{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BMIRAREOMJJGIWFPLM24CCZPHM","short_pith_number":"pith:BMIRAREO","canonical_record":{"source":{"id":"1104.3528","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-18T15:56:22Z","cross_cats_sorted":[],"title_canon_sha256":"d2a7b2fffb9702d3c9b84f06054c4e1f8637a30a0a9a91d86471e60b01377f70","abstract_canon_sha256":"2e9be13bcafcef7dd75b5cede2ee3e91ca7d68a8cfe5a7ac92343bc8b8da36c7"},"schema_version":"1.0"},"canonical_sha256":"0b1110448e62526458af5b35c10b2f3b18b9de5da7c08c6eac264ad11aaf872c","source":{"kind":"arxiv","id":"1104.3528","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.3528","created_at":"2026-05-18T01:27:55Z"},{"alias_kind":"arxiv_version","alias_value":"1104.3528v2","created_at":"2026-05-18T01:27:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3528","created_at":"2026-05-18T01:27:55Z"},{"alias_kind":"pith_short_12","alias_value":"BMIRAREOMJJG","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BMIRAREOMJJGIWFP","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BMIRAREO","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BMIRAREOMJJGIWFPLM24CCZPHM","target":"record","payload":{"canonical_record":{"source":{"id":"1104.3528","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-18T15:56:22Z","cross_cats_sorted":[],"title_canon_sha256":"d2a7b2fffb9702d3c9b84f06054c4e1f8637a30a0a9a91d86471e60b01377f70","abstract_canon_sha256":"2e9be13bcafcef7dd75b5cede2ee3e91ca7d68a8cfe5a7ac92343bc8b8da36c7"},"schema_version":"1.0"},"canonical_sha256":"0b1110448e62526458af5b35c10b2f3b18b9de5da7c08c6eac264ad11aaf872c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:55.133752Z","signature_b64":"Dn0ov3vCCHgHCDPz3S8DBEgjVxq/zmAJkVC6UHQ6K2reniYxhqUEsSv+15y0HhfhlvgYPZKWtGrVZmoh3K0qBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b1110448e62526458af5b35c10b2f3b18b9de5da7c08c6eac264ad11aaf872c","last_reissued_at":"2026-05-18T01:27:55.133110Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:55.133110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.3528","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":"2026-05-18T01:27:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QoyVYe+x+Ykj8yeyGKm5LzsI19sgDxZX518lysPsNsPARYDRNLvwGe9ju8KVX/ZJB2hJCj5qgm1imqpU2iU/AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T07:29:05.424599Z"},"content_sha256":"4c8c6a30e83d920dc91f243ba39d7bc4bb214a45394a3ab14a4f62d81ad16f31","schema_version":"1.0","event_id":"sha256:4c8c6a30e83d920dc91f243ba39d7bc4bb214a45394a3ab14a4f62d81ad16f31"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BMIRAREOMJJGIWFPLM24CCZPHM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stasheff polytopes and the coordinate ring of the cluster X-variety of type A_n","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Linhui Shen","submitted_at":"2011-04-18T15:56:22Z","abstract_excerpt":"We define Stasheff polytopes in the space of tropical points of cluster A-varieties. We study the supports of products of elements of canonical bases for cluster X-varieties. We prove that, for the cluster X-variety of type A_n, such supports are Stasheff polytopes."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3528","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":""},"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":"2026-05-18T01:27:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0R5v0D6A4dqHt4MtVTcXvpulnPnZVxbWZnzeKmrZFzJHPwYTMWBRffeM7TuPy5BlulPxrQMYjZq1yO7uSpCPBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T07:29:05.425117Z"},"content_sha256":"e1590e57f892fedbd782c3d550a3cd5a56e56ef20cb5a970fc54c71db733ec3b","schema_version":"1.0","event_id":"sha256:e1590e57f892fedbd782c3d550a3cd5a56e56ef20cb5a970fc54c71db733ec3b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BMIRAREOMJJGIWFPLM24CCZPHM/bundle.json","state_url":"https://