{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:G35FG7VCXMTC7UQBQTKYSOZAQF","short_pith_number":"pith:G35FG7VC","canonical_record":{"source":{"id":"2412.10879","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2024-12-14T16:10:26Z","cross_cats_sorted":["math.DG","math.GT"],"title_canon_sha256":"2cb0f82ed8db53f882af261467a337eb580e1cca1b942c6bdb7ea5a7d0d7d0f3","abstract_canon_sha256":"4f3b659b9c997ce039ceed7d61c4b99e2885de6d4d2e8ab8a72d9511ace067f2"},"schema_version":"1.0"},"canonical_sha256":"36fa537ea2bb262fd20184d5893b20814685d010d77e9a60bd3cc74b71a9ac8c","source":{"kind":"arxiv","id":"2412.10879","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.10879","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"arxiv_version","alias_value":"2412.10879v2","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.10879","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"pith_short_12","alias_value":"G35FG7VCXMTC","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"pith_short_16","alias_value":"G35FG7VCXMTC7UQB","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"pith_short_8","alias_value":"G35FG7VC","created_at":"2026-07-05T10:18:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:G35FG7VCXMTC7UQBQTKYSOZAQF","target":"record","payload":{"canonical_record":{"source":{"id":"2412.10879","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2024-12-14T16:10:26Z","cross_cats_sorted":["math.DG","math.GT"],"title_canon_sha256":"2cb0f82ed8db53f882af261467a337eb580e1cca1b942c6bdb7ea5a7d0d7d0f3","abstract_canon_sha256":"4f3b659b9c997ce039ceed7d61c4b99e2885de6d4d2e8ab8a72d9511ace067f2"},"schema_version":"1.0"},"canonical_sha256":"36fa537ea2bb262fd20184d5893b20814685d010d77e9a60bd3cc74b71a9ac8c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:18:11.664430Z","signature_b64":"Fn0qeJ5a6bhakB/Hf5PFmaED/Je+jxSrAuNGl1G1q/uwdSjBss609E9aOzq3dRu7MdhdIrhk/0HlDWQ14aa6CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36fa537ea2bb262fd20184d5893b20814685d010d77e9a60bd3cc74b71a9ac8c","last_reissued_at":"2026-07-05T10:18:11.663974Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:18:11.663974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.10879","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":"2026-07-05T10:18:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SFG2NSl7M75VpqG1FqFkjO39YjfUdZjrXUDItFTVbu1jIWho8V+vkn/av5/i7a7tRj3bDxvJG4s1fXQwOVkWBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T17:13:41.787509Z"},"content_sha256":"0606e172056a983af64c43440a0db490e24447498ea5d383d7373b6c09348bf3","schema_version":"1.0","event_id":"sha256:0606e172056a983af64c43440a0db490e24447498ea5d383d7373b6c09348bf3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:G35FG7VCXMTC7UQBQTKYSOZAQF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the Last Kervaire Invariant Problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG","math.GT"],"primary_cat":"math.AT","authors_text":"Guozhen Wang, Weinan Lin, Zhouli Xu","submitted_at":"2024-12-14T16:10:26Z","abstract_excerpt":"We prove that the element $h_6^2$ is a permanent cycle in the Adams spectral sequence. As a result, we establish the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, thereby resolving the final case of the Kervaire invariant problem.\n  Combining this result with the theorems of Browder, Mahowald--Tangora, Barratt--Jones--Mahowald, and Hill--Hopkins--Ravenel, we conclude that smooth framed manifolds with Kervaire invariant one exist in and only in dimensions $2, 6, 14, 30, 62$, and $126$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.10879","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/2412.10879/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":"2026-07-05T10:18:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NniE6I74ut4kNHEiWl4pe5v1Y9E0BzCPVPdt0UV6RW5ElRvrOo/8la3hhWzPzyPULWm/3daLCnq0EyOfVg6gDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-05T17:13:41.788179Z"},"content_sha256":"1a46df54875fad9c3f05d2a686368fe4356db13c0d6feed6e91895d1f1d8b92e","schema_version":"1.0","event_id":"sha256:1a46df54875fad9c3f05d2a686368fe4356db13c0d6feed6e91895d1f1d8b92e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G35FG7VCXMTC7UQBQTKYSOZAQF/bundle.json","state_url":"https://pith.science