{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:AUDJ7CCTZMHNKYABJ4TAMLB4HI","short_pith_number":"pith:AUDJ7CCT","canonical_record":{"source":{"id":"2411.10404","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-15T18:19:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3f120642c0cbeff644d283079fe6fa8e5f76ee766e3874086e2bb3d223083c16","abstract_canon_sha256":"30b3c90d3e22ffb32a1552311307a8005ddb5de0e24223de0268c03d5a7ce8d2"},"schema_version":"1.0"},"canonical_sha256":"05069f8853cb0ed560014f26062c3c3a10812f79f96b22af76b20af14f21ecf1","source":{"kind":"arxiv","id":"2411.10404","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.10404","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"arxiv_version","alias_value":"2411.10404v2","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.10404","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"pith_short_12","alias_value":"AUDJ7CCTZMHN","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"pith_short_16","alias_value":"AUDJ7CCTZMHNKYAB","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"pith_short_8","alias_value":"AUDJ7CCT","created_at":"2026-07-05T10:35:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:AUDJ7CCTZMHNKYABJ4TAMLB4HI","target":"record","payload":{"canonical_record":{"source":{"id":"2411.10404","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-15T18:19:49Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3f120642c0cbeff644d283079fe6fa8e5f76ee766e3874086e2bb3d223083c16","abstract_canon_sha256":"30b3c90d3e22ffb32a1552311307a8005ddb5de0e24223de0268c03d5a7ce8d2"},"schema_version":"1.0"},"canonical_sha256":"05069f8853cb0ed560014f26062c3c3a10812f79f96b22af76b20af14f21ecf1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:35:41.163106Z","signature_b64":"2txQkIFsNg/2EveTO8FiayRLjwWtJsyNQG2fESnxKhfdic4NsZAjsuXs3/IUrMxiusOB3HJLzUOaJJfSuGh+BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05069f8853cb0ed560014f26062c3c3a10812f79f96b22af76b20af14f21ecf1","last_reissued_at":"2026-07-05T10:35:41.162602Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:35:41.162602Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.10404","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":"2026-07-05T10:35:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wdM8AwHyRh7spkBANNjZgPomv0FjUksyoQYeabk/7TXOy8P+Q/kZmuz8885XpL4PpsOj1l0CtwgcJ+4Z8GBqCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:57:50.520451Z"},"content_sha256":"2145da8b00661808d183d5315d040a661e8338e8beb9c5e211878660f47768b1","schema_version":"1.0","event_id":"sha256:2145da8b00661808d183d5315d040a661e8338e8beb9c5e211878660f47768b1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:AUDJ7CCTZMHNKYABJ4TAMLB4HI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On commuting pairs in arbitrary sets of 2x2 matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Akshat Mudgal","submitted_at":"2024-11-15T18:19:49Z","abstract_excerpt":"Let $\\textrm{Mat}_2(\\mathbb{R})$ be the set of $2 \\times 2$ matrices with real entries. For any $\\varepsilon>0$ and any finitely--supported probability measure $\\mu$ on $\\textrm{Mat}_2(\\mathbb{R})$, we prove that either \\[ T(\\mu) = \\sum_{X, Y \\in {\\rm supp}(\\mu), XY = YX} \\mu(X) \\mu(Y) < \\varepsilon \\] or there exists some finite set ${S}$ contained in a $2$-dimensional subspace of $\\textrm{Mat}_2(\\mathbb{R})$ such that $\\mu({S}) \\geq \\varepsilon/8$. This is sharp up to the multiplicative constant. We prove quantitatively stronger results when \\[ \\mu ( (a_{i,j})_{1 \\leq i,j \\leq 2} ) = \\nu(a_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.10404","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/2411.10404/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":"2026-07-05T10:35:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"06cD3SrCPVqWfin9oETM++IMGpgkUl8GJUiBH/RRdISvfeTM4FLlSZd90HnnbiHvWOGTds9m73C2exzZVqItBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T16:57:50.521079Z"},"content_sha256":"11e73a2822f63edf91311a9f33a30a746071665a6395112b5b71743761ea8e50","schema_version":"1.0","event_id":"sha256:11e73a2822f63edf91311a9f33a30a746071665a6395112b5b71743761ea8e50"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AUDJ7CCTZMHNKYABJ4TAMLB4HI/bundle.json","state_url":"https://pith.science/pith/AUDJ7CCTZMHNKYABJ4TAMLB4HI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AUDJ7CCTZMHNKYABJ4TAMLB4HI/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":"2026-08-09T16:57:50Z","links":{"resolver":"https://pith.science/pith/AUDJ7CCTZMHNKYABJ4TAMLB4HI","bundle":"https://pith.science/pith/AUDJ7CCTZMHNKYABJ4TAMLB4HI/bundle.json","state":"https://pith.science/pith/AUDJ7CCTZMHNKYABJ4TAMLB4HI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AUDJ7CCTZMHNKYABJ4TAMLB4HI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:AUDJ7CCTZMHNKYABJ4TAMLB4HI","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":"30b3c90d3e22ffb32a1552311307a8005ddb5de0e24223de0268c03d5a7ce8d2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-15T18:19:49Z","title_canon_sha256":"3f120642c0cbeff644d283079fe6fa8e5f76ee766e3874086e2bb3d223083c16"},"schema_version":"1.0","source":{"id":"2411.10404","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.10404","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"arxiv_version","alias_value":"2411.10404v2","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.10404","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"pith_short_12","alias_value":"AUDJ7CCTZMHN","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"pith_short_16","alias_value":"AUDJ7CCTZMHNKYAB","created_at":"2026-07-05T10:35:41Z"},{"alias_kind":"pith_short_8","alias_value":"AUDJ7CCT","created_at":"2026-07-05T10:35:41Z"}],"graph_snapshots":[{"event_id":"sha256:11e73a2822f63edf91311a9f33a30a746071665a6395112b5b71743761ea8e50","target":"graph","created_at":"2026-07-05T10:35:41Z","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/2411.10404/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $\\textrm{Mat}_2(\\mathbb{R})$ be the set of $2 \\times 2$ matrices with real entries. For any $\\varepsilon>0$ and any finitely--supported probability measure $\\mu$ on $\\textrm{Mat}_2(\\mathbb{R})$, we prove that either \\[ T(\\mu) = \\sum_{X, Y \\in {\\rm supp}(\\mu), XY = YX} \\mu(X) \\mu(Y) < \\varepsilon \\] or there exists some finite set ${S}$ contained in a $2$-dimensional subspace of $\\textrm{Mat}_2(\\mathbb{R})$ such that $\\mu({S}) \\geq \\varepsilon/8$. This is sharp up to the multiplicative constant. We prove quantitatively stronger results when \\[ \\mu ( (a_{i,j})_{1 \\leq i,j \\leq 2} ) = \\nu(a_{","authors_text":"Akshat Mudgal","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-15T18:19:49Z","title":"On commuting pairs in arbitrary sets of 2x2 matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.10404","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":"sha256:2145da8b00661808d183d5315d040a661e8338e8beb9c5e211878660f47768b1","target":"record","created_at":"2026-07-05T10:35:41Z","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":"30b3c90d3e22ffb32a1552311307a8005ddb5de0e24223de0268c03d5a7ce8d2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-11-15T18:19:49Z","title_canon_sha256":"3f120642c0cbeff644d283079fe6fa8e5f76ee766e3874086e2bb3d223083c16"},"schema_version":"1.0","source":{"id":"2411.10404","kind":"arxiv","version":2}},"canonical_sha256":"05069f8853cb0ed560014f26062c3c3a10812f79f96b22af76b20af14f21ecf1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"05069f8853cb0ed560014f26062c3c3a10812f79f96b22af76b20af14f21ecf1","first_computed_at":"2026-07-05T10:35:41.162602Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:35:41.162602Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2txQkIFsNg/2EveTO8FiayRLjwWtJsyNQG2fESnxKhfdic4NsZAjsuXs3/IUrMxiusOB3HJLzUOaJJfSuGh+BA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:35:41.163106Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.10404","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2145da8b00661808d183d5315d040a661e8338e8beb9c5e211878660f47768b1","sha256:11e73a2822f63edf91311a9f33a30a746071665a6395112b5b71743761ea8e50"],"state_sha256":"811c54f392fb6e448642e2924616ae7a23252cd4e805e1dd840a17c56992d60c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aDD5Tc12Zr7dF2ju6hGO70RoZsOM+4KHdTc00Vh73DrusSjXEkOzQx7Ia3BOSQoQRx9XirCfmkQM48wA4xwhDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T16:57:50.526119Z","bundle_sha256":"e9181429773166f2dc91c26f4619694157b6fe5d4f3d3a08d7f46fbf3ea32a81"}}