{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:L7VCRMSKICK5D2PUJPG2KHAP25","short_pith_number":"pith:L7VCRMSK","canonical_record":{"source":{"id":"2604.06439","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-04-07T20:28:22Z","cross_cats_sorted":[],"title_canon_sha256":"de3a4a85f7c05e6796f29681e7986944ee950ad82e5334c7eb9084acb25c241d","abstract_canon_sha256":"4551d89704857983af055547baa8a0bd36826494c69a5cfaf8d1ab6d28c9db77"},"schema_version":"1.0"},"canonical_sha256":"5fea28b24a4095d1e9f44bcda51c0fd7595ed6c1668d57560b66d4a6f4882e76","source":{"kind":"arxiv","id":"2604.06439","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.06439","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"arxiv_version","alias_value":"2604.06439v2","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.06439","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"pith_short_12","alias_value":"L7VCRMSKICK5","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"pith_short_16","alias_value":"L7VCRMSKICK5D2PU","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"pith_short_8","alias_value":"L7VCRMSK","created_at":"2026-05-22T01:04:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:L7VCRMSKICK5D2PUJPG2KHAP25","target":"record","payload":{"canonical_record":{"source":{"id":"2604.06439","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-04-07T20:28:22Z","cross_cats_sorted":[],"title_canon_sha256":"de3a4a85f7c05e6796f29681e7986944ee950ad82e5334c7eb9084acb25c241d","abstract_canon_sha256":"4551d89704857983af055547baa8a0bd36826494c69a5cfaf8d1ab6d28c9db77"},"schema_version":"1.0"},"canonical_sha256":"5fea28b24a4095d1e9f44bcda51c0fd7595ed6c1668d57560b66d4a6f4882e76","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:01.628213Z","signature_b64":"Q/9tn29w2OwhvcuQGcT7ZrtqF0R8PBU6OKLIyOCnJBj0JYmBTXIst8/fN8xxf0mwrSdwPK2bKHAxg6uWg/tLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5fea28b24a4095d1e9f44bcda51c0fd7595ed6c1668d57560b66d4a6f4882e76","last_reissued_at":"2026-05-22T01:04:01.627509Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:01.627509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2604.06439","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":"2026-05-22T01:04:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OG2VsGNm/SnOZ+kyJbI+AN8QnQivFyjETkZhL5soQUoH/xBWsT5UtWd6ZOYsL74S7AlmIoAWawrqhFcsmVSDBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:00:04.675469Z"},"content_sha256":"75d42368ff5e069e2718838d4b9d64e25b26ad344b93144f87d5c8fe4dcbe635","schema_version":"1.0","event_id":"sha256:75d42368ff5e069e2718838d4b9d64e25b26ad344b93144f87d5c8fe4dcbe635"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:L7VCRMSKICK5D2PUJPG2KHAP25","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Greedy sparsifications of sums of positive semidefinite matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm.","cross_cats":[],"primary_cat":"math.FA","authors_text":"Grigory Ivanov","submitted_at":"2026-04-07T20:28:22Z","abstract_excerpt":"We prove a deterministic analogue of Rudelson's sampling theorem for sums of positive semidefinite matrices. Let $A_1,\\dots,A_m$ be positive semidefinite \\(d\\times d\\) matrices, and let $\\lambda_1,\\dots,\\lambda_m \\ge 0$ satisfy \\[ \\sum_{i=1}^m \\lambda_i = 1, \\qquad \\sum_{i=1}^m \\lambda_i A_i = I_d, \\qquad \\|A_i\\| \\le M \\quad\\text{for all } i=1,\\dots,m. \\] We show that there exists a deterministic sequence of indices $i_1,i_2,\\dots \\in \\{1,\\dots,m\\}$ such that for every integer $k \\ge 1$, \\[ \\left\\| \\frac{1}{k}\\sum_{r=1}^k A_{i_r} - I_d \\right\\| \\le \\begin{cases} \\displaystyle \\frac{2M\\ln(2d)}{"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that there exists a deterministic sequence of indices i1,i2,… such that for every integer k≥1, ||(1/k) sum_{r=1}^k A_{ir} - I_d|| ≤ 2M ln(2d)/k if k ≤ M ln(2d), and ≤ 3 sqrt(M ln(2d)/k) otherwise.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The result assumes there exist lambda_i ≥0 summing to 1 with sum lambda_i A_i = I_d and ||A_i||≤M for all i; if no such convex combination exists, the theorem gives no information. The proof must also construct or guarantee the sequence under only these hypotheses.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"There exists a deterministic sequence of the given PSD matrices such that the k-term average deviates from the identity by at most 2M ln(2d)/k when k is small and 3 sqrt(M ln(2d)/k) when k is large, enabling epsilon-approximations with N = O(M log d / eps^2) terms.