/pith/G35FG7VCXMTC7UQBQTKYSOZAQF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G35FG7VCXMTC7UQBQTKYSOZAQF/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":"2026-08-05T17:13:41Z","links":{"resolver":"https://pith.science/pith/G35FG7VCXMTC7UQBQTKYSOZAQF","bundle":"https://pith.science/pith/G35FG7VCXMTC7UQBQTKYSOZAQF/bundle.json","state":"https://pith.science/pith/G35FG7VCXMTC7UQBQTKYSOZAQF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G35FG7VCXMTC7UQBQTKYSOZAQF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:G35FG7VCXMTC7UQBQTKYSOZAQF","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":"4f3b659b9c997ce039ceed7d61c4b99e2885de6d4d2e8ab8a72d9511ace067f2","cross_cats_sorted":["math.DG","math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2024-12-14T16:10:26Z","title_canon_sha256":"2cb0f82ed8db53f882af261467a337eb580e1cca1b942c6bdb7ea5a7d0d7d0f3"},"schema_version":"1.0","source":{"id":"2412.10879","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.10879","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"arxiv_version","alias_value":"2412.10879v2","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.10879","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"pith_short_12","alias_value":"G35FG7VCXMTC","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"pith_short_16","alias_value":"G35FG7VCXMTC7UQB","created_at":"2026-07-05T10:18:11Z"},{"alias_kind":"pith_short_8","alias_value":"G35FG7VC","created_at":"2026-07-05T10:18:11Z"}],"graph_snapshots":[{"event_id":"sha256:1a46df54875fad9c3f05d2a686368fe4356db13c0d6feed6e91895d1f1d8b92e","target":"graph","created_at":"2026-07-05T10:18:11Z","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/2412.10879/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that the element $h_6^2$ is a permanent cycle in the Adams spectral sequence. As a result, we establish the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, thereby resolving the final case of the Kervaire invariant problem.\n  Combining this result with the theorems of Browder, Mahowald--Tangora, Barratt--Jones--Mahowald, and Hill--Hopkins--Ravenel, we conclude that smooth framed manifolds with Kervaire invariant one exist in and only in dimensions $2, 6, 14, 30, 62$, and $126$.","authors_text":"Guozhen Wang, Weinan Lin, Zhouli Xu","cross_cats":["math.DG","math.GT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2024-12-14T16:10:26Z","title":"On the Last Kervaire Invariant Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.10879","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":"sha256:0606e172056a983af64c43440a0db490e24447498ea5d383d7373b6c09348bf3","target":"record","created_at":"2026-07-05T10:18:11Z","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":"4f3b659b9c997ce039ceed7d61c4b99e2885de6d4d2e8ab8a72d9511ace067f2","cross_cats_sorted":["math.DG","math.GT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2024-12-14T16:10:26Z","title_canon_sha256":"2cb0f82ed8db53f882af261467a337eb580e1cca1b942c6bdb7ea5a7d0d7d0f3"},"schema_version":"1.0","source":{"id":"2412.10879","kind":"arxiv","version":2}},"canonical_sha256":"36fa537ea2bb262fd20184d5893b20814685d010d77e9a60bd3cc74b71a9ac8c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"36fa537ea2bb262fd20184d5893b20814685d010d77e9a60bd3cc74b71a9ac8c","first_computed_at":"2026-07-05T10:18:11.663974Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:18:11.663974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fn0qeJ5a6bhakB/Hf5PFmaED/Je+jxSrAuNGl1G1q/uwdSjBss609E9aOzq3dRu7MdhdIrhk/0HlDWQ14aa6CA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:18:11.664430Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.10879","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0606e172056a983af64c43440a0db490e24447498ea5d383d7373b6c09348bf3","sha256:1a46df54875fad9c3f05d2a686368fe4356db13c0d6feed6e91895d1f1d8b92e"],"state_sha256":"ce6b4ed3c7f11d1873c6348980a366e9489ead3e8e294ca7ee0b8e74e42abcdc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"58sNijVgB+l7j/cmVO0EKMWbcEPGjENPxeR++8OFluAVHnB8E7vpgDcQ2CsurO/Z95zzICSjWjog8wS9513TCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-05T17:13:41.791584Z","bundle_sha256":"5f9811d09f3b69edf330a42954cbb9e7072a9329a3da723a69c7bca4527fe13b"}}