{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:TBNQUAMT7C6RLLL6POJZPM7Z73","short_pith_number":"pith:TBNQUAMT","canonical_record":{"source":{"id":"2103.15810","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-03-29T17:58:49Z","cross_cats_sorted":[],"title_canon_sha256":"f552225c8bf7f5bb84525b12420cbbb565e1d755b443bf1be9588cb16db361c0","abstract_canon_sha256":"e3a19dcd402b51e42bba2b16924ea80f82cec18505ff4cecbe7be8a1330965fd"},"schema_version":"1.0"},"canonical_sha256":"985b0a0193f8bd15ad7e7b9397b3f9fed456f708a257a832e06103cc27547d73","source":{"kind":"arxiv","id":"2103.15810","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2103.15810","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"2103.15810v3","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2103.15810","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"TBNQUAMT7C6R","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"pith_short_16","alias_value":"TBNQUAMT7C6RLLL6","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"pith_short_8","alias_value":"TBNQUAMT","created_at":"2026-07-05T08:54:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:TBNQUAMT7C6RLLL6POJZPM7Z73","target":"record","payload":{"canonical_record":{"source":{"id":"2103.15810","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-03-29T17:58:49Z","cross_cats_sorted":[],"title_canon_sha256":"f552225c8bf7f5bb84525b12420cbbb565e1d755b443bf1be9588cb16db361c0","abstract_canon_sha256":"e3a19dcd402b51e42bba2b16924ea80f82cec18505ff4cecbe7be8a1330965fd"},"schema_version":"1.0"},"canonical_sha256":"985b0a0193f8bd15ad7e7b9397b3f9fed456f708a257a832e06103cc27547d73","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:54:39.942329Z","signature_b64":"sZVMBcwq6dHnyEA5Wa02juqT5X6xaP79MZcnzmBDdXOM8ypojUNTM+ifE+IfceTQJAhayHPrtCk8TYKspSr5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"985b0a0193f8bd15ad7e7b9397b3f9fed456f708a257a832e06103cc27547d73","last_reissued_at":"2026-07-05T08:54:39.941871Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:54:39.941871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2103.15810","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M+4NlwJ6i+v9b76DXoq0xBTZjLJx+P5q6DrNWQKIPC49seDVVVJjMMKAjs1V6dPTUdFPYV5KurKB1o1dnzyVCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T20:00:59.917838Z"},"content_sha256":"760c4446d7ed76ed2095c7fe8725118a28377e186798d84effba7e77c6eab49a","schema_version":"1.0","event_id":"sha256:760c4446d7ed76ed2095c7fe8725118a28377e186798d84effba7e77c6eab49a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:TBNQUAMT7C6RLLL6POJZPM7Z73","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kinematic numerators from the worldsheet: cubic trees from labelled trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"Jintian Tian, Linghui Hou, Song He, Yong Zhang","submitted_at":"2021-03-29T17:58:49Z","abstract_excerpt":"In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called \"Cayley functions\", which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.15810","kind":"arxiv","version":3},"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/2103.15810/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":"2026-07-05T08:54:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"i3523IKaPAw1/ZqahEym3P2ejNyOmV/ZGGmIaQYguV7zVCaaXqeEyrhLPnUMWC3hbTi9XBukBOGdfe778ipzBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T20:00:59.918349Z"},"content_sha256":"9bad126456f3c94b8bc51834915ed1df2c2cc9fb3745d00332040176982c3f0d","schema_version":"1.0","event_id":"sha256:9bad126456f3c94b8bc51834915ed1df2c2cc9fb3745d00332040176982c3f0d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TBNQUAMT7C6RLLL6POJZPM7Z73/bundle.json","state_url":"https://pith.science/pith/TBNQUAMT7C6RLLL6POJZPM7Z73/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TBNQUAMT7C6RLLL6POJZPM7Z73/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":"2026-08-04T20:00:59Z","links":{"resolver":"https://pith.science/pith/TBNQUAMT7C6RLLL6POJZPM7Z73","bundle":"https://pith.science/pith/TBNQUAMT