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7e6fc82bf66d089e2b3d383478e9d1ae5838cdbb1a0d322674e8cb2b43c8b5f0"},"source":{"id":"2604.06439","kind":"arxiv","version":2},"verdict":{"id":"e11ebe43-e902-4c08-ba18-d245c6777c81","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T17:56:56.515356Z","strongest_claim":"We show that there exists a deterministic sequence of indices i1,i2,… such that for every integer k≥1, ||(1/k) sum_{r=1}^k A_{ir} - I_d|| ≤ 2M ln(2d)/k if k ≤ M ln(2d), and ≤ 3 sqrt(M ln(2d)/k) otherwise.","one_line_summary":"There exists a deterministic sequence of the given PSD matrices such that the k-term average deviates from the identity by at most 2M ln(2d)/k when k is small and 3 sqrt(M ln(2d)/k) when k is large, enabling epsilon-approximations with N = O(M log d / eps^2) terms.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The result assumes there exist lambda_i ≥0 summing to 1 with sum lambda_i A_i = I_d and ||A_i||≤M for all i; if no such convex combination exists, the theorem gives no information. The proof must also construct or guarantee the sequence under only these hypotheses.","pith_extraction_headline":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.06439/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":2,"snapshot_sha256":"17411bbec0eab82c2f3ca5497c0545128c05dec28a708869c55ff710a3b06e3a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":"e11ebe43-e902-4c08-ba18-d245c6777c81"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-22T01:04:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VlSwhjODxhxykEqK1o2QOPiR7JVIexNwFm2zMz50SZmws/GfyC/4VyFTFvmbWGbJOF0iBLqToyOGgZ81i4WbDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:00:04.676050Z"},"content_sha256":"37ade332dcc0e8f6911f0f59fa6da4e7fad6fe51bdb73aff0a89470615bad2fc","schema_version":"1.0","event_id":"sha256:37ade332dcc0e8f6911f0f59fa6da4e7fad6fe51bdb73aff0a89470615bad2fc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L7VCRMSKICK5D2PUJPG2KHAP25/bundle.json","state_url":"https://pith.science/pith/L7VCRMSKICK5D2PUJPG2KHAP25/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L7VCRMSKICK5D2PUJPG2KHAP25/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":"2026-06-02T08:00:04Z","links":{"resolver":"https://pith.science/pith/L7VCRMSKICK5D2PUJPG2KHAP25","bundle":"https://pith.science/pith/L7VCRMSKICK5D2PUJPG2KHAP25/bundle.json","state":"https://pith.science/pith/L7VCRMSKICK5D2PUJPG2KHAP25/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L7VCRMSKICK5D2PUJPG2KHAP25/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:L7VCRMSKICK5D2PUJPG2KHAP25","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":"4551d89704857983af055547baa8a0bd36826494c69a5cfaf8d1ab6d28c9db77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-04-07T20:28:22Z","title_canon_sha256":"de3a4a85f7c05e6796f29681e7986944ee950ad82e5334c7eb9084acb25c241d"},"schema_version":"1.0","source":{"id":"2604.06439","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2604.06439","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"arxiv_version","alias_value":"2604.06439v2","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.06439","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"pith_short_12","alias_value":"L7VCRMSKICK5","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"pith_short_16","alias_value":"L7VCRMSKICK5D2PU","created_at":"2026-05-22T01:04:01Z"},{"alias_kind":"pith_short_8","alias_value":"L7VCRMSK","created_at":"2026-05-22T01:04:01Z"}],"graph_snapshots":[{"event_id":"sha256:37ade332dcc0e8f6911f0f59fa6da4e7fad6fe51bdb73aff0a89470615bad2fc","target":"graph","created_at":"2026-05-22T01:04:01Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We show that there exists a deterministic sequence of indices i1,i2,… such that for every integer k≥1, ||(1/k) sum_{r=1}^k A_{ir} - I_d|| ≤ 2M ln(2d)/k if k ≤ M ln(2d), and ≤ 3 sqrt(M ln(2d)/k) otherwise."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The result assumes there exist lambda_i ≥0 summing to 1 with sum lambda_i A_i = I_d and ||A_i||≤M for all i; if no such convex combination exists, the theorem gives no information. The proof must also construct or guarantee the sequence under only these hypotheses."