{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:BM5KFY42CM46MJRB3WGETAXXEO","short_pith_number":"pith:BM5KFY42","canonical_record":{"source":{"id":"2010.09708","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-10-19T17:57:04Z","cross_cats_sorted":["hep-th","math-ph","math.AG","math.MP"],"title_canon_sha256":"ea8020fb45f03952c6dc94738aac7be91377d9a23b880fb8aa911145aa9e05a4","abstract_canon_sha256":"9aaee4325cf7113f4a7775f7c79950957642c659b2cb77061a9a489491e5e5ad"},"schema_version":"1.0"},"canonical_sha256":"0b3aa2e39a1339e62621dd8c4982f72386c16b3796c617ccaf1dcb162b49846b","source":{"kind":"arxiv","id":"2010.09708","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.09708","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"2010.09708v2","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.09708","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"BM5KFY42CM46","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"pith_short_16","alias_value":"BM5KFY42CM46MJRB","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"pith_short_8","alias_value":"BM5KFY42","created_at":"2026-07-05T04:11:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:BM5KFY42CM46MJRB3WGETAXXEO","target":"record","payload":{"canonical_record":{"source":{"id":"2010.09708","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-10-19T17:57:04Z","cross_cats_sorted":["hep-th","math-ph","math.AG","math.MP"],"title_canon_sha256":"ea8020fb45f03952c6dc94738aac7be91377d9a23b880fb8aa911145aa9e05a4","abstract_canon_sha256":"9aaee4325cf7113f4a7775f7c79950957642c659b2cb77061a9a489491e5e5ad"},"schema_version":"1.0"},"canonical_sha256":"0b3aa2e39a1339e62621dd8c4982f72386c16b3796c617ccaf1dcb162b49846b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:11:23.633877Z","signature_b64":"tn0IA+kin4UpmMI4301sDpyC59DtPqaskMFH5aCIAt8V9u/FqRCGZXWJI12syFF2X82XrxDd45OBpox1RZQYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b3aa2e39a1339e62621dd8c4982f72386c16b3796c617ccaf1dcb162b49846b","last_reissued_at":"2026-07-05T04:11:23.633478Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:11:23.633478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2010.09708","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T04:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ldwLHB+8nvmdB8vDH81IDF1dF2jeBQkbdhouTcoUDa1aZp1k3nwmq3xasz96L5aG3e2QYLZZg7yHmYBdhEHSAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T21:29:58.388586Z"},"content_sha256":"d03e0812365c81e32ec221cfbc709b3ada1102ad1921c43e7b2cd192ee67b246","schema_version":"1.0","event_id":"sha256:d03e0812365c81e32ec221cfbc709b3ada1102ad1921c43e7b2cd192ee67b246"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:BM5KFY42CM46MJRB3WGETAXXEO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math-ph","math.AG","math.MP"],"primary_cat":"math.CO","authors_text":"Freddy Cachazo, Nick Early","submitted_at":"2020-10-19T17:57:04Z","abstract_excerpt":"In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\\{1,\\ldots, n\\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.09708","kind":"arxiv","version":2},"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/2010.09708/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":"2026-07-05T04:11:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"On1b4MUf3bGyh3map97LIUhMFmJnh8y4lcWNztWlqOqoUW+zmKGkAgyCfh6A1GWr2WUPSPzmB3QuEGxalpWEBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T21:29:58.389079Z"},"content_sha256":"8c8c1e4fb9ddda3f800aecc3269713a41fd517fc29ba8e55017bebfaa44842ad","schema_version":"1.0","event_id":"sha256:8c8c1e4fb9ddda3f800aecc3269713a41fd517fc29ba8e55017bebfaa44842ad"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BM5KFY42CM46MJRB3WGETAXXEO/bundle.json","state_url":"https://pith.science/pith/BM5KFY42CM46MJRB3WGETAXXEO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BM5KFY42CM46MJRB3WGETAXXEO/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":"2026-08-04T21:29:58Z","links":{"resolver":"https://pith.science/pith/BM5KFY42CM46MJRB3WGETAXXEO","bundle":"https://pith.science/pith/BM5KFY42CM46MJRB3WGETAXXEO/bundle.json","state":"https://pith.science/pith/BM5KFY42CM46MJRB3WGETAXXEO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BM5KFY42CM46MJRB3WGETAXXEO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:BM5KFY42CM46MJRB3WGETAXXEO","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":"9aaee4325cf7113f4a7775f7c79950957642c659b2cb77061a9a489491e5e5ad","cross_cats_sorted":["hep-th","math-ph","math.AG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-10-19T17:57:04Z","title_canon_sha256":"ea8020fb45f03952c6dc94738aac7be91377d9a23b880fb8aa911145aa9e05a4"},"schema_version":"1.0","source":{"id":"2010.09708","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2010.09708","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"arxiv_version","alias_value":"2010.09708v2","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2010.09708","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"pith_short_12","alias_value":"BM5KFY42CM46","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"pith_short_16","alias_value":"BM5KFY42CM46MJRB","created_at":"2026-07-05T04:11:23Z"},{"alias_kind":"pith_short_8","alias_value":"BM5KFY42","created_at":"2026-07-05T04:11:23Z"}],"graph_snapshots":[{"event_id":"sha256:8c8c1e4fb9ddda3f800aecc3269713a41fd517fc29ba8e55017bebfaa44842ad","target":"graph","created_at":"2026-07-05T04:11:23Z","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/2010.09708/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\\{1,\\ldots, n\\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $","authors_text":"Freddy Cachazo, Nick Early","cross_cats":["hep-th","math-ph","math.AG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-10-19T17:57:04Z","title":"Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2010.09708","kind":"arxiv","version":2},"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:d03e0812365c81e32ec221cfbc709b3ada1102ad1921c43e7b2cd192ee67b246","target":"record","created_at":"2026-07-05T04:11:23Z","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":"9aaee4325cf7113f4a7775f7c79950957642c659b2cb77061a9a489491e5e5ad","cross_cats_sorted":["hep-th","math-ph","math.AG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-10-19T17:57:04Z","title_canon_sha256":"ea8020fb45f03952c6dc94738aac7be91377d9a23b880fb8aa911145aa9e05a4"},"schema_version":"1.0","source":{"id":"2010.09708","kind":"arxiv","version":2}},"canonical_sha256":"0b3aa2e39a1339e62621dd8c4982f72386c16b3796c617ccaf1dcb162b49846b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0b3aa2e39a1339e62621dd8c4982f72386c16b3796c617ccaf1dcb162b49846b","first_computed_at":"2026-07-05T04:11:23.633478Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:11:23.633478Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tn0IA+kin4UpmMI4301sDpyC59DtPqaskMFH5aCIAt8V9u/FqRCGZXWJI12syFF2X82XrxDd45OBpox1RZQYDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T04:11:23.633877Z","signed_message":"canonical_sha256_bytes"},"source_id":"2010.09708","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d03e0812365c81e32ec221cfbc709b3ada1102ad1921c43e7b2cd192ee67b246","sha256:8c8c1e4fb9ddda3f800aecc3269713a41fd517fc29ba8e55017bebfaa44842ad"],"state_sha256":"de612d5a66e9d92e71fdbece73b5523e39be1dc5e8302c26200a5d74aec97916"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ND0K2/XPMb4XJGdvSwouG5s1EkOx+8PqMHLm4akfLECmiGVf78TOQVNrkGFHwuoYjlbhoPuquieJH4u0nH3eBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T21:29:58.392124Z","bundle_sha256":"13ed9d90bda2f6a4a97270e12e5b6453ac0f4de48d52552d7f159286bb9e5b91"}}