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":"2026-05-21T01:04:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5czDrM1inBa65t/86uHIWxltqMJzAdtqgEuljQicnoL1A6wI+eBQQBbaFBrRiIYioJMOVBned+kAOK2rlSymBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T21:00:26.870850Z"},"content_sha256":"618dde3317e588b7888e7d1f151beb1fb961bccbb672b92ac051187700766bda","schema_version":"1.0","event_id":"sha256:618dde3317e588b7888e7d1f151beb1fb961bccbb672b92ac051187700766bda"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TPNQSK5ICWQUODUQVBJOIINPID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exponentiable Virtual Double Categories and Representability of Exponentials","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CT","authors_text":"Ea E Thompson, Kevin D. Carlson","submitted_at":"2026-05-20T00:48:45Z","abstract_excerpt":"Virtual double categories provide an effective framework for formal category theory. Recent work has investigated the question of higher morphisms between virtual double categories, following on from work on higher morphisms between double categories, and building up to Arkor's recent conjecture on exponentiable virtual double categories--those virtual double categories, morphisms out of which can themselves be enriched to a whole virtual double category.\n  In this paper we resolve Arkor's conjecture by providing a number of equivalent characterizations of the exponentiable virtual double cate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20586","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/2605.20586/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":"2026-05-21T01:04:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"62Y7n4YlOK+hL9GjbTqRyHyQ6PV1ut1E37wVvG/HT5ex4YuuvdExZHC5FjBnY+s4i2nMCTIFxxnt3MvcoGbNDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-24T21:00:26.871569Z"},"content_sha256":"06226fc09ae3280672434391d2d9a1fc29afae9fe500b84f1b98d4df4067a40a","schema_version":"1.0","event_id":"sha256:06226fc09ae3280672434391d2d9a1fc29afae9fe500b84f1b98d4df4067a40a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TPNQSK5ICWQUODUQVBJOIINPID/bundle.json","state_url":"https://pith.science/pith/TPNQSK5ICWQUODUQVBJOIINPID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TPNQSK5ICWQUODUQVBJOIINPID/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":"2026-05-24T21:00:26Z","links":{"resolver":"https://pith.science/pith/TPNQSK5ICWQUODUQVBJOIINPID","bundle":"https://pith.science/pith/TPNQSK5ICWQUODUQVBJOIINPID/bundle.json","state":"https://pith.science/pith/TPNQSK5ICWQUODUQVBJOIINPID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TPNQSK5ICWQUODUQVBJOIINPID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TPNQSK5ICWQUODUQVBJOIINPID","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":"4383a20c0c786c4767b9d32e7b163d509e8ddeed5bc1dd4dba5b6a077407bc57","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CT","submitted_at":"2026-05-20T00:48:45Z","title_canon_sha256":"e222c584d71edfff7d97a7ae560200c1e764528c84134efa10bebdc58616758c"},"schema_version":"1.0","source":{"id":"2605.20586","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.20586","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"arxiv_version","alias_value":"2605.20586v1","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.20586","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"pith_short_12","alias_value":"TPNQSK5ICWQU","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"pith_short_16","alias_value":"TPNQSK5ICWQUODUQ","created_at":"2026-05-21T01:04:43Z"},{"alias_kind":"pith_short_8","alias_value":"TPNQSK5I","created_at":"2026-05-21T01:04:43Z"}],"graph_snapshots":[{"event_id":"sha256:06226fc09ae3280672434391d2d9a1fc29afae9fe500b84f1b98d4df4067a40a","target":"graph","created_at":"2026-05-21T01:04:43Z","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/2605.20586/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Virtual double categories provide an effective framework for formal category theory. Recent work has investigated the question of higher morphisms between virtual double categories, following on from work on higher morphisms between double categories, and building up to Arkor's recent conjecture on exponentiable virtual double categories--those virtual double categories, morphisms out of which can themselves be enriched to a whole virtual double category.\n  In this paper we resolve Arkor's conjecture by providing a number of equivalent characterizations of the exponentiable virtual double cate","authors_text":"Ea E Thompson, Kevin D. Carlson","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CT","submitted_at":"2026-05-20T00:48:45Z","title":"Exponentiable Virtual Double Categories and Representability of Exponentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20586","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":"sha256:618dde3317e588b7888e7d1f151beb1fb961bccbb672b92ac051187700766bda","target":"record","created_at":"2026-05-21T01:04:43Z","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":"4383a20c0c786c4767b9d32e7b163d509e8ddeed5bc1dd4dba5b6a077407bc57","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CT","submitted_at":"2026-05-20T00:48:45Z","title_canon_sha256":"e222c584d71edfff7d97a7ae560200c1e764528c84134efa10bebdc58616758c"},"schema_version":"1.0","source":{"id":"2605.20586","kind":"arxiv","version":1}},"canonical_sha256":"9bdb092ba815a1470e90a852e421af40c562d904ec5d3909ded72c102bd83f04","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9bdb092ba815a1470e90a852e421af40c562d904ec5d3909ded72c102bd83f04","first_computed_at":"2026-05-21T01:04:43.543010Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-21T01:04:43.543010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OAgogMjyqUHJaSFZZKIVE7ZL4oivgwcFG1NL4bTTLhnMNwhoc25SUD8NpwEqXfhhAo7RD8f1d225RNWw/GpoDw==","signature_status":"signed_v1","signed_at":"2026-05-21T01:04:43.543462Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.20586","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:618dde3317e588b7888e7d1f151beb1fb961bccbb672b92ac051187700766bda","sha256:06226fc09ae3280672434391d2d9a1fc29afae9fe500b84f1b98d4df4067a40a"],"state_sha256":"add625d0c081fe2e0da320de652410d96201c154e310cd4024902472b5bbcd36"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VJE+3V4wESULD81hLz8f9fNLBsubuSuQc7lN78QYsDMaygwi1lXIvB5A6EbuS2g503et05Uu5ndgmvVxfdJ/BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-24T21:00:26.876082Z","bundle_sha256":"6a110c70624e43d5c7de86d780f07022d7a94afb396a5f41578b44a3fe64355e"}}