{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:O6B7BGDWWMJ7J45UT5EFX6F43O","short_pith_number":"pith:O6B7BGDW","canonical_record":{"source":{"id":"1506.01725","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-06-04T20:07:30Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"abcd6a4e6cf8fd56ffd78902f228f7dd0dbac30759716b28695e851ddf8bf14d","abstract_canon_sha256":"5183c21503545da2d5cfb644b6179c3e08780104d4b21b982a2bacf5965e7b78"},"schema_version":"1.0"},"canonical_sha256":"7783f09876b313f4f3b49f485bf8bcdb9c85d9cf633e424ab210b9c8e2e12651","source":{"kind":"arxiv","id":"1506.01725","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.01725","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"arxiv_version","alias_value":"1506.01725v1","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.01725","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"pith_short_12","alias_value":"O6B7BGDWWMJ7","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"O6B7BGDWWMJ7J45U","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"O6B7BGDW","created_at":"2026-05-18T12:29:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:O6B7BGDWWMJ7J45UT5EFX6F43O","target":"record","payload":{"canonical_record":{"source":{"id":"1506.01725","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-06-04T20:07:30Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"abcd6a4e6cf8fd56ffd78902f228f7dd0dbac30759716b28695e851ddf8bf14d","abstract_canon_sha256":"5183c21503545da2d5cfb644b6179c3e08780104d4b21b982a2bacf5965e7b78"},"schema_version":"1.0"},"canonical_sha256":"7783f09876b313f4f3b49f485bf8bcdb9c85d9cf633e424ab210b9c8e2e12651","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:37.160095Z","signature_b64":"fLrSO8wCuFTgc3HUoMSaszmDDfc2MiHhKC4jQRalaNzlNxZ9UcZ/Gmi0aGxVEK2Ab+bbiV9K2PdUNhTcvhLMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7783f09876b313f4f3b49f485bf8bcdb9c85d9cf633e424ab210b9c8e2e12651","last_reissued_at":"2026-05-17T23:54:37.159488Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:37.159488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.01725","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":"2026-05-17T23:54:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ESOb6xbuEd1z60PPloa3p84PW7+DxKjSPfVVtCTYWvkyfL+hQ38p6z3k0hFy0rZWMNE7CHgQun9vatmSdQ3MCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T09:09:58.424638Z"},"content_sha256":"0f2f613bc10104173bb1977810b70bf58779250dea824dc5e697f082ee0c1d8d","schema_version":"1.0","event_id":"sha256:0f2f613bc10104173bb1977810b70bf58779250dea824dc5e697f082ee0c1d8d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:O6B7BGDWWMJ7J45UT5EFX6F43O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some Bi-Matrix Models for Bi-Free Limit Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.OA","authors_text":"Paul Skoufranis","submitted_at":"2015-06-04T20:07:30Z","abstract_excerpt":"In this paper, an analogue of matrix models from free probability is developed in the bi-free setting. A bi-matrix model is not simply a pair of matrix models, but a pair of matrix models where one element in the pair acts by left-multiplication on matrices and the other element acts via a `twisted'-right action. The asymptotic distributions of bi-matrix models of Gaussian random variables tend to bi-free central limit distributions with certain covariance matrices. Furthermore, many classical random matrix results immediately generalize to the bi-free setting. For example, bi-matrix models of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01725","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":""},"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":"2026-05-17T23:54:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sUuvWgIbxBsjZ9eXNhMr5ez7X4ZGFgVZtipQkcgWMGlpbCnN0Dk7ZKOVa8ii+04LD7VCDdoC+VaYAa0L3gvRAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T09:09:58.424973Z"},"content_sha256":"f2b1182e0c4cbf79723eebf0b28da7bdc20d1b8af553403a2aa9fa5030f02b13","schema_version":"1.0","event_id":"sha256:f2b1182e0c4cbf79723eebf0b28da7bdc20d1b8af553403a2aa9fa5030f02b13"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/O6B7BGDWWMJ7J45UT5EFX6F43O/bundle.json","state_