7C6RLLL6POJZPM7Z73/bundle.json","state":"https://pith.science/pith/TBNQUAMT7C6RLLL6POJZPM7Z73/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TBNQUAMT7C6RLLL6POJZPM7Z73/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:TBNQUAMT7C6RLLL6POJZPM7Z73","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":"e3a19dcd402b51e42bba2b16924ea80f82cec18505ff4cecbe7be8a1330965fd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-03-29T17:58:49Z","title_canon_sha256":"f552225c8bf7f5bb84525b12420cbbb565e1d755b443bf1be9588cb16db361c0"},"schema_version":"1.0","source":{"id":"2103.15810","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2103.15810","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"arxiv_version","alias_value":"2103.15810v3","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2103.15810","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"pith_short_12","alias_value":"TBNQUAMT7C6R","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"pith_short_16","alias_value":"TBNQUAMT7C6RLLL6","created_at":"2026-07-05T08:54:39Z"},{"alias_kind":"pith_short_8","alias_value":"TBNQUAMT","created_at":"2026-07-05T08:54:39Z"}],"graph_snapshots":[{"event_id":"sha256:9bad126456f3c94b8bc51834915ed1df2c2cc9fb3745d00332040176982c3f0d","target":"graph","created_at":"2026-07-05T08:54:39Z","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/2103.15810/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note we revisit the problem of explicitly computing tree-level scattering amplitudes in various theories in any dimension from worldsheet formulas. The latter are known to produce cubic-tree expansion of tree amplitudes with kinematic numerators automatically satisfying Jacobi-identities, once any half-integrand on the worldsheet is reduced to logarithmic functions. We review a natural class of worldsheet functions called \"Cayley functions\", which are in one-to-one correspondence with labelled trees, and natural expansions of known half-integrands onto them with coefficients that are p","authors_text":"Jintian Tian, Linghui Hou, Song He, Yong Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-03-29T17:58:49Z","title":"Kinematic numerators from the worldsheet: cubic trees from labelled trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.15810","kind":"arxiv","version":3},"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":"sha256:760c4446d7ed76ed2095c7fe8725118a28377e186798d84effba7e77c6eab49a","target":"record","created_at":"2026-07-05T08:54:39Z","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":"e3a19dcd402b51e42bba2b16924ea80f82cec18505ff4cecbe7be8a1330965fd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-03-29T17:58:49Z","title_canon_sha256":"f552225c8bf7f5bb84525b12420cbbb565e1d755b443bf1be9588cb16db361c0"},"schema_version":"1.0","source":{"id":"2103.15810","kind":"arxiv","version":3}},"canonical_sha256":"985b0a0193f8bd15ad7e7b9397b3f9fed456f708a257a832e06103cc27547d73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"985b0a0193f8bd15ad7e7b9397b3f9fed456f708a257a832e06103cc27547d73","first_computed_at":"2026-07-05T08:54:39.941871Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:54:39.941871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sZVMBcwq6dHnyEA5Wa02juqT5X6xaP79MZcnzmBDdXOM8ypojUNTM+ifE+IfceTQJAhayHPrtCk8TYKspSr5Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T08:54:39.942329Z","signed_message":"canonical_sha256_bytes"},"source_id":"2103.15810","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:760c4446d7ed76ed2095c7fe8725118a28377e186798d84effba7e77c6eab49a","sha256:9bad126456f3c94b8bc51834915ed1df2c2cc9fb3745d00332040176982c3f0d"],"state_sha256":"b6b878ffa082a6ef8919bd2ba02f9b9d042ad2636e6b07729c40ba9e0bbdcb3a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VCeSmYfMLWCABkj5Xqznt0C4KGX4MAcyLy7f+vEw6pjl+ae5i6LmzhosWU9FSvHzoMceR0YLkrGAJ7jsWccfDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T20:00:59.923636Z","bundle_sha256":"46db9306a8f47cc5d6154c02d601b38d9a3b9cfbb8ec0d90ce17bc1213747dd5"}}