{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:NGJGKSE3TN4WL5SRWZ6VTQTRWF","short_pith_number":"pith:NGJGKSE3","canonical_record":{"source":{"id":"2608.06344","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T17:51:00Z","cross_cats_sorted":[],"title_canon_sha256":"427acbc77fce2e53783440b658f31132690b1b80367c469219d1135035e38bf0","abstract_canon_sha256":"189aca2a089ded97a87b09e308575b1ce632f60b5ad4ea8c1f1627ed9ebc2dff"},"schema_version":"1.0"},"canonical_sha256":"699265489b9b7965f651b67d59c271b17fedf098dc83d9229b9916e0b23b54a0","source":{"kind":"arxiv","id":"2608.06344","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.06344","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"2608.06344v1","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.06344","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"NGJGKSE3TN4W","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"pith_short_16","alias_value":"NGJGKSE3TN4WL5SR","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"pith_short_8","alias_value":"NGJGKSE3","created_at":"2026-08-07T01:41:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:NGJGKSE3TN4WL5SRWZ6VTQTRWF","target":"record","payload":{"canonical_record":{"source":{"id":"2608.06344","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T17:51:00Z","cross_cats_sorted":[],"title_canon_sha256":"427acbc77fce2e53783440b658f31132690b1b80367c469219d1135035e38bf0","abstract_canon_sha256":"189aca2a089ded97a87b09e308575b1ce632f60b5ad4ea8c1f1627ed9ebc2dff"},"schema_version":"1.0"},"canonical_sha256":"699265489b9b7965f651b67d59c271b17fedf098dc83d9229b9916e0b23b54a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-07T01:41:26.855243Z","signature_b64":"im+zm4Qy5yfDQodzXSzfNS5FM2x+1OKl96sZaAM0CQYilbacsRCnwmJaoMPLscKOTaRyglxjzegjVrW9QLIaAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"699265489b9b7965f651b67d59c271b17fedf098dc83d9229b9916e0b23b54a0","last_reissued_at":"2026-08-07T01:41:26.853074Z","signature_status":"signed_v1","first_computed_at":"2026-08-07T01:41:26.853074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.06344","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":"2026-08-07T01:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZDKRMTwqsvkZo3T1mf7lrkrCAnw5dlwwZYVIqx8AQZzIeXnzaLTWM4FceFCfyTqp4pX9e86N0uL3zU/+fZFbAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T21:04:49.152315Z"},"content_sha256":"b6389a7472bc0cedab74b7b3a597dd66901708fdf00116b317039bc04a922f8d","schema_version":"1.0","event_id":"sha256:b6389a7472bc0cedab74b7b3a597dd66901708fdf00116b317039bc04a922f8d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:NGJGKSE3TN4WL5SRWZ6VTQTRWF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A General Aubry-Mather Theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nassif Ghoussoub","submitted_at":"2026-08-06T17:51:00Z","abstract_excerpt":"This paper reproduces the front matter --- preface, overview and table of contents --- of a monograph by the author, submitted for publication under the title {\\it Skew Linear Entropies and Kantorovich Operators: A General Aubry-Mather Theory}. The book isolates a class of non-linear operators, which we call {\\it Kantorovich operators}, that are ubiquitous in analysis, probability, dynamical systems, mathematical economics and finance. We develop aspects of their ergodic theory in a way that extends classical ones involving Markov operators, free-energy transfers, or the Hopf--Lax--Oleinik sem"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06344","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/2608.06344/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":"2026-08-07T01:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vzrFtnSGgPpkzyY94jnWm6XPWHQC0f/vLx3C8cVyyBIEoIwQE4RehoOU75e7ixrnqI1Icfa5cADK4S7USLm5Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T21:04:49.152827Z"},"content_sha256":"6c06210134a1be7eb8c21ccfa6142315e4883bb89296c6658f7305227bb89716","schema_version":"1.0","event_id":"sha256:6c06210134a1be7eb8c21ccfa6142315e4883bb89296c6658f7305227bb89716"},{"event_type":"integrity_finding","subject_pith_number":"pith:2026:NGJGKSE3TN4WL5SRWZ6VTQTRWF","target":"integrity","payload":{"note":"DOI in the printed bibliography is fragmented by whitespace or line breaks. A longer candidate (10.1007/s00526-019-1624-y) was visible in the surrounding text but could not be confirmed against doi.org as printed.","snippet":"J. Backhoff Veraguas, M. Beiglb¨ ock and G. Pammer,Existence, Duality, and Cyclic Monotonic- ity for Weak Transport Costs,Calc. Var. 58, 203 (2019). https://doi.org/10.1007/s00526-019- 