{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:ZTPZOJJVVYJAISO5T77AYX23BU","short_pith_number":"pith:ZTPZOJJV","canonical_record":{"source":{"id":"2506.16657","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-20T00:05:48Z","cross_cats_sorted":["math.CT","math.DG","math.GT"],"title_canon_sha256":"9520d889a3ba0949f1b854b3888da2477d7d3b46ebe8b57f3f8da3052aa0a54e","abstract_canon_sha256":"2e6ed12b7e4275fbf41e896ccc2c763fb916573addabd71c1f80b079d18de8cc"},"schema_version":"1.0"},"canonical_sha256":"ccdf972535ae120449dd9ffe0c5f5b0d239f6c52cabe5323df78347815a76ce1","source":{"kind":"arxiv","id":"2506.16657","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.16657","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"arxiv_version","alias_value":"2506.16657v1","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.16657","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"pith_short_12","alias_value":"ZTPZOJJVVYJA","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"pith_short_16","alias_value":"ZTPZOJJVVYJAISO5","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"pith_short_8","alias_value":"ZTPZOJJV","created_at":"2026-07-05T11:24:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:ZTPZOJJVVYJAISO5T77AYX23BU","target":"record","payload":{"canonical_record":{"source":{"id":"2506.16657","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-20T00:05:48Z","cross_cats_sorted":["math.CT","math.DG","math.GT"],"title_canon_sha256":"9520d889a3ba0949f1b854b3888da2477d7d3b46ebe8b57f3f8da3052aa0a54e","abstract_canon_sha256":"2e6ed12b7e4275fbf41e896ccc2c763fb916573addabd71c1f80b079d18de8cc"},"schema_version":"1.0"},"canonical_sha256":"ccdf972535ae120449dd9ffe0c5f5b0d239f6c52cabe5323df78347815a76ce1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:24:40.299295Z","signature_b64":"J9PzV/8AVwWnuBTp5asO9/vLmNlEHQrjIs728Oe02y4MBvqtkQIgLXJj5UhUtv2l0LaX/TFHVjkkt0793VubAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccdf972535ae120449dd9ffe0c5f5b0d239f6c52cabe5323df78347815a76ce1","last_reissued_at":"2026-07-05T11:24:40.298804Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:24:40.298804Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.16657","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":"2026-07-05T11:24:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"51SMUJf/MRthBy5fOs4jFDeun+6l+TQoN3G8ahnG6AshkHv+FGEb+xHTEnR4WNwHzBmO/3XH7+JGK914YKKTCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:06:04.385471Z"},"content_sha256":"e427978d78ff6da48d1a968b95861c21bfa44dcff1dd6bacc7f34c9b0aa452ba","schema_version":"1.0","event_id":"sha256:e427978d78ff6da48d1a968b95861c21bfa44dcff1dd6bacc7f34c9b0aa452ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:ZTPZOJJVVYJAISO5T77AYX23BU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Thin homotopy and the signature of piecewise linear surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.DG","math.GT"],"primary_cat":"math.AT","authors_text":"Darrick Lee, Francis Bischoff","submitted_at":"2025-06-20T00:05:48Z","abstract_excerpt":"We introduce a crossed module of piecewise linear surfaces and study the signature homomorphism, defined as the surface holonomy of a universal translation invariant $2$-connection. This provides a transform whereby surfaces are represented by formal series of tensors. Our main result is that the signature uniquely characterizes a surface up to translation and thin homotopy, also known as tree-like equivalence in the case of paths. This generalizes a result of Chen and positively answers a question of Kapranov in the setting of piecewise linear surfaces. As part of this work, we provide severa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.16657","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/2506.16657/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":"2026-07-05T11:24:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x4s7M/1VYjgAwn2WiCSVJBga45ow+GafP3Y0q3MR/d+3MP+8WMC5YVgiXgNxSkAzioseNVuMoqj0VC88MwejDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:06:04.386023Z"},"content_sha256":"c5e29c943efc1e21086ffa3c0b0c2a93b279f7c1dde68d4bb30d0818da610455","schema_version":"1.0","event_id":"sha256:c5e29c943efc1e21086ffa3c0b0c2a93b279f7c1dde68d4bb30d0818da610455"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZTPZOJJVVYJAISO5T77AYX23BU/bundle.json","state_url":"https://pith.science