{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KSTBQEP7PVRAQ2VDGRDKUK72VW","short_pith_number":"pith:KSTBQEP7","canonical_record":{"source":{"id":"2607.00665","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-01T09:13:24Z","cross_cats_sorted":[],"title_canon_sha256":"3983980468a51a2a603b4d24e4000aa56bbda9c10b1614b2bf8a58c39d959acb","abstract_canon_sha256":"6d477814ff5cf0aad56ea7eccb121c5a55443f73b957f6c8128e56653df17d98"},"schema_version":"1.0"},"canonical_sha256":"54a61811ff7d62086aa33446aa2bfaada66144ce6117b23398485aaf2e39caaf","source":{"kind":"arxiv","id":"2607.00665","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.00665","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"arxiv_version","alias_value":"2607.00665v1","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00665","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"pith_short_12","alias_value":"KSTBQEP7PVRA","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"pith_short_16","alias_value":"KSTBQEP7PVRAQ2VD","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"pith_short_8","alias_value":"KSTBQEP7","created_at":"2026-07-02T01:17:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KSTBQEP7PVRAQ2VDGRDKUK72VW","target":"record","payload":{"canonical_record":{"source":{"id":"2607.00665","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-01T09:13:24Z","cross_cats_sorted":[],"title_canon_sha256":"3983980468a51a2a603b4d24e4000aa56bbda9c10b1614b2bf8a58c39d959acb","abstract_canon_sha256":"6d477814ff5cf0aad56ea7eccb121c5a55443f73b957f6c8128e56653df17d98"},"schema_version":"1.0"},"canonical_sha256":"54a61811ff7d62086aa33446aa2bfaada66144ce6117b23398485aaf2e39caaf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:17:50.850718Z","signature_b64":"VjzVc4F+ekC6KiNuDbsiI3d7aNdpIn8dabCDQW4EequMW4GEh77H99MrlJAVk9RwA78OW5KLFLxuqnETSINJAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54a61811ff7d62086aa33446aa2bfaada66144ce6117b23398485aaf2e39caaf","last_reissued_at":"2026-07-02T01:17:50.850174Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:17:50.850174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.00665","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":"2026-07-02T01:17:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VRL4oTIZSD/ojpcIqlYOk6KTbTkH17zPMYfBsgUpYDuCx1C+2R/HaUoWe4/Hvpjt+MGR5E6sWlLD9QzAE3tHDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T18:23:40.229288Z"},"content_sha256":"5ba05b824423fff7ccc08d4bce4ab2e59da9402ce82c233182b2b151985fa049","schema_version":"1.0","event_id":"sha256:5ba05b824423fff7ccc08d4bce4ab2e59da9402ce82c233182b2b151985fa049"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KSTBQEP7PVRAQ2VDGRDKUK72VW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Effective dynamics of the Sinkhorn algorithm in the regime of low entropy regularization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Guillaume Wang","submitted_at":"2026-07-01T09:13:24Z","abstract_excerpt":"The Sinkhorn algorithm is the de facto standard method for numerically solving entropy-regularized optimal transport problems over finite sets. In this work, we investigate a phenomenon arising when Sinkhorn is applied with a small regularization parameter $\\tau$: the evolution of the dual variables (the logarithm of the scaling factors) is approximately piecewise-linear, while the primal variables (the approximate transport plans) exhibit a saddle-to-saddle type behavior. We prove that as $\\tau \\to 0$, the Sinkhorn iterates indeed converge to a continuous-time curve consistent with these obse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00665","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/2607.00665/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":"2026-07-02T01:17:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nfcqn90BwwS2AHbgNiZy2FLmAa38IQwapktjhqEwuOcLz47alfUgNP6SrOhaLUeRPmaayx8Q4iPuvKIKsD6mCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T18:23:40.229678Z"},"content_sha256":"e941e77be48693f92fbb8d55ac46b52f9ae327b2035b2a4e86ea3b956570a620","schema_version":"1.0","event_id":"sha256:e941e77be48693f92fbb8d55ac46b52f9ae327b2035b2a4e86ea3b956570a620"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KSTBQEP7PVRAQ2VDGRDKUK72VW/bundle.json","state_url":"https://pith.science