{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:7Z7OHVHFMVDGBYHWLGOXEUGJVT","short_pith_number":"pith:7Z7OHVHF","canonical_record":{"source":{"id":"2607.06343","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-07T14:40:09Z","cross_cats_sorted":[],"title_canon_sha256":"e334cebe0477aae0e72e9b9f98d818190e33c3926f74f37a87d58472d9b82de1","abstract_canon_sha256":"075f31e230a7666c58b9c9a9b5e3f97b4f3f975a5a1d281d41c33086e10b7b4c"},"schema_version":"1.0"},"canonical_sha256":"fe7ee3d4e5654660e0f6599d7250c9acec4be63f730536031f50f14a7a4b3081","source":{"kind":"arxiv","id":"2607.06343","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06343","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06343v1","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06343","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"7Z7OHVHFMVDG","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"pith_short_16","alias_value":"7Z7OHVHFMVDGBYHW","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"pith_short_8","alias_value":"7Z7OHVHF","created_at":"2026-07-08T01:19:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:7Z7OHVHFMVDGBYHWLGOXEUGJVT","target":"record","payload":{"canonical_record":{"source":{"id":"2607.06343","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-07T14:40:09Z","cross_cats_sorted":[],"title_canon_sha256":"e334cebe0477aae0e72e9b9f98d818190e33c3926f74f37a87d58472d9b82de1","abstract_canon_sha256":"075f31e230a7666c58b9c9a9b5e3f97b4f3f975a5a1d281d41c33086e10b7b4c"},"schema_version":"1.0"},"canonical_sha256":"fe7ee3d4e5654660e0f6599d7250c9acec4be63f730536031f50f14a7a4b3081","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:19:22.093480Z","signature_b64":"pxDMN7Dm/MqQjN2aK/P3Pfbz+xrf/FcmCc+d2QnZGKWts+kuBNj5wBzb3KNluQgAC0DY/OXyA6S6Rded+/pFCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe7ee3d4e5654660e0f6599d7250c9acec4be63f730536031f50f14a7a4b3081","last_reissued_at":"2026-07-08T01:19:22.093056Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:19:22.093056Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.06343","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":"2026-07-08T01:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KZkMYYvBGFMxmxzYFSx1ImXFX7Y9hb63clnHiYKN30qpCFrHgDs9bJxRxj4zd/6V+THFucKC2CB9ywcov8OXCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T05:07:39.462687Z"},"content_sha256":"4d7794d5d424369e4294a658fdbf2e1cdda3c071cfde055b21924acd5ba2107e","schema_version":"1.0","event_id":"sha256:4d7794d5d424369e4294a658fdbf2e1cdda3c071cfde055b21924acd5ba2107e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:7Z7OHVHFMVDGBYHWLGOXEUGJVT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp Logarithmic Thresholds for Cut Schedules in an Abstract Branch-and-Cut Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Hongyi Jiang","submitted_at":"2026-07-07T14:40:09Z","abstract_excerpt":"Branch-and-cut interleaves branching with cutting-plane generation. How the two operations share the work of proving a bound is a basic theoretical question. We study an abstract model in which a tree certifies a target bound $Z$. Each branch node improves the bound by $\\ell$ on one child and by $r$ on the other, where $0<\\ell\\le r$. The $i$th cut along a root-to-node path improves it by $c_i\\ge0$, with cumulative improvement $C_k=\\sum_{i=1}^k c_i$.\n  Asymmetric branching enters through the rate $\\lambda^{\\star}>0$ defined by $e^{-\\lambda^{\\star}\\ell}+e^{-\\lambda^{\\star}r}=1$. We establish uni"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06343","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/2607.06343/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":"2026-07-08T01:19:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2DUM/sI3XhIaaoE24jBFYSxXWBjgI9IfO//w21zg/NjJu0a4lJM3RNPM6VARH7R4o2dov0V84dRwfkwi7wvRCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T05:07:39.463616Z"},"content_sha256":"af6bca88f30f833b47ec991f12990c8972a8e5c3d3908add6939def2043890be","schema_version":"1.0","event_id":"sha256:af6bca88f30f833b47ec991f12990c8972a8e5c3d3908add6939def2043890be"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7Z7OHVHFMVDGBYHWLGOXEUGJVT/bundle.json","state_url":"https://pith.science/pith/7Z7OHVHFMVDGBYHWLGOXEUGJVT