{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7IPACMNUKSCYE63UUI3LTSAKFA","short_pith_number":"pith:7IPACMNU","canonical_record":{"source":{"id":"1611.05558","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-11-17T04:12:19Z","cross_cats_sorted":[],"title_canon_sha256":"18ad71239070c27227f01bd44991a8d8e70deffa5872da1be1c19df1de5af937","abstract_canon_sha256":"91a45b8ee7022e98641f58de65e507618ec42ec29584c57fa0fd9b9a0cc5d8dc"},"schema_version":"1.0"},"canonical_sha256":"fa1e0131b45485827b74a236b9c80a280adfd23e5c755deda1b51128998c5295","source":{"kind":"arxiv","id":"1611.05558","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.05558","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"arxiv_version","alias_value":"1611.05558v2","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05558","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"pith_short_12","alias_value":"7IPACMNUKSCY","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7IPACMNUKSCYE63U","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7IPACMNU","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7IPACMNUKSCYE63UUI3LTSAKFA","target":"record","payload":{"canonical_record":{"source":{"id":"1611.05558","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-11-17T04:12:19Z","cross_cats_sorted":[],"title_canon_sha256":"18ad71239070c27227f01bd44991a8d8e70deffa5872da1be1c19df1de5af937","abstract_canon_sha256":"91a45b8ee7022e98641f58de65e507618ec42ec29584c57fa0fd9b9a0cc5d8dc"},"schema_version":"1.0"},"canonical_sha256":"fa1e0131b45485827b74a236b9c80a280adfd23e5c755deda1b51128998c5295","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:45.202567Z","signature_b64":"xAbDZkSkYKiaAOZtFWku00fLqTdz5WddDhcLxhwf81tRlsfkOCUNDtuVNWTN/ab54+U8a3SQL4VY1j1/DmGABg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa1e0131b45485827b74a236b9c80a280adfd23e5c755deda1b51128998c5295","last_reissued_at":"2026-05-18T00:24:45.202059Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:45.202059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.05558","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":"2026-05-18T00:24:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v/OFTrGCFOKugd0MPJJOT3BoYeulOYRzSqOBKWx2d7IzCzEiueyV/ayPyO8po8hI8ALG/NQKfUZiNssA1oDRCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:22:31.822127Z"},"content_sha256":"644a294fd15c6b6b00ad4fc96b7e16ea83090fcfdeec27e5d2391d677c764cf6","schema_version":"1.0","event_id":"sha256:644a294fd15c6b6b00ad4fc96b7e16ea83090fcfdeec27e5d2391d677c764cf6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7IPACMNUKSCYE63UUI3LTSAKFA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Probabilistic Rank and Matrix Rigidity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Josh Alman, Ryan Williams","submitted_at":"2016-11-17T04:12:19Z","abstract_excerpt":"We consider a notion of probabilistic rank and probabilistic sign-rank of a matrix, which measures the extent to which a matrix can be probabilistically represented by low-rank matrices. We demonstrate several connections with matrix rigidity, communication complexity, and circuit lower bounds, including:\n  The Walsh-Hadamard Transform is Not Very Rigid. We give surprising upper bounds on the rigidity of a family of matrices whose rigidity has been extensively studied, and was conjectured to be highly rigid. For the $2^n \\times 2^n$ Walsh-Hadamard transform $H_n$ (a.k.a. Sylvester matrices, or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05558","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":"2026-05-18T00:24:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GWolf01Yk93fnz4WVZN2I8gpw2L1Sgu+NAKVQdg+LMBZAJMlx/0Vf6UQ/v1Ht//JFFLZkxdnYZ6pIyFHi3AeBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:22:31.822787Z"},"content_sha256":"3dcfd03f1a8d67a373c5f3d86a8a8ac4783d1b45c82b3709cf5dd2d105fe38fb","schema_version":"1.0","event_id":"sha256:3dcfd03f1a8d67a373c5f3d86a8a8ac4783d1b45c82b3709cf5dd2d105fe38fb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7IPACMNUKSCYE63UUI3LTSAKFA/bundle.json","state_url":"https://pith.science