{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:D4HG33Z3IUK5VROTXDLWQWZQPK","short_pith_number":"pith:D4HG33Z3","canonical_record":{"source":{"id":"2201.06154","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-01-16T23:30:35Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2a809533056c7116f41017df3dbc083cfe55ad69abd65c76d17517a0a741461c","abstract_canon_sha256":"d71289cf27387aaa1e76e014b6dc7f5049d01a4249ac74a9dd77de01aba8ef0f"},"schema_version":"1.0"},"canonical_sha256":"1f0e6def3b4515dac5d3b8d7685b307abb681d68d6e92e6051ddb8e68b4a23b2","source":{"kind":"arxiv","id":"2201.06154","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.06154","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"arxiv_version","alias_value":"2201.06154v1","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.06154","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"pith_short_12","alias_value":"D4HG33Z3IUK5","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"pith_short_16","alias_value":"D4HG33Z3IUK5VROT","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"pith_short_8","alias_value":"D4HG33Z3","created_at":"2026-07-05T03:48:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:D4HG33Z3IUK5VROTXDLWQWZQPK","target":"record","payload":{"canonical_record":{"source":{"id":"2201.06154","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-01-16T23:30:35Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"2a809533056c7116f41017df3dbc083cfe55ad69abd65c76d17517a0a741461c","abstract_canon_sha256":"d71289cf27387aaa1e76e014b6dc7f5049d01a4249ac74a9dd77de01aba8ef0f"},"schema_version":"1.0"},"canonical_sha256":"1f0e6def3b4515dac5d3b8d7685b307abb681d68d6e92e6051ddb8e68b4a23b2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:48:55.324364Z","signature_b64":"uw511AMdgDCmlV1PF0iUVoEgAsMxmwvZCxPm/oL0tmq+xlk/Wo0LhUeDHxMck59b2Tsm22EeJR31FefKXik8DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1f0e6def3b4515dac5d3b8d7685b307abb681d68d6e92e6051ddb8e68b4a23b2","last_reissued_at":"2026-07-05T03:48:55.323904Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:48:55.323904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2201.06154","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":"2026-07-05T03:48:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fJgBuyZdu5a1ztB0+Q/UEQ3IrkIhOqHIDxaPLeG+Pi1MvACXzMy8cZH956ZLHkGeBVusTyPXp7+pwTpjX1GCAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T03:53:38.555735Z"},"content_sha256":"66898d2a51344ec25288969add8049129d1cb87ef2684f5ba2e6c7874b67ce15","schema_version":"1.0","event_id":"sha256:66898d2a51344ec25288969add8049129d1cb87ef2684f5ba2e6c7874b67ce15"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:D4HG33Z3IUK5VROTXDLWQWZQPK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Min-max minimal hypersurfaces with higher multiplicity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Xin Zhou, Zhichao Wang","submitted_at":"2022-01-16T23:30:35Z","abstract_excerpt":"We exhibit the first set of examples of non-bumpy metrics on the $(n+1)$-sphere ($2\\leq n\\leq 6$) in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$-sphere. This is proved by a new area-and-separation estimate for certain minimal hypersurfaces with Morse index two inspired by an early work of Colding-Minicozzi. We also construct non-bumpy projective spaces in which the first min-max hypersurfaces are one-sided, and non-bumpy balls in which the free boundary min-max hypersurfaces are improper."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.06154","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/2201.06154/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":"2026-07-05T03:48:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nvJ8G8eqC9eapo+5se8YwbDgOkFt+1liCbOY7hsEH1afbGCWnTNvut/HBXprykWoEehKWTUqE6+H4NvgnPBADA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T03:53:38.556224Z"},"content_sha256":"52af30e4fb1a0dff833d8e03ccdd4af090fff4b62153e8fcd633888c48c54c10","schema_version":"1.0","event_id":"sha256:52af30e4fb1a0dff833d8e03ccdd4af090fff4b62153e8fcd633888c48c54c10"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/D4HG33Z3IUK5VROTXDLWQWZQPK/bundle.json","state_url":"https://pith.science/pith/D