url":"https://pith.science/pith/O6B7BGDWWMJ7J45UT5EFX6F43O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/O6B7BGDWWMJ7J45UT5EFX6F43O/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":"2026-06-26T09:09:58Z","links":{"resolver":"https://pith.science/pith/O6B7BGDWWMJ7J45UT5EFX6F43O","bundle":"https://pith.science/pith/O6B7BGDWWMJ7J45UT5EFX6F43O/bundle.json","state":"https://pith.science/pith/O6B7BGDWWMJ7J45UT5EFX6F43O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/O6B7BGDWWMJ7J45UT5EFX6F43O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:O6B7BGDWWMJ7J45UT5EFX6F43O","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":"5183c21503545da2d5cfb644b6179c3e08780104d4b21b982a2bacf5965e7b78","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-06-04T20:07:30Z","title_canon_sha256":"abcd6a4e6cf8fd56ffd78902f228f7dd0dbac30759716b28695e851ddf8bf14d"},"schema_version":"1.0","source":{"id":"1506.01725","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.01725","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"arxiv_version","alias_value":"1506.01725v1","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.01725","created_at":"2026-05-17T23:54:37Z"},{"alias_kind":"pith_short_12","alias_value":"O6B7BGDWWMJ7","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"O6B7BGDWWMJ7J45U","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"O6B7BGDW","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:f2b1182e0c4cbf79723eebf0b28da7bdc20d1b8af553403a2aa9fa5030f02b13","target":"graph","created_at":"2026-05-17T23:54:37Z","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":"In this paper, an analogue of matrix models from free probability is developed in the bi-free setting. A bi-matrix model is not simply a pair of matrix models, but a pair of matrix models where one element in the pair acts by left-multiplication on matrices and the other element acts via a `twisted'-right action. The asymptotic distributions of bi-matrix models of Gaussian random variables tend to bi-free central limit distributions with certain covariance matrices. Furthermore, many classical random matrix results immediately generalize to the bi-free setting. For example, bi-matrix models of","authors_text":"Paul Skoufranis","cross_cats":["math.FA","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-06-04T20:07:30Z","title":"Some Bi-Matrix Models for Bi-Free Limit Distributions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01725","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":"sha256:0f2f613bc10104173bb1977810b70bf58779250dea824dc5e697f082ee0c1d8d","target":"record","created_at":"2026-05-17T23:54:37Z","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":"5183c21503545da2d5cfb644b6179c3e08780104d4b21b982a2bacf5965e7b78","cross_cats_sorted":["math.FA","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-06-04T20:07:30Z","title_canon_sha256":"abcd6a4e6cf8fd56ffd78902f228f7dd0dbac30759716b28695e851ddf8bf14d"},"schema_version":"1.0","source":{"id":"1506.01725","kind":"arxiv","version":1}},"canonical_sha256":"7783f09876b313f4f3b49f485bf8bcdb9c85d9cf633e424ab210b9c8e2e12651","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7783f09876b313f4f3b49f485bf8bcdb9c85d9cf633e424ab210b9c8e2e12651","first_computed_at":"2026-05-17T23:54:37.159488Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:37.159488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fLrSO8wCuFTgc3HUoMSaszmDDfc2MiHhKC4jQRalaNzlNxZ9UcZ/Gmi0aGxVEK2Ab+bbiV9K2PdUNhTcvhLMCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:37.160095Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.01725","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f2f613bc10104173bb1977810b70bf58779250dea824dc5e697f082ee0c1d8d","sha256:f2b1182e0c4cbf79723eebf0b28da7bdc20d1b8af553403a2aa9fa5030f02b13"],"state_sha256":"02381b138e741a54453e1b820816b2f87f9629b53f400bc74fa431fdb0eab37e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YWFaKHSgj7xguiXNr4LMnJgRhejNjg70TNFM/HYPq4lbAzTaPaRZadAO8jSB7ttgzYKjxlTQwGx9ucqq+wwJBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T09:09:58.426741Z","bundle_sha256":"59e89d6a55ff3a049a6fe0b0f057b97d4bbbea971acafddd6f5c32b43db81cf9"}}