{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:PBCKP2S5P4MMCTN2WMGR4CZH7W","short_pith_number":"pith:PBCKP2S5","canonical_record":{"source":{"id":"2606.13224","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-11T11:38:51Z","cross_cats_sorted":[],"title_canon_sha256":"a23fb0d1f1d74fec69235d73c11a85c55d5c25d169f99c71bf93927b5322c914","abstract_canon_sha256":"32fb3341989ffa8585e3930b4b7d2664f518001ab8fa1f3d3149ea2399e8ae04"},"schema_version":"1.0"},"canonical_sha256":"7844a7ea5d7f18c14dbab30d1e0b27fd9557ec23e19b5b256f21d6c9d4f7f618","source":{"kind":"arxiv","id":"2606.13224","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13224","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13224v1","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13224","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"pith_short_12","alias_value":"PBCKP2S5P4MM","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"pith_short_16","alias_value":"PBCKP2S5P4MMCTN2","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"pith_short_8","alias_value":"PBCKP2S5","created_at":"2026-06-12T01:09:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:PBCKP2S5P4MMCTN2WMGR4CZH7W","target":"record","payload":{"canonical_record":{"source":{"id":"2606.13224","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-11T11:38:51Z","cross_cats_sorted":[],"title_canon_sha256":"a23fb0d1f1d74fec69235d73c11a85c55d5c25d169f99c71bf93927b5322c914","abstract_canon_sha256":"32fb3341989ffa8585e3930b4b7d2664f518001ab8fa1f3d3149ea2399e8ae04"},"schema_version":"1.0"},"canonical_sha256":"7844a7ea5d7f18c14dbab30d1e0b27fd9557ec23e19b5b256f21d6c9d4f7f618","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:47.477922Z","signature_b64":"mTFmgHDmaCcNqlFwWDtcEmYeDADTtIMtef31yoWaqtV8hZAr9ID1H7KaVkpVO2YJobtwNODWksUWHn7SOgR8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7844a7ea5d7f18c14dbab30d1e0b27fd9557ec23e19b5b256f21d6c9d4f7f618","last_reissued_at":"2026-06-12T01:09:47.477098Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:47.477098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.13224","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":"2026-06-12T01:09:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+wMAscafwyHA4j//+hPmmEZ+FOhFMO3QZ+zUGNNlAw1BlJNtKvJxK/A3xfcKbcQ3OZXyrOBgnTYUmDuLporTAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:29:50.139594Z"},"content_sha256":"fba9d499cb47b967c19c4fd26e01c9bd7e7187723424f6a64def120e87887f2c","schema_version":"1.0","event_id":"sha256:fba9d499cb47b967c19c4fd26e01c9bd7e7187723424f6a64def120e87887f2c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:PBCKP2S5P4MMCTN2WMGR4CZH7W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bott-Chern and Aeppli homotopy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Jiahao Hu","submitted_at":"2026-06-11T11:38:51Z","abstract_excerpt":"This paper introduces Bott-Chern and Aeppli homotopy sets for a fibrant class of bisimplicial sets and establishes their basic properties. In positive bidegrees, Bott-Chern homotopy sets carry natural monoid structures, while Aeppli homotopy sets carry natural group structures. They are related by a loop-space comparison: after a bidegree shift, the Aeppli homotopy groups of X are naturally identified with the Bott-Chern homotopy monoids of the loop space of X. In particular, the Bott-Chern homotopy monoids of loop spaces are groups.\n  To justify our definitions, we show that the Bott-Chern ho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13224","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/2606.13224/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":"2026-06-12T01:09:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0eWdvzLeogBe0dDIaLpYc//nRgPTiKqvYOPh53Rz/DVmmtRIuAfUPwgz9Pt8wCyv/tBPDV4MkJNtU1IMsL3NAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T18:29:50.139966Z"},"content_sha256":"a38e372d0c451792ce14b38ae366d06a50f3dff4d9d988fe4594b0c7041fe4e3","schema_version":"1.0","event_id":"sha256:a38e372d0c451792ce14b38ae366d06a50f3dff4d9d988fe4594b0c7041fe4e3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PBCKP2S5P4MMCTN2WMGR4CZH7W/bundle.json","state_url":"https://pith.science