4HG33Z3IUK5VROTXDLWQWZQPK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/D4HG33Z3IUK5VROTXDLWQWZQPK/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":"2026-08-06T03:53:38Z","links":{"resolver":"https://pith.science/pith/D4HG33Z3IUK5VROTXDLWQWZQPK","bundle":"https://pith.science/pith/D4HG33Z3IUK5VROTXDLWQWZQPK/bundle.json","state":"https://pith.science/pith/D4HG33Z3IUK5VROTXDLWQWZQPK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/D4HG33Z3IUK5VROTXDLWQWZQPK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:D4HG33Z3IUK5VROTXDLWQWZQPK","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":"d71289cf27387aaa1e76e014b6dc7f5049d01a4249ac74a9dd77de01aba8ef0f","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-01-16T23:30:35Z","title_canon_sha256":"2a809533056c7116f41017df3dbc083cfe55ad69abd65c76d17517a0a741461c"},"schema_version":"1.0","source":{"id":"2201.06154","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2201.06154","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"arxiv_version","alias_value":"2201.06154v1","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2201.06154","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"pith_short_12","alias_value":"D4HG33Z3IUK5","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"pith_short_16","alias_value":"D4HG33Z3IUK5VROT","created_at":"2026-07-05T03:48:55Z"},{"alias_kind":"pith_short_8","alias_value":"D4HG33Z3","created_at":"2026-07-05T03:48:55Z"}],"graph_snapshots":[{"event_id":"sha256:52af30e4fb1a0dff833d8e03ccdd4af090fff4b62153e8fcd633888c48c54c10","target":"graph","created_at":"2026-07-05T03:48:55Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2201.06154/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We exhibit the first set of examples of non-bumpy metrics on the $(n+1)$-sphere ($2\\leq n\\leq 6$) in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$-sphere. This is proved by a new area-and-separation estimate for certain minimal hypersurfaces with Morse index two inspired by an early work of Colding-Minicozzi. We also construct non-bumpy projective spaces in which the first min-max hypersurfaces are one-sided, and non-bumpy balls in which the free boundary min-max hypersurfaces are improper.","authors_text":"Xin Zhou, Zhichao Wang","cross_cats":["math.AP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-01-16T23:30:35Z","title":"Min-max minimal hypersurfaces with higher multiplicity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.06154","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":"sha256:66898d2a51344ec25288969add8049129d1cb87ef2684f5ba2e6c7874b67ce15","target":"record","created_at":"2026-07-05T03:48:55Z","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":"d71289cf27387aaa1e76e014b6dc7f5049d01a4249ac74a9dd77de01aba8ef0f","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2022-01-16T23:30:35Z","title_canon_sha256":"2a809533056c7116f41017df3dbc083cfe55ad69abd65c76d17517a0a741461c"},"schema_version":"1.0","source":{"id":"2201.06154","kind":"arxiv","version":1}},"canonical_sha256":"1f0e6def3b4515dac5d3b8d7685b307abb681d68d6e92e6051ddb8e68b4a23b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1f0e6def3b4515dac5d3b8d7685b307abb681d68d6e92e6051ddb8e68b4a23b2","first_computed_at":"2026-07-05T03:48:55.323904Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:48:55.323904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uw511AMdgDCmlV1PF0iUVoEgAsMxmwvZCxPm/oL0tmq+xlk/Wo0LhUeDHxMck59b2Tsm22EeJR31FefKXik8DA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:48:55.324364Z","signed_message":"canonical_sha256_bytes"},"source_id":"2201.06154","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:66898d2a51344ec25288969add8049129d1cb87ef2684f5ba2e6c7874b67ce15","sha256:52af30e4fb1a0dff833d8e03ccdd4af090fff4b62153e8fcd633888c48c54c10"],"state_sha256":"77515a568886fedddede522289a77215a1ea3cdf7ad20593ffc9ec515c5d6310"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E4hfawc1skXD3TONQJqVpmKpKqL+XYOFoHo5eOwb8vQb13D1G4bBmPkyyBYvzwASLD3jCC8GECe8dxvTOyf9CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T03:53:38.559787Z","bundle_sha256":"f8d458301289909c9a627a138798360a16290d726eb04c51ad3ed6ae9aea1cab"}}