{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:73JG6VT5JXLE34V6WCU6WKSNDU","short_pith_number":"pith:73JG6VT5","canonical_record":{"source":{"id":"2509.19598","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-09-23T21:44:54Z","cross_cats_sorted":["cs.DS","math.IT"],"title_canon_sha256":"97966d960ba7a10f84f8fa2f126625c2ea084cf65311f291bcff50709c42fd85","abstract_canon_sha256":"c21b3175b4bf317a8b19e40613017d675a0f8abc47c987d9c52a4102c1d5fd2b"},"schema_version":"1.0"},"canonical_sha256":"fed26f567d4dd64df2beb0a9eb2a4d1d372742cb2b1b28c5658e4ee2704101a6","source":{"kind":"arxiv","id":"2509.19598","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.19598","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"arxiv_version","alias_value":"2509.19598v3","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.19598","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"pith_short_12","alias_value":"73JG6VT5JXLE","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"pith_short_16","alias_value":"73JG6VT5JXLE34V6","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"pith_short_8","alias_value":"73JG6VT5","created_at":"2026-06-19T16:12:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:73JG6VT5JXLE34V6WCU6WKSNDU","target":"record","payload":{"canonical_record":{"source":{"id":"2509.19598","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-09-23T21:44:54Z","cross_cats_sorted":["cs.DS","math.IT"],"title_canon_sha256":"97966d960ba7a10f84f8fa2f126625c2ea084cf65311f291bcff50709c42fd85","abstract_canon_sha256":"c21b3175b4bf317a8b19e40613017d675a0f8abc47c987d9c52a4102c1d5fd2b"},"schema_version":"1.0"},"canonical_sha256":"fed26f567d4dd64df2beb0a9eb2a4d1d372742cb2b1b28c5658e4ee2704101a6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:15.103721Z","signature_b64":"MA/9upbYrAcWiSKJwV6fUunWgW0PUFD9hyjnR511azJ1PmXDv0hUbrbzQez15TgTkkU+6oXRFjUydkDnRU4gBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fed26f567d4dd64df2beb0a9eb2a4d1d372742cb2b1b28c5658e4ee2704101a6","last_reissued_at":"2026-06-19T16:12:15.103328Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:15.103328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.19598","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-19T16:12:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YcsgrJzwuTg0zvLhClb8geBmkBpGyGrj7x/MDwtqa233Lacv6YmycfFhSMcUwc5oMe9+uOCjJ9ckTgQkJbCkDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T11:03:59.068403Z"},"content_sha256":"e6451dc2cf2532d19c2a44cd86ada1ca7ac2a59a60388b21d57adbab337a904c","schema_version":"1.0","event_id":"sha256:e6451dc2cf2532d19c2a44cd86ada1ca7ac2a59a60388b21d57adbab337a904c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:73JG6VT5JXLE34V6WCU6WKSNDU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Efficient $\\varepsilon$-approximate minimum-entropy couplings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.IT"],"primary_cat":"cs.IT","authors_text":"Spencer Compton","submitted_at":"2025-09-23T21:44:54Z","abstract_excerpt":"Given $m \\ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $H(\\operatorname{ALG}) \\le H(\\operatorname{OPT}) + c$, where $c \\approx 0.53$ for $m=2$, and $c \\approx 1.22$ for general $m$ [CKQGK '23].\n  A main open question is whether this"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.19598","kind":"arxiv","version":3},"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/2509.19598/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":"2026-06-19T16:12:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0GT0jVvb3kBmwKScW5f6XUgUEkQsjG6X0MICm52Df2GLhM6aX2k42pQBoFAmsWLlZdNN5eMUl6Pp15+0o6nvBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T11:03:59.068793Z"},"content_sha256":"3bac2534a2d8de7758543320f80021edc7805d3a0f29ece5adbd59a385daf945","schema_version":"1.0","event_id":"sha256:3bac2534a2d8de7758543320f80021edc7805d3a0f29ece5adbd59a385daf945"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/73JG6VT5JXLE34V6WCU6WKSNDU/bundle.json","state_url":"https://pith.science/pith/73JG