1624-y","arxiv_id":"2608.06344","detector":"doi_compliance","evidence":{"ref_index":11,"verdict_class":"incontrovertible","resolved_title":null,"printed_excerpt":"10.1007/s00526-019-","reconstructed_doi":"10.1007/s00526-019-1624-y"},"severity":"advisory","ref_index":11,"audited_at":"2026-08-07T05:08:04.442762Z","event_type":"pith.integrity.v1","detected_doi":"10.1007/s00526-019-1624-y","detector_url":"https://pith.science/pith-integrity-protocol#doi_compliance","external_url":null,"finding_type":"recoverable_identifier","evidence_hash":"98680bda701a3d83a0286dda03f7ca68c8f7fb821e37ca2f21800fc5e361f6b1","paper_version":1,"verdict_class":"incontrovertible","resolved_title":null,"detector_version":"1.1.0","detected_arxiv_id":null,"integrity_event_id":18449,"payload_sha256":"0c45e6c8d2494a25258486e485174c53345a4ac3a172b7a84cacf093c96eff16","signature_b64":"sUD2Ro18ncBzwqrrJu27Jrd+vebbkCy6EPHofCWK279ZaDctzRtgD9PQD38mzGiCuVR1GAOzZzy21X3ZoXjYBg==","signing_key_id":"pith-v1-2026-05"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-08-07T05:08:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X4s2odVImlmq2JafMxy+nqrFykOoqQCm+k2NaD0NHxnr4Ox8tiVQZxbbv9OEoPvfZvtoOwnlghZ7/JJFuczDCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T21:04:49.155663Z"},"content_sha256":"32665059f83cbad7449a5fe78f7cdaf3f92146308f4eb8069d5639825fc5d660","schema_version":"1.0","event_id":"sha256:32665059f83cbad7449a5fe78f7cdaf3f92146308f4eb8069d5639825fc5d660"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NGJGKSE3TN4WL5SRWZ6VTQTRWF/bundle.json","state_url":"https://pith.science/pith/NGJGKSE3TN4WL5SRWZ6VTQTRWF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NGJGKSE3TN4WL5SRWZ6VTQTRWF/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":"2026-08-12T21:04:49Z","links":{"resolver":"https://pith.science/pith/NGJGKSE3TN4WL5SRWZ6VTQTRWF","bundle":"https://pith.science/pith/NGJGKSE3TN4WL5SRWZ6VTQTRWF/bundle.json","state":"https://pith.science/pith/NGJGKSE3TN4WL5SRWZ6VTQTRWF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NGJGKSE3TN4WL5SRWZ6VTQTRWF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:NGJGKSE3TN4WL5SRWZ6VTQTRWF","merge_version":"pith-open-graph-merge-v1","event_count":3,"valid_event_count":3,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"189aca2a089ded97a87b09e308575b1ce632f60b5ad4ea8c1f1627ed9ebc2dff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T17:51:00Z","title_canon_sha256":"427acbc77fce2e53783440b658f31132690b1b80367c469219d1135035e38bf0"},"schema_version":"1.0","source":{"id":"2608.06344","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.06344","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"2608.06344v1","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.06344","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"NGJGKSE3TN4W","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"pith_short_16","alias_value":"NGJGKSE3TN4WL5SR","created_at":"2026-08-07T01:41:26Z"},{"alias_kind":"pith_short_8","alias_value":"NGJGKSE3","created_at":"2026-08-07T01:41:26Z"}],"graph_snapshots":[{"event_id":"sha256:6c06210134a1be7eb8c21ccfa6142315e4883bb89296c6658f7305227bb89716","target":"graph","created_at":"2026-08-07T01:41:26Z","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/2608.06344/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper reproduces the front matter --- preface, overview and table of contents --- of a monograph by the author, submitted for publication under the title {\\it Skew Linear Entropies and Kantorovich Operators: A General Aubry-Mather Theory}. The book isolates a class of non-linear operators, which we call {\\it Kantorovich operators}, that are ubiquitous in analysis, probability, dynamical systems, mathematical economics and finance. We develop aspects of their ergodic theory in a way that extends classical ones involving Markov operators, free-energy transfers, or the Hopf--Lax--Oleinik sem","authors_text":"Nassif Ghoussoub","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T17:51:00Z","title":"A General Aubry-Mather Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06344","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:b6389a7472bc0cedab74b7b3a597dd66901708fdf00116b317039bc04a922f8d","target":"record","created_at":"2026-08-07T01:41:26Z","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":"189aca2a089ded97a87b09e308575b1ce632f60b5ad4ea8c1f1627ed9ebc2dff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-08-06T17:51:00Z","title_canon_sha256":"427acbc77fce2e53783440b658f31132690b1b80367c469219d1135035e38bf0"},"schema_version":"1.0","source":{"id":"2608.06344","kind":"arxiv","version":1}},"canonical_sha256":"699265489b9b7965f651b67d59c271b17fedf098dc83d9229b9916e0b23b54a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"699265489b9b7965f651b67d59c271b17fedf098dc83d9229b9916e0b23b54a0","first_computed_at":"2026-08-07T01:41:26.853074Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-07T01:41:26.853074Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"im+zm4Qy5yfDQodzXSzfNS5FM2x+1OKl96sZaAM0CQYilbacsRCnwmJaoMPLscKOTaRyglxjzegjVrW9QLIaAg==","signature_status":"signed_v1","signed_at":"2026-08-07T01:41:26.855243Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.06344","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b6389a7472bc0cedab74b7b3a597dd66901708fdf00116b317039bc04a922f8d","sha256:6c06210134a1be7eb8c21ccfa6142315e4883bb89296c6658f7305227bb89716","sha256:32665059f83cbad7449a5fe78f7cdaf3f92146308f4eb8069d5639825fc5d660"],"state_sha256":"dac709d78c59ab3cdf95a1e8a6504d822da04f7226561708079ee3ccd31d674e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eStRyvkTIROiWULk3kf7Xb1X8cyhx32zDj6elRcR+ln60DIGIsKrU+rjlyxH2WW6Qjz/4iOMHOk5v6j5MmFQBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T21:04:49.157984Z","bundle_sha256":"99eece096605be04ddf2e08820b2cd5661fe86e892ba54259996cde9260974b7"}}