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7Z7OHVHFMVDGBYHWLGOXEUGJVT/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":"2026-08-20T05:07:39Z","links":{"resolver":"https://pith.science/pith/7Z7OHVHFMVDGBYHWLGOXEUGJVT","bundle":"https://pith.science/pith/7Z7OHVHFMVDGBYHWLGOXEUGJVT/bundle.json","state":"https://pith.science/pith/7Z7OHVHFMVDGBYHWLGOXEUGJVT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7Z7OHVHFMVDGBYHWLGOXEUGJVT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7Z7OHVHFMVDGBYHWLGOXEUGJVT","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":"075f31e230a7666c58b9c9a9b5e3f97b4f3f975a5a1d281d41c33086e10b7b4c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-07T14:40:09Z","title_canon_sha256":"e334cebe0477aae0e72e9b9f98d818190e33c3926f74f37a87d58472d9b82de1"},"schema_version":"1.0","source":{"id":"2607.06343","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.06343","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"arxiv_version","alias_value":"2607.06343v1","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06343","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"pith_short_12","alias_value":"7Z7OHVHFMVDG","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"pith_short_16","alias_value":"7Z7OHVHFMVDGBYHW","created_at":"2026-07-08T01:19:22Z"},{"alias_kind":"pith_short_8","alias_value":"7Z7OHVHF","created_at":"2026-07-08T01:19:22Z"}],"graph_snapshots":[{"event_id":"sha256:af6bca88f30f833b47ec991f12990c8972a8e5c3d3908add6939def2043890be","target":"graph","created_at":"2026-07-08T01:19:22Z","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/2607.06343/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Branch-and-cut interleaves branching with cutting-plane generation. How the two operations share the work of proving a bound is a basic theoretical question. We study an abstract model in which a tree certifies a target bound $Z$. Each branch node improves the bound by $\\ell$ on one child and by $r$ on the other, where $0<\\ell\\le r$. The $i$th cut along a root-to-node path improves it by $c_i\\ge0$, with cumulative improvement $C_k=\\sum_{i=1}^k c_i$.\n  Asymmetric branching enters through the rate $\\lambda^{\\star}>0$ defined by $e^{-\\lambda^{\\star}\\ell}+e^{-\\lambda^{\\star}r}=1$. We establish uni","authors_text":"Hongyi Jiang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-07T14:40:09Z","title":"Sharp Logarithmic Thresholds for Cut Schedules in an Abstract Branch-and-Cut Model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06343","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":"sha256:4d7794d5d424369e4294a658fdbf2e1cdda3c071cfde055b21924acd5ba2107e","target":"record","created_at":"2026-07-08T01:19:22Z","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":"075f31e230a7666c58b9c9a9b5e3f97b4f3f975a5a1d281d41c33086e10b7b4c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-07T14:40:09Z","title_canon_sha256":"e334cebe0477aae0e72e9b9f98d818190e33c3926f74f37a87d58472d9b82de1"},"schema_version":"1.0","source":{"id":"2607.06343","kind":"arxiv","version":1}},"canonical_sha256":"fe7ee3d4e5654660e0f6599d7250c9acec4be63f730536031f50f14a7a4b3081","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe7ee3d4e5654660e0f6599d7250c9acec4be63f730536031f50f14a7a4b3081","first_computed_at":"2026-07-08T01:19:22.093056Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-08T01:19:22.093056Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pxDMN7Dm/MqQjN2aK/P3Pfbz+xrf/FcmCc+d2QnZGKWts+kuBNj5wBzb3KNluQgAC0DY/OXyA6S6Rded+/pFCw==","signature_status":"signed_v1","signed_at":"2026-07-08T01:19:22.093480Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.06343","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4d7794d5d424369e4294a658fdbf2e1cdda3c071cfde055b21924acd5ba2107e","sha256:af6bca88f30f833b47ec991f12990c8972a8e5c3d3908add6939def2043890be"],"state_sha256":"79bea80eb5bec78072dcd7853311cf8fadbba326efa754c0ce1875660ea292e3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0vDnVEIYeRQVKtOc3587bFqRazkQghyxy5/W6rmsy3Zo2cR9DDFomQuz7XBh/2TSawBzWLjFm1peKPzRPtd+Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T05:07:39.469276Z","bundle_sha256":"ba213f8eba02fdc6f336eda299a950bae6dbcd1e8dc2f7d826b3d9fd344fd1b7"}}