/pith/PBCKP2S5P4MMCTN2WMGR4CZH7W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PBCKP2S5P4MMCTN2WMGR4CZH7W/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":"2026-06-25T18:29:50Z","links":{"resolver":"https://pith.science/pith/PBCKP2S5P4MMCTN2WMGR4CZH7W","bundle":"https://pith.science/pith/PBCKP2S5P4MMCTN2WMGR4CZH7W/bundle.json","state":"https://pith.science/pith/PBCKP2S5P4MMCTN2WMGR4CZH7W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PBCKP2S5P4MMCTN2WMGR4CZH7W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:PBCKP2S5P4MMCTN2WMGR4CZH7W","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":"32fb3341989ffa8585e3930b4b7d2664f518001ab8fa1f3d3149ea2399e8ae04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-11T11:38:51Z","title_canon_sha256":"a23fb0d1f1d74fec69235d73c11a85c55d5c25d169f99c71bf93927b5322c914"},"schema_version":"1.0","source":{"id":"2606.13224","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13224","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13224v1","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13224","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"pith_short_12","alias_value":"PBCKP2S5P4MM","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"pith_short_16","alias_value":"PBCKP2S5P4MMCTN2","created_at":"2026-06-12T01:09:47Z"},{"alias_kind":"pith_short_8","alias_value":"PBCKP2S5","created_at":"2026-06-12T01:09:47Z"}],"graph_snapshots":[{"event_id":"sha256:a38e372d0c451792ce14b38ae366d06a50f3dff4d9d988fe4594b0c7041fe4e3","target":"graph","created_at":"2026-06-12T01:09:47Z","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/2606.13224/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper introduces Bott-Chern and Aeppli homotopy sets for a fibrant class of bisimplicial sets and establishes their basic properties. In positive bidegrees, Bott-Chern homotopy sets carry natural monoid structures, while Aeppli homotopy sets carry natural group structures. They are related by a loop-space comparison: after a bidegree shift, the Aeppli homotopy groups of X are naturally identified with the Bott-Chern homotopy monoids of the loop space of X. In particular, the Bott-Chern homotopy monoids of loop spaces are groups.\n  To justify our definitions, we show that the Bott-Chern ho","authors_text":"Jiahao Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-11T11:38:51Z","title":"Bott-Chern and Aeppli homotopy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13224","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:fba9d499cb47b967c19c4fd26e01c9bd7e7187723424f6a64def120e87887f2c","target":"record","created_at":"2026-06-12T01:09:47Z","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":"32fb3341989ffa8585e3930b4b7d2664f518001ab8fa1f3d3149ea2399e8ae04","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2026-06-11T11:38:51Z","title_canon_sha256":"a23fb0d1f1d74fec69235d73c11a85c55d5c25d169f99c71bf93927b5322c914"},"schema_version":"1.0","source":{"id":"2606.13224","kind":"arxiv","version":1}},"canonical_sha256":"7844a7ea5d7f18c14dbab30d1e0b27fd9557ec23e19b5b256f21d6c9d4f7f618","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7844a7ea5d7f18c14dbab30d1e0b27fd9557ec23e19b5b256f21d6c9d4f7f618","first_computed_at":"2026-06-12T01:09:47.477098Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-12T01:09:47.477098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mTFmgHDmaCcNqlFwWDtcEmYeDADTtIMtef31yoWaqtV8hZAr9ID1H7KaVkpVO2YJobtwNODWksUWHn7SOgR8CQ==","signature_status":"signed_v1","signed_at":"2026-06-12T01:09:47.477922Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.13224","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fba9d499cb47b967c19c4fd26e01c9bd7e7187723424f6a64def120e87887f2c","sha256:a38e372d0c451792ce14b38ae366d06a50f3dff4d9d988fe4594b0c7041fe4e3"],"state_sha256":"14036412d32bfe34ff386a44753792018e4c54459ca819f889f1493ad3041fa1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WeYpxqSshmOyM/Wz75ouFwuAJ0YTQ+NCvYJm3VYA2obg7x7rFylp0S8SbCNnXspcgOH5r5n9PL6pXW5jPrSoCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T18:29:50.142189Z","bundle_sha256":"dd47e7c4728a1ef571cf17bb60e7d9c932701886282789d7e148800734f85ec4"}}