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"There exists a deterministic sequence of the given PSD matrices such that the k-term average deviates from the identity by at most 2M ln(2d)/k when k is small and 3 sqrt(M ln(2d)/k) when k is large, enabling epsilon-approximations with N = O(M log d / eps^2) terms."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm."}],"snapshot_sha256":"7e6fc82bf66d089e2b3d383478e9d1ae5838cdbb1a0d322674e8cb2b43c8b5f0"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"17411bbec0eab82c2f3ca5497c0545128c05dec28a708869c55ff710a3b06e3a"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.06439/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove a deterministic analogue of Rudelson's sampling theorem for sums of positive semidefinite matrices. Let $A_1,\\dots,A_m$ be positive semidefinite \\(d\\times d\\) matrices, and let $\\lambda_1,\\dots,\\lambda_m \\ge 0$ satisfy \\[ \\sum_{i=1}^m \\lambda_i = 1, \\qquad \\sum_{i=1}^m \\lambda_i A_i = I_d, \\qquad \\|A_i\\| \\le M \\quad\\text{for all } i=1,\\dots,m. \\] We show that there exists a deterministic sequence of indices $i_1,i_2,\\dots \\in \\{1,\\dots,m\\}$ such that for every integer $k \\ge 1$, \\[ \\left\\| \\frac{1}{k}\\sum_{r=1}^k A_{i_r} - I_d \\right\\| \\le \\begin{cases} \\displaystyle \\frac{2M\\ln(2d)}{","authors_text":"Grigory Ivanov","cross_cats":[],"headline":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-04-07T20:28:22Z","title":"Greedy sparsifications of sums of positive semidefinite matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.06439","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T17:56:56.515356Z","id":"e11ebe43-e902-4c08-ba18-d245c6777c81","model_set":{"reader":"grok-4.3"},"one_line_summary":"There exists a deterministic sequence of the given PSD matrices such that the k-term average deviates from the identity by at most 2M ln(2d)/k when k is small and 3 sqrt(M ln(2d)/k) when k is large, enabling epsilon-approximations with N = O(M log d / eps^2) terms.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"There exists a deterministic sequence of positive semidefinite matrices whose partial averages converge to the identity with explicit error bounds in operator norm.","strongest_claim":"We show that there exists a deterministic sequence of indices i1,i2,… such that for every integer k≥1, ||(1/k) sum_{r=1}^k A_{ir} - I_d|| ≤ 2M ln(2d)/k if k ≤ M ln(2d), and ≤ 3 sqrt(M ln(2d)/k) otherwise.","weakest_assumption":"The result assumes there exist lambda_i ≥0 summing to 1 with sum lambda_i A_i = I_d and ||A_i||≤M for all i; if no such convex combination exists, the theorem gives no information. The proof must also construct or guarantee the sequence under only these hypotheses."}},"verdict_id":"e11ebe43-e902-4c08-ba18-d245c6777c81"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:75d42368ff5e069e2718838d4b9d64e25b26ad344b93144f87d5c8fe4dcbe635","target":"record","created_at":"2026-05-22T01:04:01Z","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":"4551d89704857983af055547baa8a0bd36826494c69a5cfaf8d1ab6d28c9db77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-04-07T20:28:22Z","title_canon_sha256":"de3a4a85f7c05e6796f29681e7986944ee950ad82e5334c7eb9084acb25c241d"},"schema_version":"1.0","source":{"id":"2604.06439","kind":"arxiv","version":2}},"canonical_sha256":"5fea28b24a4095d1e9f44bcda51c0fd7595ed6c1668d57560b66d4a6f4882e76","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5fea28b24a4095d1e9f44bcda51c0fd7595ed6c1668d57560b66d4a6f4882e76","first_computed_at":"2026-05-22T01:04:01.627509Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:04:01.627509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q/9tn29w2OwhvcuQGcT7ZrtqF0R8PBU6OKLIyOCnJBj0JYmBTXIst8/fN8xxf0mwrSdwPK2bKHAxg6uWg/tLBw==","signature_status":"signed_v1","signed_at":"2026-05-22T01:04:01.628213Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.06439","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:75d42368ff5e069e2718838d4b9d64e25b26ad344b93144f87d5c8fe4dcbe635","sha256:37ade332dcc0e8f6911f0f59fa6da4e7fad6fe51bdb73aff0a89470615bad2fc"],"state_sha256":"2a2018d8c203b03c9495afb505af6ff278807a734e37bf76af650daab1ece68f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n0s8AveGJRzf+tnYlV6P6yrw0vS0sxmEhvAkRDSkF990kTKB9mCHowjSSEGzqPOCkhhAL5IBpcNu2x5L8WPODw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T08:00:04.678541Z","bundle_sha256":"df7571bc1fbd70c9330a74cb127edd3c066715eadd88d850b23425efdeb0b14e"}}