{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:M5V5ASP7SFWU3OTYXHLD4DG42O","short_pith_number":"pith:M5V5ASP7","canonical_record":{"source":{"id":"2605.24692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-23T18:10:12Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"e12f7ee780222643b5534259a9be10f767d09388dbddc9cb3093043f6380056e","abstract_canon_sha256":"c11f982ca3d519e54dcc6a8b4bac7d5497578d2fc5e0a62a0550d72e29ccc2e5"},"schema_version":"1.0"},"canonical_sha256":"676bd049ff916d4dba78b9d63e0cdcd3900fb21bb3dc36d469619ece658d65bf","source":{"kind":"arxiv","id":"2605.24692","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.24692","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"arxiv_version","alias_value":"2605.24692v1","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24692","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"pith_short_12","alias_value":"M5V5ASP7SFWU","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"pith_short_16","alias_value":"M5V5ASP7SFWU3OTY","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"pith_short_8","alias_value":"M5V5ASP7","created_at":"2026-05-26T01:03:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:M5V5ASP7SFWU3OTYXHLD4DG42O","target":"record","payload":{"canonical_record":{"source":{"id":"2605.24692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-23T18:10:12Z","cross_cats_sorted":["cs.NA"],"title_canon_sha256":"e12f7ee780222643b5534259a9be10f767d09388dbddc9cb3093043f6380056e","abstract_canon_sha256":"c11f982ca3d519e54dcc6a8b4bac7d5497578d2fc5e0a62a0550d72e29ccc2e5"},"schema_version":"1.0"},"canonical_sha256":"676bd049ff916d4dba78b9d63e0cdcd3900fb21bb3dc36d469619ece658d65bf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T01:03:53.570368Z","signature_b64":"R7qcwvjYtSe8ak9ds/whmXSGqvg1aPQTY5h3JvN2zJ9ZTRQhpRnihTkcvjN0Zo+6uwq3KNgppE676soy7cykDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"676bd049ff916d4dba78b9d63e0cdcd3900fb21bb3dc36d469619ece658d65bf","last_reissued_at":"2026-05-26T01:03:53.569757Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T01:03:53.569757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.24692","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":"2026-05-26T01:03:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oNqI1m9eg6nC4nKskKACKT7mKKNROUxGLLkNthtYtQlzs4bQZGd/uMQbYzXovtB3gHDWXFFfkHWaRXSokfM6CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T13:33:24.685031Z"},"content_sha256":"29cdf912c0394010ae5e983496486031db2e7c73d1e90c976b74cfba747160c3","schema_version":"1.0","event_id":"sha256:29cdf912c0394010ae5e983496486031db2e7c73d1e90c976b74cfba747160c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:M5V5ASP7SFWU3OTYXHLD4DG42O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp Convergence Rates and Optimal Weights for Cimmino's Reflection Algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Hemant Sharma","submitted_at":"2026-05-23T18:10:12Z","abstract_excerpt":"In this paper, Cimmino's classical reflection algorithm for solving the $n\\times n$ nonsingular linear system $A\\bx=\\bb$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as $\\e^{(\\nu+1)}=M_w\\,\\e^{(\\nu)}$, where $M_w = I - A^\\top D_w A$, the error is shown to contract by the spectral radius $\\sprad(M_w)$ at every step, with a sharp, asymptotically tight bound. For $n=2$, a closed-form expression for the contraction factor is derived, \\[\n  \\sprad(M_w) \\;=\\; |1-\\mu|\n  + \\tfrac{1}{2}\\sqrt{(w_1-w_2)^2 + 4w_1w_2\\cos^2\\!\\theta}, \\] where $\\mu=(w_1+w_2)/2$ and $\\th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24692","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.24692/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":"2026-05-26T01:03:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0OIjWpf0Cv8AI+H0xFycHyLHO2Xe20Uz3GmANh4OqxL0tHg+Roh7rmY8a6ghNlQWpA9HBCBskEJ3h01FbhmYDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T13:33:24.685430Z"},"content_sha256":"afe5094df3c72426ba64906d6b9d7f51e29db19a85923a3abd8e060a4f28e083","schema_version":"1.0","event_id":"sha256:afe5094df3c72426ba64906d6b9d7f51e29db19a85923a3abd8e060a4f28e083"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/bundle.json","state_url":