{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:A3RPZWPG5LCF3AIAZVMUNC6GEI","short_pith_number":"pith:A3RPZWPG","canonical_record":{"source":{"id":"2412.21189","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-th","submitted_at":"2024-12-30T18:55:53Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"a544775da81bdb46600693602d1f2b6e591e5104763a86520353b12a5d9f8734","abstract_canon_sha256":"92e26686d43056813e552bdaa6816ef2c9b9699c1fac2e0aefc7b456618a354a"},"schema_version":"1.0"},"canonical_sha256":"06e2fcd9e6eac45d8100cd59468bc6222ffec134f9f8bd271426385abbe98092","source":{"kind":"arxiv","id":"2412.21189","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.21189","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"arxiv_version","alias_value":"2412.21189v1","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.21189","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"pith_short_12","alias_value":"A3RPZWPG5LCF","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"pith_short_16","alias_value":"A3RPZWPG5LCF3AIA","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"pith_short_8","alias_value":"A3RPZWPG","created_at":"2026-07-05T09:55:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:A3RPZWPG5LCF3AIAZVMUNC6GEI","target":"record","payload":{"canonical_record":{"source":{"id":"2412.21189","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-th","submitted_at":"2024-12-30T18:55:53Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"a544775da81bdb46600693602d1f2b6e591e5104763a86520353b12a5d9f8734","abstract_canon_sha256":"92e26686d43056813e552bdaa6816ef2c9b9699c1fac2e0aefc7b456618a354a"},"schema_version":"1.0"},"canonical_sha256":"06e2fcd9e6eac45d8100cd59468bc6222ffec134f9f8bd271426385abbe98092","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:55:28.598440Z","signature_b64":"5yKab5oqz7l9y6qEERop45ekYDfoADUzBGbhlKHOQ37e4fpkxSBtDhfqYlm7ZFzW3+bIqISvUi52XMptbEIwAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06e2fcd9e6eac45d8100cd59468bc6222ffec134f9f8bd271426385abbe98092","last_reissued_at":"2026-07-05T09:55:28.597938Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:55:28.597938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.21189","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":"2026-07-05T09:55:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tnD5bUTCWjL7jQF9Pj8U92TOpnefb7VCbtO5MCT4Vjh60mMS3LFFANUI0nzqN612kuLMFIgSTInATkQCvQpRBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T13:05:22.154577Z"},"content_sha256":"b680ba5017d9755ca801b57c0bfd6e060f8ee76ec556c668f08bd5dfed7bb025","schema_version":"1.0","event_id":"sha256:b680ba5017d9755ca801b57c0bfd6e060f8ee76ec556c668f08bd5dfed7bb025"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:A3RPZWPG5LCF3AIAZVMUNC6GEI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Many Colours of Amplitudes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"hep-th","authors_text":"Cristian Vergu, Jacob L. Bourjaily, Michael Plesser","submitted_at":"2024-12-30T18:55:53Z","abstract_excerpt":"We study the colour-dependence of scattering amplitudes in Yang-Mills theory with arbitrary (but fixed) gauge group and various representations of charged matter. When the rank of the gauge theory is taken arbitrarily large compared to the number of particles involved in an amplitude, it is well known that the number of independent colour-structure tensors grows factorially with multiplicity; however, for any fixed gauge group, this number grows at most exponentially with multiplicity. We review how this counting arises in representation theory and survey its implications for a wide variety of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.21189","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/2412.21189/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":"2026-07-05T09:55:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tdGyWbGyYu7ZbcJ0br493srk0e7nDqJXP6Tx7MFK+8eoaA+KSRpjZfiPWMakMKjXurYPJPezoAkZoW8eJ5YUBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T13:05:22.155133Z"},"content_sha256":"96595acbf0086948c652ab7cb9f78d198718d34c43c7b19f357a9e9a5fdaa682","schema_version":"1.0","event_id":"sha256:96595acbf0086948c652ab7cb9f78d198718d34c43c7b19f357a9e9a5fdaa682"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A3RPZWPG5LCF3AIAZVMUNC6GEI/bundle.json","state_url":"https://pith.science/pith/A3RPZWPG5LCF3