6VT5JXLE34V6WCU6WKSNDU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/73JG6VT5JXLE34V6WCU6WKSNDU/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":"2026-06-30T11:03:59Z","links":{"resolver":"https://pith.science/pith/73JG6VT5JXLE34V6WCU6WKSNDU","bundle":"https://pith.science/pith/73JG6VT5JXLE34V6WCU6WKSNDU/bundle.json","state":"https://pith.science/pith/73JG6VT5JXLE34V6WCU6WKSNDU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/73JG6VT5JXLE34V6WCU6WKSNDU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:73JG6VT5JXLE34V6WCU6WKSNDU","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":"c21b3175b4bf317a8b19e40613017d675a0f8abc47c987d9c52a4102c1d5fd2b","cross_cats_sorted":["cs.DS","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-09-23T21:44:54Z","title_canon_sha256":"97966d960ba7a10f84f8fa2f126625c2ea084cf65311f291bcff50709c42fd85"},"schema_version":"1.0","source":{"id":"2509.19598","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.19598","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"arxiv_version","alias_value":"2509.19598v3","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.19598","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"pith_short_12","alias_value":"73JG6VT5JXLE","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"pith_short_16","alias_value":"73JG6VT5JXLE34V6","created_at":"2026-06-19T16:12:15Z"},{"alias_kind":"pith_short_8","alias_value":"73JG6VT5","created_at":"2026-06-19T16:12:15Z"}],"graph_snapshots":[{"event_id":"sha256:3bac2534a2d8de7758543320f80021edc7805d3a0f29ece5adbd59a385daf945","target":"graph","created_at":"2026-06-19T16:12:15Z","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/2509.19598/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given $m \\ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $H(\\operatorname{ALG}) \\le H(\\operatorname{OPT}) + c$, where $c \\approx 0.53$ for $m=2$, and $c \\approx 1.22$ for general $m$ [CKQGK '23].\n  A main open question is whether this","authors_text":"Spencer Compton","cross_cats":["cs.DS","math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-09-23T21:44:54Z","title":"Efficient $\\varepsilon$-approximate minimum-entropy couplings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.19598","kind":"arxiv","version":3},"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:e6451dc2cf2532d19c2a44cd86ada1ca7ac2a59a60388b21d57adbab337a904c","target":"record","created_at":"2026-06-19T16:12:15Z","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":"c21b3175b4bf317a8b19e40613017d675a0f8abc47c987d9c52a4102c1d5fd2b","cross_cats_sorted":["cs.DS","math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2025-09-23T21:44:54Z","title_canon_sha256":"97966d960ba7a10f84f8fa2f126625c2ea084cf65311f291bcff50709c42fd85"},"schema_version":"1.0","source":{"id":"2509.19598","kind":"arxiv","version":3}},"canonical_sha256":"fed26f567d4dd64df2beb0a9eb2a4d1d372742cb2b1b28c5658e4ee2704101a6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fed26f567d4dd64df2beb0a9eb2a4d1d372742cb2b1b28c5658e4ee2704101a6","first_computed_at":"2026-06-19T16:12:15.103328Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:12:15.103328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MA/9upbYrAcWiSKJwV6fUunWgW0PUFD9hyjnR511azJ1PmXDv0hUbrbzQez15TgTkkU+6oXRFjUydkDnRU4gBg==","signature_status":"signed_v1","signed_at":"2026-06-19T16:12:15.103721Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.19598","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6451dc2cf2532d19c2a44cd86ada1ca7ac2a59a60388b21d57adbab337a904c","sha256:3bac2534a2d8de7758543320f80021edc7805d3a0f29ece5adbd59a385daf945"],"state_sha256":"1a8b8c1b36cd8848ab6e587b0c4496c688a3dc569511a11c35c0d9f2aadfee63"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hkt9XoFjEB9844bJl3zzHhukosXVLrviuE06LkuRQv85d4ZXyYH2tuwE3SBZ/tYP7SasSxStLbV17Ru3+WxFAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T11:03:59.070761Z","bundle_sha256":"f978b2b075d7e1cc97c30bf7604ef4229c02eb972f7737cd7bbce464c134dee8"}}