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This gives an exact formula for the largest p for which a well-covered graph belongs to W_p.","weakest_assumption":"The three descriptions (graph-theoretic fibers, ridge degrees in the independence complex, and (r-1)-clique codegrees in the complement) are equivalent for α-critical members of W_p, relying on the prior definitions of α-criticality and the class W_p without additional verification steps shown in the abstract."}},"verdict_id":"4a50035d-8dc1-4181-b5d8-f1af6a5f5d6e"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:47d700ba3c1176b59e002ad321c4abb4e59bd0bfd0ab05d107d98e2a8a722c07","target":"record","created_at":"2026-05-20T00:03:25Z","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":"3aa8f44668e110176887b93fc9b18b08695b9510e7cf54ba13bee961f38ea9f9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-16T06:43:26Z","title_canon_sha256":"e0a0e21c87ed69b775f8ab0282c88e388b86b81a3f02275e7e881c5d44ab8353"},"schema_version":"1.0","source":{"id":"2605.16838","kind":"arxiv","version":1}},"canonical_sha256":"ccbd0005a7bb7efd44da428510a4e18eddd887983671373ef56357bae2f15aef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ccbd0005a7bb7efd44da428510a4e18eddd887983671373ef56357bae2f15aef","first_computed_at":"2026-05-20T00:03:25.443892Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:25.443892Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RUZS2rJg6oVtLTV7WQQIZsqMTXVBI64aMYdLGDrbTZMCh/G0N2gGwXMviePQakOEg5Mh55kNyWIltGYqgj7UAg==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:25.444837Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.16838","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47d700ba3c1176b59e002ad321c4abb4e59bd0bfd0ab05d107d98e2a8a722c07","sha256:dda93a6f1b2f8949c654217bc3c34fa8ae07c9ae4d61a0db6d77f1e49070cad4"],"state_sha256":"0500e94862493105364c258c7635cabb30208bd6b93ce4fc2a6b532a3310abcf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fUHIqzDkld9wMarFJBelMnFl5MlFIs+hdhTxhgXg4meh8qNwJWDi+k3sWGIZfskTJfs+joJLfGRBrLJvCoeKAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:38:31.412544Z","bundle_sha256":"3dfb5510b1bc92317e0695aad4cc281a63e927e6fda9fa7ccdbb4667f4beaa49"}}