{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:VBVCKEZTZ3YRTJPEEPDESPVPVK","short_pith_number":"pith:VBVCKEZT","canonical_record":{"source":{"id":"2607.26450","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-29T04:03:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"71edca6a9defa578c0621f26d5a024abb402ec1933cd6e11f10eea239e402cdf","abstract_canon_sha256":"be26ad790cb7bab201f942383e16f99366e93a37d064cb4e57ace5580473b7bf"},"schema_version":"1.0"},"canonical_sha256":"a86a251333cef119a5e423c6493eafaa91ecffae84d47f6a2b02a3dc89265528","source":{"kind":"arxiv","id":"2607.26450","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26450","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26450v1","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26450","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"VBVCKEZTZ3YR","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"VBVCKEZTZ3YRTJPE","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"VBVCKEZT","created_at":"2026-07-30T01:20:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:VBVCKEZTZ3YRTJPEEPDESPVPVK","target":"record","payload":{"canonical_record":{"source":{"id":"2607.26450","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-29T04:03:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"71edca6a9defa578c0621f26d5a024abb402ec1933cd6e11f10eea239e402cdf","abstract_canon_sha256":"be26ad790cb7bab201f942383e16f99366e93a37d064cb4e57ace5580473b7bf"},"schema_version":"1.0"},"canonical_sha256":"a86a251333cef119a5e423c6493eafaa91ecffae84d47f6a2b02a3dc89265528","receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a86a251333cef119a5e423c6493eafaa91ecffae84d47f6a2b02a3dc89265528","last_reissued_at":"2026-07-30T01:20:32.268506Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-30T01:20:32.268506Z"},"source_kind":"arxiv","source_id":"2607.26450","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":"2026-07-30T01:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TMNNvGfkCfEiN2Z9aWvpB2BdNun3dQZM6YeqOehRhiE/uwyBWn3axw635brIp/vYtMwD3g3Arq86l6eatx2hCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T20:34:09.157099Z"},"content_sha256":"0c0f68ecc09458ce72f687869db933c39e0cda97b04535e5650d6c43b19f0f36","schema_version":"1.0","event_id":"sha256:0c0f68ecc09458ce72f687869db933c39e0cda97b04535e5650d6c43b19f0f36"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:VBVCKEZTZ3YRTJPEEPDESPVPVK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Improved Bounds for Distinct Multiples in Intervals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kaizhe Chen","submitted_at":"2026-07-29T04:03:50Z","abstract_excerpt":"In this note, we study two functions introduced by Erd\\H{o}s and Pomerance. For any positive integer $n$, let $F(n)$ be the smallest integer $F>0$ such that any $F$ consecutive integers contain a distinct multiple for each positive integer at most $n$, and let $h_{\\mathbb{P}}(n)$ be the smallest integer $H>0$ such that any $H$ consecutive integers contain a distinct multiple for each prime at most $n$. Based on the square-residue digit construction of Green and Ruzsa, we prove \\[\n  F(n)\\ge h_{\\mathbb P}(n)\\ge n\\exp\\!\\left(\\frac{1}{50}\\frac{\\log n}{\\log\\log n}\\right), \\] for sufficiently large "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26450","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/2607.26450/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":"2026-07-30T01:20:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IC/sTXeuNRy3jHhHdC7t1pJ3ewC+MGDwzUkM+jJyINkadgRG66j9hehIu+bSIi8LOuKgdpXE/HcK7VzzXmygDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T20:34:09.157686Z"},"content_sha256":"89e5b70b977ec477ffe52220afa7e3d5633a5fd4b63313b3657f4943d65d1ba4","schema_version":"1.0","event_id":"sha256:89e5b70b977ec477ffe52220afa7e3d5633a5fd4b63313b3657f4943d65d1ba4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/bundle.json","state_url":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/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":"2026-08-11T20:34:09Z","links":{"resolver":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK","bundle":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/bundle.json","state":"https://pith.science/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VBVCKEZTZ3YRTJPEEPDESPVPVK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:VBVCKEZTZ3YRTJPEEPDESPVPVK","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":"be26ad790cb7bab201f942383e16f99366e93a37d064cb4e57ace5580473b7bf","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-29T04:03:50Z","title_canon_sha256":"71edca6a9defa578c0621f26d5a024abb402ec1933cd6e11f10eea239e402cdf"},"schema_version":"1.0","source":{"id":"2607.26450","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.26450","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"arxiv_version","alias_value":"2607.26450v1","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.26450","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"pith_short_12","alias_value":"VBVCKEZTZ3YR","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"pith_short_16","alias_value":"VBVCKEZTZ3YRTJPE","created_at":"2026-07-30T01:20:32Z"},{"alias_kind":"pith_short_8","alias_value":"VBVCKEZT","created_at":"2026-07-30T01:20:32Z"}],"graph_snapshots":[{"event_id":"sha256:89e5b70b977ec477ffe52220afa7e3d5633a5fd4b63313b3657f4943d65d1ba4","target":"graph","created_at":"2026-07-30T01:20:32Z","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/2607.26450/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this note, we study two functions introduced by Erd\\H{o}s and Pomerance. For any positive integer $n$, let $F(n)$ be the smallest integer $F>0$ such that any $F$ consecutive integers contain a distinct multiple for each positive integer at most $n$, and let $h_{\\mathbb{P}}(n)$ be the smallest integer $H>0$ such that any $H$ consecutive integers contain a distinct multiple for each prime at most $n$. Based on the square-residue digit construction of Green and Ruzsa, we prove \\[\n  F(n)\\ge h_{\\mathbb P}(n)\\ge n\\exp\\!\\left(\\frac{1}{50}\\frac{\\log n}{\\log\\log n}\\right), \\] for sufficiently large ","authors_text":"Kaizhe Chen","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-29T04:03:50Z","title":"Improved Bounds for Distinct Multiples in Intervals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.26450","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":"sha256:0c0f68ecc09458ce72f687869db933c39e0cda97b04535e5650d6c43b19f0f36","target":"record","created_at":"2026-07-30T01:20:32Z","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":"be26ad790cb7bab201f942383e16f99366e93a37d064cb4e57ace5580473b7bf","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-07-29T04:03:50Z","title_canon_sha256":"71edca6a9defa578c0621f26d5a024abb402ec1933cd6e11f10eea239e402cdf"},"schema_version":"1.0","source":{"id":"2607.26450","kind":"arxiv","version":1}},"canonical_sha256":"a86a251333cef119a5e423c6493eafaa91ecffae84d47f6a2b02a3dc89265528","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a86a251333cef119a5e423c6493eafaa91ecffae84d47f6a2b02a3dc89265528","first_computed_at":"2026-07-30T01:20:32.268506Z","kind":"pith_receipt","last_reissued_at":"2026-07-30T01:20:32.268506Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.26450","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0c0f68ecc09458ce72f687869db933c39e0cda97b04535e5650d6c43b19f0f36","sha256:89e5b70b977ec477ffe52220afa7e3d5633a5fd4b63313b3657f4943d65d1ba4"],"state_sha256":"479dd6b055819e4948fdb1a52ecf5fb916ee1ae0d0cbcc81955d0bd04ee4b50e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qZ6nZgN2ZytzHjV4bn+WpT0KcWYtasfRJ/UkndIQSLW5ds86UkoBHk7kV7kZ1eBSZfGKzI4LpTw3qvMnqHAVAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T20:34:09.163313Z","bundle_sha256":"671d6fac0107cc6c5a120572470c49ab5b52fa2c7c44edb602ccdd6d84502f35"}}