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/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":"2026-07-04T13:33:24Z","links":{"resolver":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O","bundle":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/bundle.json","state":"https://pith.science/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/M5V5ASP7SFWU3OTYXHLD4DG42O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:M5V5ASP7SFWU3OTYXHLD4DG42O","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":"c11f982ca3d519e54dcc6a8b4bac7d5497578d2fc5e0a62a0550d72e29ccc2e5","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-23T18:10:12Z","title_canon_sha256":"e12f7ee780222643b5534259a9be10f767d09388dbddc9cb3093043f6380056e"},"schema_version":"1.0","source":{"id":"2605.24692","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.24692","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"arxiv_version","alias_value":"2605.24692v1","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.24692","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"pith_short_12","alias_value":"M5V5ASP7SFWU","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"pith_short_16","alias_value":"M5V5ASP7SFWU3OTY","created_at":"2026-05-26T01:03:53Z"},{"alias_kind":"pith_short_8","alias_value":"M5V5ASP7","created_at":"2026-05-26T01:03:53Z"}],"graph_snapshots":[{"event_id":"sha256:afe5094df3c72426ba64906d6b9d7f51e29db19a85923a3abd8e060a4f28e083","target":"graph","created_at":"2026-05-26T01:03:53Z","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/2605.24692/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, Cimmino's classical reflection algorithm for solving the $n\\times n$ nonsingular linear system $A\\bx=\\bb$ is analysed through the lens of spectral theory. Reformulating the weighted iteration as $\\e^{(\\nu+1)}=M_w\\,\\e^{(\\nu)}$, where $M_w = I - A^\\top D_w A$, the error is shown to contract by the spectral radius $\\sprad(M_w)$ at every step, with a sharp, asymptotically tight bound. For $n=2$, a closed-form expression for the contraction factor is derived, \\[\n  \\sprad(M_w) \\;=\\; |1-\\mu|\n  + \\tfrac{1}{2}\\sqrt{(w_1-w_2)^2 + 4w_1w_2\\cos^2\\!\\theta}, \\] where $\\mu=(w_1+w_2)/2$ and $\\th","authors_text":"Hemant Sharma","cross_cats":["cs.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-23T18:10:12Z","title":"Sharp Convergence Rates and Optimal Weights for Cimmino's Reflection Algorithm"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24692","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":"sha256:29cdf912c0394010ae5e983496486031db2e7c73d1e90c976b74cfba747160c3","target":"record","created_at":"2026-05-26T01:03:53Z","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":"c11f982ca3d519e54dcc6a8b4bac7d5497578d2fc5e0a62a0550d72e29ccc2e5","cross_cats_sorted":["cs.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2026-05-23T18:10:12Z","title_canon_sha256":"e12f7ee780222643b5534259a9be10f767d09388dbddc9cb3093043f6380056e"},"schema_version":"1.0","source":{"id":"2605.24692","kind":"arxiv","version":1}},"canonical_sha256":"676bd049ff916d4dba78b9d63e0cdcd3900fb21bb3dc36d469619ece658d65bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"676bd049ff916d4dba78b9d63e0cdcd3900fb21bb3dc36d469619ece658d65bf","first_computed_at":"2026-05-26T01:03:53.569757Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T01:03:53.569757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R7qcwvjYtSe8ak9ds/whmXSGqvg1aPQTY5h3JvN2zJ9ZTRQhpRnihTkcvjN0Zo+6uwq3KNgppE676soy7cykDw==","signature_status":"signed_v1","signed_at":"2026-05-26T01:03:53.570368Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.24692","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:29cdf912c0394010ae5e983496486031db2e7c73d1e90c976b74cfba747160c3","sha256:afe5094df3c72426ba64906d6b9d7f51e29db19a85923a3abd8e060a4f28e083"],"state_sha256":"5662f5afe460fdec6b9300febbd8a19c5b652a38edb8fff62b1326cd997db6e8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WomjKujT3yb6F9pgeAzJOUj+oQ8Mev0uAED76dgodzxug7g/r/gDVCZ/IuwjIasqh+XoVWvpMVMBg3k51MobCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T13:33:24.687469Z","bundle_sha256":"97028c74105c0b4e2377ae4df9a582ce695ded342900dea4df31e60a668814fd"}}