/pith/7IPACMNUKSCYE63UUI3LTSAKFA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7IPACMNUKSCYE63UUI3LTSAKFA/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":"2026-05-30T06:22:31Z","links":{"resolver":"https://pith.science/pith/7IPACMNUKSCYE63UUI3LTSAKFA","bundle":"https://pith.science/pith/7IPACMNUKSCYE63UUI3LTSAKFA/bundle.json","state":"https://pith.science/pith/7IPACMNUKSCYE63UUI3LTSAKFA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7IPACMNUKSCYE63UUI3LTSAKFA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7IPACMNUKSCYE63UUI3LTSAKFA","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":"91a45b8ee7022e98641f58de65e507618ec42ec29584c57fa0fd9b9a0cc5d8dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-11-17T04:12:19Z","title_canon_sha256":"18ad71239070c27227f01bd44991a8d8e70deffa5872da1be1c19df1de5af937"},"schema_version":"1.0","source":{"id":"1611.05558","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.05558","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"arxiv_version","alias_value":"1611.05558v2","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.05558","created_at":"2026-05-18T00:24:45Z"},{"alias_kind":"pith_short_12","alias_value":"7IPACMNUKSCY","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7IPACMNUKSCYE63U","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7IPACMNU","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:3dcfd03f1a8d67a373c5f3d86a8a8ac4783d1b45c82b3709cf5dd2d105fe38fb","target":"graph","created_at":"2026-05-18T00:24:45Z","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 a notion of probabilistic rank and probabilistic sign-rank of a matrix, which measures the extent to which a matrix can be probabilistically represented by low-rank matrices. We demonstrate several connections with matrix rigidity, communication complexity, and circuit lower bounds, including:\n  The Walsh-Hadamard Transform is Not Very Rigid. We give surprising upper bounds on the rigidity of a family of matrices whose rigidity has been extensively studied, and was conjectured to be highly rigid. For the $2^n \\times 2^n$ Walsh-Hadamard transform $H_n$ (a.k.a. Sylvester matrices, or","authors_text":"Josh Alman, Ryan Williams","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-11-17T04:12:19Z","title":"Probabilistic Rank and Matrix Rigidity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05558","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":"sha256:644a294fd15c6b6b00ad4fc96b7e16ea83090fcfdeec27e5d2391d677c764cf6","target":"record","created_at":"2026-05-18T00:24:45Z","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":"91a45b8ee7022e98641f58de65e507618ec42ec29584c57fa0fd9b9a0cc5d8dc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-11-17T04:12:19Z","title_canon_sha256":"18ad71239070c27227f01bd44991a8d8e70deffa5872da1be1c19df1de5af937"},"schema_version":"1.0","source":{"id":"1611.05558","kind":"arxiv","version":2}},"canonical_sha256":"fa1e0131b45485827b74a236b9c80a280adfd23e5c755deda1b51128998c5295","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa1e0131b45485827b74a236b9c80a280adfd23e5c755deda1b51128998c5295","first_computed_at":"2026-05-18T00:24:45.202059Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:45.202059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xAbDZkSkYKiaAOZtFWku00fLqTdz5WddDhcLxhwf81tRlsfkOCUNDtuVNWTN/ab54+U8a3SQL4VY1j1/DmGABg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:45.202567Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.05558","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:644a294fd15c6b6b00ad4fc96b7e16ea83090fcfdeec27e5d2391d677c764cf6","sha256:3dcfd03f1a8d67a373c5f3d86a8a8ac4783d1b45c82b3709cf5dd2d105fe38fb"],"state_sha256":"8adc6901570c9e4e0dbf4764d5aed2ccffdff3961cd82629f48b26512f33ceb8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RYFCE+J9R/hxIrJ0KWWwEwO4pbKKFKnlkYJY8U8Ze/wMdeJK5TQFpaMuFGECqgM+/GsnIntbuN/D4OC1nIrMBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T06:22:31.826064Z","bundle_sha256":"efd292e09364748fc6b25e11de1707a30da225502c5db20fbdc8b2ca7781d923"}}