{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:XDBBFLWC7V52QZPRU3TOH3RM5K","short_pith_number":"pith:XDBBFLWC","schema_version":"1.0","canonical_sha256":"b8c212aec2fd7ba865f1a6e6e3ee2cea95b504525862ecf7c7d861f0b41cc1d3","source":{"kind":"arxiv","id":"1804.04650","version":1},"attestation_state":"computed","paper":{"title":"On the number of hard ball collisions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Krzysztof Burdzy, Mauricio Duarte","submitted_at":"2018-04-12T17:56:26Z","abstract_excerpt":"We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are $n$ balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than $n^{-n}$, then the total number of collisions is bounded by $n^{(5/2+\\varepsilon)n}$, for any fixed $\\varepsilon>0$ and large $n$. We also show that if there is a number of collisions larger than $n^{cn}$ for an appropriate $c>0$, then a large number of these collisions occur within a "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1804.04650","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-04-12T17:56:26Z","cross_cats_sorted":[],"title_canon_sha256":"82c4a7d6cc827afa2eaf25ad72042bee72e51d75e4c8402feac7533461df61f7","abstract_canon_sha256":"65bb44daf8c0e62f9ae56d51c0c94672f7627ef22560c57d5432d434c8708e94"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:36.626395Z","signature_b64":"0HWBnAJo2GE8rnAcnzFxcpLwpbNzl1KQVMKSlPxeucJWse6c5hJdwyIuXXDEerQnbITR01ff0xTg4bO0XscxDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8c212aec2fd7ba865f1a6e6e3ee2cea95b504525862ecf7c7d861f0b41cc1d3","last_reissued_at":"2026-05-18T00:18:36.625653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:36.625653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of hard ball collisions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Krzysztof Burdzy, Mauricio Duarte","submitted_at":"2018-04-12T17:56:26Z","abstract_excerpt":"We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are $n$ balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than $n^{-n}$, then the total number of collisions is bounded by $n^{(5/2+\\varepsilon)n}$, for any fixed $\\varepsilon>0$ and large $n$. We also show that if there is a number of collisions larger than $n^{cn}$ for an appropriate $c>0$, then a large number of these collisions occur within a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.04650","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1804.04650","created_at":"2026-05-18T00:18:36.625755+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.04650v1","created_at":"2026-05-18T00:18:36.625755+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.04650","created_at":"2026-05-18T00:18:36.625755+00:00"},{"alias_kind":"pith_short_12","alias_value":"XDBBFLWC7V52","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_16","alias_value":"XDBBFLWC7V52QZPR","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_8","alias_value":"XDBBFLWC","created_at":"2026-05-18T12:33:01.666342+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K","json":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K.json","graph_json":"https://pith.science/api/pith-number/XDBBFLWC7V52QZPRU3TOH3RM5K/graph.json","events_json":"https://pith.science/api/pith-number/XDBBFLWC7V52QZPRU3TOH3RM5K/events.json","paper":"https://pith.science/paper/XDBBFLWC"},"agent_actions":{"view_html":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K","download_json":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K.json","view_paper":"https://pith.science/paper/XDBBFLWC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.04650&json=true","fetch_graph":"https://pith.science/api/pith-number/XDBBFLWC7V52QZPRU3TOH3RM5K/graph.json","fetch_events":"https://pith.science/api/pith-number/XDBBFLWC7V52QZPRU3TOH3RM5K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K/action/storage_attestation","attest_author":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K/action/author_attestation","sign_citation":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K/action/citation_signature","submit_replication":"https://pith.science/pith/XDBBFLWC7V52QZPRU3TOH3RM5K/action/replication_record"}},"created_at":"2026-05-18T00:18:36.625755+00:00","updated_at":"2026-05-18T00:18:36.625755+00:00"}