AIAZVMUNC6GEI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A3RPZWPG5LCF3AIAZVMUNC6GEI/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":"2026-08-09T13:05:22Z","links":{"resolver":"https://pith.science/pith/A3RPZWPG5LCF3AIAZVMUNC6GEI","bundle":"https://pith.science/pith/A3RPZWPG5LCF3AIAZVMUNC6GEI/bundle.json","state":"https://pith.science/pith/A3RPZWPG5LCF3AIAZVMUNC6GEI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A3RPZWPG5LCF3AIAZVMUNC6GEI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:A3RPZWPG5LCF3AIAZVMUNC6GEI","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":"92e26686d43056813e552bdaa6816ef2c9b9699c1fac2e0aefc7b456618a354a","cross_cats_sorted":["math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-th","submitted_at":"2024-12-30T18:55:53Z","title_canon_sha256":"a544775da81bdb46600693602d1f2b6e591e5104763a86520353b12a5d9f8734"},"schema_version":"1.0","source":{"id":"2412.21189","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.21189","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"arxiv_version","alias_value":"2412.21189v1","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.21189","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"pith_short_12","alias_value":"A3RPZWPG5LCF","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"pith_short_16","alias_value":"A3RPZWPG5LCF3AIA","created_at":"2026-07-05T09:55:28Z"},{"alias_kind":"pith_short_8","alias_value":"A3RPZWPG","created_at":"2026-07-05T09:55:28Z"}],"graph_snapshots":[{"event_id":"sha256:96595acbf0086948c652ab7cb9f78d198718d34c43c7b19f357a9e9a5fdaa682","target":"graph","created_at":"2026-07-05T09:55:28Z","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/2412.21189/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the colour-dependence of scattering amplitudes in Yang-Mills theory with arbitrary (but fixed) gauge group and various representations of charged matter. When the rank of the gauge theory is taken arbitrarily large compared to the number of particles involved in an amplitude, it is well known that the number of independent colour-structure tensors grows factorially with multiplicity; however, for any fixed gauge group, this number grows at most exponentially with multiplicity. We review how this counting arises in representation theory and survey its implications for a wide variety of","authors_text":"Cristian Vergu, Jacob L. Bourjaily, Michael Plesser","cross_cats":["math.RT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-th","submitted_at":"2024-12-30T18:55:53Z","title":"The Many Colours of Amplitudes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.21189","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:b680ba5017d9755ca801b57c0bfd6e060f8ee76ec556c668f08bd5dfed7bb025","target":"record","created_at":"2026-07-05T09:55:28Z","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":"92e26686d43056813e552bdaa6816ef2c9b9699c1fac2e0aefc7b456618a354a","cross_cats_sorted":["math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"hep-th","submitted_at":"2024-12-30T18:55:53Z","title_canon_sha256":"a544775da81bdb46600693602d1f2b6e591e5104763a86520353b12a5d9f8734"},"schema_version":"1.0","source":{"id":"2412.21189","kind":"arxiv","version":1}},"canonical_sha256":"06e2fcd9e6eac45d8100cd59468bc6222ffec134f9f8bd271426385abbe98092","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06e2fcd9e6eac45d8100cd59468bc6222ffec134f9f8bd271426385abbe98092","first_computed_at":"2026-07-05T09:55:28.597938Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:55:28.597938Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5yKab5oqz7l9y6qEERop45ekYDfoADUzBGbhlKHOQ37e4fpkxSBtDhfqYlm7ZFzW3+bIqISvUi52XMptbEIwAA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:55:28.598440Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.21189","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b680ba5017d9755ca801b57c0bfd6e060f8ee76ec556c668f08bd5dfed7bb025","sha256:96595acbf0086948c652ab7cb9f78d198718d34c43c7b19f357a9e9a5fdaa682"],"state_sha256":"ea6b9a2f7583de7e49cc9ee937eab534abbf07e77519cdd1056d8a9e499a6ac7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RmoPmszXLq0hQF3ToM9W2fKN8KAsDzzd8MFgDR8GYO1FVtVrcCnX3kg/LjEh7LuAj6hdRkZy6OPgQreY74QdCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T13:05:22.160969Z","bundle_sha256":"c29bff2417b384cd7b86aae55d8a3b209e07286ba476fd9b49e22f4452579396"}}