{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:Q2ILGCCXRWA5RH63X6OGTTB4PA","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":"af8f9f6e7a336c6f58c3f74f972c5b4c5811a6e567716efcba160640a5da4b33","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2021-07-21T21:45:24Z","title_canon_sha256":"ea560a1cc720f84164500899839a9a95db28f6ae40dfba1e72c8ad9e2624742e"},"schema_version":"1.0","source":{"id":"2107.10362","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2107.10362","created_at":"2026-07-05T03:49:15Z"},{"alias_kind":"arxiv_version","alias_value":"2107.10362v2","created_at":"2026-07-05T03:49:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2107.10362","created_at":"2026-07-05T03:49:15Z"},{"alias_kind":"pith_short_12","alias_value":"Q2ILGCCXRWA5","created_at":"2026-07-05T03:49:15Z"},{"alias_kind":"pith_short_16","alias_value":"Q2ILGCCXRWA5RH63","created_at":"2026-07-05T03:49:15Z"},{"alias_kind":"pith_short_8","alias_value":"Q2ILGCCX","created_at":"2026-07-05T03:49:15Z"}],"graph_snapshots":[{"event_id":"sha256:7d83cddb5f2df82855d34929eaae7dedaa3bde85fcc059a2f78c2a0dc107daff","target":"graph","created_at":"2026-07-05T03:49:15Z","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/2107.10362/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a new upper bound $K_+$ on the number of totally elastic collisions of $n$ hard spheres with equal radii and equal masses in $R^d$. Our bound satisfies $\\log K_+ \\leq c(d) n \\log n$.","authors_text":"Krzysztof Burdzy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2021-07-21T21:45:24Z","title":"An improved upper bound on the number of billiard ball collisions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.10362","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":"sha256:462fda2366855da8c13c3b986992450bc53d034b46a3df485649928e71617919","target":"record","created_at":"2026-07-05T03:49:15Z","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":"af8f9f6e7a336c6f58c3f74f972c5b4c5811a6e567716efcba160640a5da4b33","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2021-07-21T21:45:24Z","title_canon_sha256":"ea560a1cc720f84164500899839a9a95db28f6ae40dfba1e72c8ad9e2624742e"},"schema_version":"1.0","source":{"id":"2107.10362","kind":"arxiv","version":2}},"canonical_sha256":"8690b308578d81d89fdbbf9c69cc3c7821b6f3e77b01a31b7ad76448595cab9a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8690b308578d81d89fdbbf9c69cc3c7821b6f3e77b01a31b7ad76448595cab9a","first_computed_at":"2026-07-05T03:49:15.311558Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:49:15.311558Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4jz285XoTVcD2XGMxUmTTRlBSrhs59La+igLbd5JcjVl+x6Z83H3PFAaejCZeLgyE1o13u969xZUDT0QLKNiBA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:49:15.312029Z","signed_message":"canonical_sha256_bytes"},"source_id":"2107.10362","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:462fda2366855da8c13c3b986992450bc53d034b46a3df485649928e71617919","sha256:7d83cddb5f2df82855d34929eaae7dedaa3bde85fcc059a2f78c2a0dc107daff"],"state_sha256":"98b58ef54c36059e62767cc3980f11bee7a49bcdcb156c72c89f9818f87e79c2"}