/pith/KSTBQEP7PVRAQ2VDGRDKUK72VW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KSTBQEP7PVRAQ2VDGRDKUK72VW/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":"2026-07-02T18:23:40Z","links":{"resolver":"https://pith.science/pith/KSTBQEP7PVRAQ2VDGRDKUK72VW","bundle":"https://pith.science/pith/KSTBQEP7PVRAQ2VDGRDKUK72VW/bundle.json","state":"https://pith.science/pith/KSTBQEP7PVRAQ2VDGRDKUK72VW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KSTBQEP7PVRAQ2VDGRDKUK72VW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KSTBQEP7PVRAQ2VDGRDKUK72VW","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":"6d477814ff5cf0aad56ea7eccb121c5a55443f73b957f6c8128e56653df17d98","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-01T09:13:24Z","title_canon_sha256":"3983980468a51a2a603b4d24e4000aa56bbda9c10b1614b2bf8a58c39d959acb"},"schema_version":"1.0","source":{"id":"2607.00665","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.00665","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"arxiv_version","alias_value":"2607.00665v1","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00665","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"pith_short_12","alias_value":"KSTBQEP7PVRA","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"pith_short_16","alias_value":"KSTBQEP7PVRAQ2VD","created_at":"2026-07-02T01:17:50Z"},{"alias_kind":"pith_short_8","alias_value":"KSTBQEP7","created_at":"2026-07-02T01:17:50Z"}],"graph_snapshots":[{"event_id":"sha256:e941e77be48693f92fbb8d55ac46b52f9ae327b2035b2a4e86ea3b956570a620","target":"graph","created_at":"2026-07-02T01:17:50Z","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/2607.00665/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Sinkhorn algorithm is the de facto standard method for numerically solving entropy-regularized optimal transport problems over finite sets. In this work, we investigate a phenomenon arising when Sinkhorn is applied with a small regularization parameter $\\tau$: the evolution of the dual variables (the logarithm of the scaling factors) is approximately piecewise-linear, while the primal variables (the approximate transport plans) exhibit a saddle-to-saddle type behavior. We prove that as $\\tau \\to 0$, the Sinkhorn iterates indeed converge to a continuous-time curve consistent with these obse","authors_text":"Guillaume Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-01T09:13:24Z","title":"Effective dynamics of the Sinkhorn algorithm in the regime of low entropy regularization"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00665","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":"sha256:5ba05b824423fff7ccc08d4bce4ab2e59da9402ce82c233182b2b151985fa049","target":"record","created_at":"2026-07-02T01:17:50Z","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":"6d477814ff5cf0aad56ea7eccb121c5a55443f73b957f6c8128e56653df17d98","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-07-01T09:13:24Z","title_canon_sha256":"3983980468a51a2a603b4d24e4000aa56bbda9c10b1614b2bf8a58c39d959acb"},"schema_version":"1.0","source":{"id":"2607.00665","kind":"arxiv","version":1}},"canonical_sha256":"54a61811ff7d62086aa33446aa2bfaada66144ce6117b23398485aaf2e39caaf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"54a61811ff7d62086aa33446aa2bfaada66144ce6117b23398485aaf2e39caaf","first_computed_at":"2026-07-02T01:17:50.850174Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-02T01:17:50.850174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VjzVc4F+ekC6KiNuDbsiI3d7aNdpIn8dabCDQW4EequMW4GEh77H99MrlJAVk9RwA78OW5KLFLxuqnETSINJAQ==","signature_status":"signed_v1","signed_at":"2026-07-02T01:17:50.850718Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.00665","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5ba05b824423fff7ccc08d4bce4ab2e59da9402ce82c233182b2b151985fa049","sha256:e941e77be48693f92fbb8d55ac46b52f9ae327b2035b2a4e86ea3b956570a620"],"state_sha256":"a08f2625f07f67edfe4da63db2288d26d454753ac3344612de1c798644dbe83e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V/rYNK8GP57Kw2PpPZzNwyMBkjKvtjU3knCXviE7orqEdTOfhog641zDUyEvYhO82KDVMNBBX1xHNnH0ljUBBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T18:23:40.231680Z","bundle_sha256":"1075d28392a041e3688d45c7d621a0777c94673380d2fb28e4c6b1dc39265ee7"}}