pith.science/pith/BMIRAREOMJJGIWFPLM24CCZPHM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BMIRAREOMJJGIWFPLM24CCZPHM/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":"2026-05-31T07:29:05Z","links":{"resolver":"https://pith.science/pith/BMIRAREOMJJGIWFPLM24CCZPHM","bundle":"https://pith.science/pith/BMIRAREOMJJGIWFPLM24CCZPHM/bundle.json","state":"https://pith.science/pith/BMIRAREOMJJGIWFPLM24CCZPHM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BMIRAREOMJJGIWFPLM24CCZPHM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BMIRAREOMJJGIWFPLM24CCZPHM","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":"2e9be13bcafcef7dd75b5cede2ee3e91ca7d68a8cfe5a7ac92343bc8b8da36c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-18T15:56:22Z","title_canon_sha256":"d2a7b2fffb9702d3c9b84f06054c4e1f8637a30a0a9a91d86471e60b01377f70"},"schema_version":"1.0","source":{"id":"1104.3528","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.3528","created_at":"2026-05-18T01:27:55Z"},{"alias_kind":"arxiv_version","alias_value":"1104.3528v2","created_at":"2026-05-18T01:27:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3528","created_at":"2026-05-18T01:27:55Z"},{"alias_kind":"pith_short_12","alias_value":"BMIRAREOMJJG","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BMIRAREOMJJGIWFP","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BMIRAREO","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:e1590e57f892fedbd782c3d550a3cd5a56e56ef20cb5a970fc54c71db733ec3b","target":"graph","created_at":"2026-05-18T01:27:55Z","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"},"paper":{"abstract_excerpt":"We define Stasheff polytopes in the space of tropical points of cluster A-varieties. We study the supports of products of elements of canonical bases for cluster X-varieties. We prove that, for the cluster X-variety of type A_n, such supports are Stasheff polytopes.","authors_text":"Linhui Shen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-18T15:56:22Z","title":"Stasheff polytopes and the coordinate ring of the cluster X-variety of type A_n"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3528","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":"sha256:4c8c6a30e83d920dc91f243ba39d7bc4bb214a45394a3ab14a4f62d81ad16f31","target":"record","created_at":"2026-05-18T01:27:55Z","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":"2e9be13bcafcef7dd75b5cede2ee3e91ca7d68a8cfe5a7ac92343bc8b8da36c7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2011-04-18T15:56:22Z","title_canon_sha256":"d2a7b2fffb9702d3c9b84f06054c4e1f8637a30a0a9a91d86471e60b01377f70"},"schema_version":"1.0","source":{"id":"1104.3528","kind":"arxiv","version":2}},"canonical_sha256":"0b1110448e62526458af5b35c10b2f3b18b9de5da7c08c6eac264ad11aaf872c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0b1110448e62526458af5b35c10b2f3b18b9de5da7c08c6eac264ad11aaf872c","first_computed_at":"2026-05-18T01:27:55.133110Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:27:55.133110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Dn0ov3vCCHgHCDPz3S8DBEgjVxq/zmAJkVC6UHQ6K2reniYxhqUEsSv+15y0HhfhlvgYPZKWtGrVZmoh3K0qBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:27:55.133752Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.3528","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c8c6a30e83d920dc91f243ba39d7bc4bb214a45394a3ab14a4f62d81ad16f31","sha256:e1590e57f892fedbd782c3d550a3cd5a56e56ef20cb5a970fc54c71db733ec3b"],"state_sha256":"a912f9a45e02ef75f7d2f87f25a46b8d0e2442570b401fe33651cd06c879eb86"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9GPuyhtAGC5hyybVF/ppFY3LM4OWfRnqfRlzmwSJbzeLCk+kdCXzWy7KY9Fas1Mrl2oJq2k5cirTbICWKLedCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T07:29:05.428215Z","bundle_sha256":"2cddf2472c9772a58145bbe59018b49dc1e892c61f70746f2e90bf8776c88385"}}