/pith/ZTPZOJJVVYJAISO5T77AYX23BU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZTPZOJJVVYJAISO5T77AYX23BU/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":"2026-08-16T05:06:04Z","links":{"resolver":"https://pith.science/pith/ZTPZOJJVVYJAISO5T77AYX23BU","bundle":"https://pith.science/pith/ZTPZOJJVVYJAISO5T77AYX23BU/bundle.json","state":"https://pith.science/pith/ZTPZOJJVVYJAISO5T77AYX23BU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZTPZOJJVVYJAISO5T77AYX23BU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:ZTPZOJJVVYJAISO5T77AYX23BU","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":"2e6ed12b7e4275fbf41e896ccc2c763fb916573addabd71c1f80b079d18de8cc","cross_cats_sorted":["math.CT","math.DG","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-20T00:05:48Z","title_canon_sha256":"9520d889a3ba0949f1b854b3888da2477d7d3b46ebe8b57f3f8da3052aa0a54e"},"schema_version":"1.0","source":{"id":"2506.16657","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.16657","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"arxiv_version","alias_value":"2506.16657v1","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.16657","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"pith_short_12","alias_value":"ZTPZOJJVVYJA","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"pith_short_16","alias_value":"ZTPZOJJVVYJAISO5","created_at":"2026-07-05T11:24:40Z"},{"alias_kind":"pith_short_8","alias_value":"ZTPZOJJV","created_at":"2026-07-05T11:24:40Z"}],"graph_snapshots":[{"event_id":"sha256:c5e29c943efc1e21086ffa3c0b0c2a93b279f7c1dde68d4bb30d0818da610455","target":"graph","created_at":"2026-07-05T11:24:40Z","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/2506.16657/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We introduce a crossed module of piecewise linear surfaces and study the signature homomorphism, defined as the surface holonomy of a universal translation invariant $2$-connection. This provides a transform whereby surfaces are represented by formal series of tensors. Our main result is that the signature uniquely characterizes a surface up to translation and thin homotopy, also known as tree-like equivalence in the case of paths. This generalizes a result of Chen and positively answers a question of Kapranov in the setting of piecewise linear surfaces. As part of this work, we provide severa","authors_text":"Darrick Lee, Francis Bischoff","cross_cats":["math.CT","math.DG","math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-20T00:05:48Z","title":"Thin homotopy and the signature of piecewise linear surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.16657","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:e427978d78ff6da48d1a968b95861c21bfa44dcff1dd6bacc7f34c9b0aa452ba","target":"record","created_at":"2026-07-05T11:24:40Z","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":"2e6ed12b7e4275fbf41e896ccc2c763fb916573addabd71c1f80b079d18de8cc","cross_cats_sorted":["math.CT","math.DG","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2025-06-20T00:05:48Z","title_canon_sha256":"9520d889a3ba0949f1b854b3888da2477d7d3b46ebe8b57f3f8da3052aa0a54e"},"schema_version":"1.0","source":{"id":"2506.16657","kind":"arxiv","version":1}},"canonical_sha256":"ccdf972535ae120449dd9ffe0c5f5b0d239f6c52cabe5323df78347815a76ce1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ccdf972535ae120449dd9ffe0c5f5b0d239f6c52cabe5323df78347815a76ce1","first_computed_at":"2026-07-05T11:24:40.298804Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:24:40.298804Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J9PzV/8AVwWnuBTp5asO9/vLmNlEHQrjIs728Oe02y4MBvqtkQIgLXJj5UhUtv2l0LaX/TFHVjkkt0793VubAA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:24:40.299295Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.16657","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e427978d78ff6da48d1a968b95861c21bfa44dcff1dd6bacc7f34c9b0aa452ba","sha256:c5e29c943efc1e21086ffa3c0b0c2a93b279f7c1dde68d4bb30d0818da610455"],"state_sha256":"46c34e5bdfaa044a778ad77fc54bfa380582f5b606dfe633f704bd0cdc066728"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eI+jQaVW/z0MOUsnM6FjVE9TJ1cOke1LHEI0kMavKy9pia61QYLQMm5ExonJUDQ7y4EgkjR4M/HggbVcGwphAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T05:06:04.401706Z","bundle_sha256":"4c16ae57e41662fe5c60e525da699780705e3100e87e9a884ec88c2135186a8d"}}