{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:UNDUISBL66WADFVYH5IQIUBHOY","short_pith_number":"pith:UNDUISBL","canonical_record":{"source":{"id":"1306.4429","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-06-19T05:33:51Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"6ecbf7321d9f08c6c16e29bd311e4652182e01c259b3dc16a936149aa042ff21","abstract_canon_sha256":"e24d23bfc9a4741e4549443b46e5f43789606cfe84ee591164529e6262566cd6"},"schema_version":"1.0"},"canonical_sha256":"a34744482bf7ac0196b83f510450277606de84e564cb50d0f16843c8cb4d5b2a","source":{"kind":"arxiv","id":"1306.4429","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.4429","created_at":"2026-05-18T03:18:42Z"},{"alias_kind":"arxiv_version","alias_value":"1306.4429v3","created_at":"2026-05-18T03:18:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.4429","created_at":"2026-05-18T03:18:42Z"},{"alias_kind":"pith_short_12","alias_value":"UNDUISBL66WA","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"UNDUISBL66WADFVY","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"UNDUISBL","created_at":"2026-05-18T12:28:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:UNDUISBL66WADFVYH5IQIUBHOY","target":"record","payload":{"canonical_record":{"source":{"id":"1306.4429","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-06-19T05:33:51Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"6ecbf7321d9f08c6c16e29bd311e4652182e01c259b3dc16a936149aa042ff21","abstract_canon_sha256":"e24d23bfc9a4741e4549443b46e5f43789606cfe84ee591164529e6262566cd6"},"schema_version":"1.0"},"canonical_sha256":"a34744482bf7ac0196b83f510450277606de84e564cb50d0f16843c8cb4d5b2a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:42.057571Z","signature_b64":"QKyUXnBqXIUyA7HJSoxazuMtT4zX5VgTXrBi5HM1JTY9YSnk4ANmX+SZgV3MoHtuHCfqeHI2axFDpJQTae6VCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a34744482bf7ac0196b83f510450277606de84e564cb50d0f16843c8cb4d5b2a","last_reissued_at":"2026-05-18T03:18:42.056995Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:42.056995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1306.4429","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":"2026-05-18T03:18:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kGIEohqOE7RhDzTP9uGKplsChor/OdI7x/yg732DV4EahCWppeBfazwU1SVHfGAXiYhTewoNFvbJA+bFGDEoAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:58:31.522532Z"},"content_sha256":"5204d25f301215abce73988a9b3991b69843f05a95f63403a71bc72e4bc89b7c","schema_version":"1.0","event_id":"sha256:5204d25f301215abce73988a9b3991b69843f05a95f63403a71bc72e4bc89b7c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:UNDUISBL66WADFVYH5IQIUBHOY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the number of populations of critical points of master functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.AG","authors_text":"Alexander Varchenko, Evgeny Mukhin","submitted_at":"2013-06-19T05:33:51Z","abstract_excerpt":"We consider the master functions associated with one irreducible integrable highest weight representation of a Kac-Moody algebra. We study the generation procedure of new critical points from a given critical point of one of these master functions. We show that all critical points of all these master functions can be generated from the critical point of the master function with no variables. In particular this means that the set of all critical points of all these master functions form a single population of critical points. We formulate a conjecture that the number of populations of critical "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4429","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":""},"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":"2026-05-18T03:18:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PEWrerL2EKA3zM/AdKyxCJUwK4MYch25vUb1CM0/cjLY9ZpU+NG07wUrOBL3YF3HGcPcoviX9Fy5OPxoZP73Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:58:31.522881Z"},"content_sha256":"5f2b4e5910703ff1e35946c5ee5da0b607325e587383d58ee1bca53800fc3f4b","schema_version":"1.0","event_id":"sha256:5f2b4e5910703ff1e35946c5ee5da0b607325e587383d58ee1bca53800fc3f4b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UNDUISBL66WADFVYH5IQIUBHOY/bundle.json","state_url":"https://pith.science/pith/UNDUISBL66WADFVYH5IQIUBHOY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UNDUISBL66WADFVYH5IQIUBHOY/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":"2026-06-20T02:58:31Z","links":{"resolver":"https://pith.science/pith/UNDUISBL66WADFVYH5IQIUBHOY","bundle":"https://pith.science/pith/UNDUISBL66WADFVYH5IQIUBHOY/bundle.json","state":"https://pith.science/pith/UNDUISBL66WADFVYH5IQIUBHOY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UNDUISBL66WADFVYH5IQIUBHOY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:UNDUISBL66WADFVYH5IQIUBHOY","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":"e24d23bfc9a4741e4549443b46e5f43789606cfe84ee591164529e6262566cd6","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-06-19T05:33:51Z","title_canon_sha256":"6ecbf7321d9f08c6c16e29bd311e4652182e01c259b3dc16a936149aa042ff21"},"schema_version":"1.0","source":{"id":"1306.4429","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.4429","created_at":"2026-05-18T03:18:42Z"},{"alias_kind":"arxiv_version","alias_value":"1306.4429v3","created_at":"2026-05-18T03:18:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.4429","created_at":"2026-05-18T03:18:42Z"},{"alias_kind":"pith_short_12","alias_value":"UNDUISBL66WA","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"UNDUISBL66WADFVY","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"UNDUISBL","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:5f2b4e5910703ff1e35946c5ee5da0b607325e587383d58ee1bca53800fc3f4b","target":"graph","created_at":"2026-05-18T03:18:42Z","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 consider the master functions associated with one irreducible integrable highest weight representation of a Kac-Moody algebra. We study the generation procedure of new critical points from a given critical point of one of these master functions. We show that all critical points of all these master functions can be generated from the critical point of the master function with no variables. In particular this means that the set of all critical points of all these master functions form a single population of critical points. We formulate a conjecture that the number of populations of critical ","authors_text":"Alexander Varchenko, Evgeny Mukhin","cross_cats":["math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-06-19T05:33:51Z","title":"On the number of populations of critical points of master functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.4429","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":"sha256:5204d25f301215abce73988a9b3991b69843f05a95f63403a71bc72e4bc89b7c","target":"record","created_at":"2026-05-18T03:18:42Z","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":"e24d23bfc9a4741e4549443b46e5f43789606cfe84ee591164529e6262566cd6","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-06-19T05:33:51Z","title_canon_sha256":"6ecbf7321d9f08c6c16e29bd311e4652182e01c259b3dc16a936149aa042ff21"},"schema_version":"1.0","source":{"id":"1306.4429","kind":"arxiv","version":3}},"canonical_sha256":"a34744482bf7ac0196b83f510450277606de84e564cb50d0f16843c8cb4d5b2a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a34744482bf7ac0196b83f510450277606de84e564cb50d0f16843c8cb4d5b2a","first_computed_at":"2026-05-18T03:18:42.056995Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:42.056995Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QKyUXnBqXIUyA7HJSoxazuMtT4zX5VgTXrBi5HM1JTY9YSnk4ANmX+SZgV3MoHtuHCfqeHI2axFDpJQTae6VCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:42.057571Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.4429","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5204d25f301215abce73988a9b3991b69843f05a95f63403a71bc72e4bc89b7c","sha256:5f2b4e5910703ff1e35946c5ee5da0b607325e587383d58ee1bca53800fc3f4b"],"state_sha256":"622b3652a9c54d1624f3976bc4dedf6eede2e175f2e0da15967ef835002f3109"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0tvZHsuvbT4iTgpl95fCt2BX/8DKxnWken9F/O97NtzLT2CFfMqfj7q+jR9h2E9TonBgBcojANKb3iICGiJGBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T02:58:31.524831Z","bundle_sha256":"4a6354ac45f4ac96b67b7383a26878b915bd1e4a19347f7116eed335b5a87337"}}