{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:5YFEDKSLB5YPSCDQLZT4DBEK3K","short_pith_number":"pith:5YFEDKSL","canonical_record":{"source":{"id":"1901.07712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-23T03:54:51Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"9276f167f41a57d5fa8121a735db2a24469ed5a00810c8ea98f3ff792f6eccea","abstract_canon_sha256":"a2bd5ec7aacbf0a8674e12446fbcbc06d3c5a3ce9dbb47cc514113740a179df8"},"schema_version":"1.0"},"canonical_sha256":"ee0a41aa4b0f70f908705e67c1848ada8ec60254af42ca5e667fd3a642b391a9","source":{"kind":"arxiv","id":"1901.07712","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.07712","created_at":"2026-05-17T23:55:41Z"},{"alias_kind":"arxiv_version","alias_value":"1901.07712v1","created_at":"2026-05-17T23:55:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.07712","created_at":"2026-05-17T23:55:41Z"},{"alias_kind":"pith_short_12","alias_value":"5YFEDKSLB5YP","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"5YFEDKSLB5YPSCDQ","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"5YFEDKSL","created_at":"2026-05-18T12:33:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:5YFEDKSLB5YPSCDQLZT4DBEK3K","target":"record","payload":{"canonical_record":{"source":{"id":"1901.07712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-23T03:54:51Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"9276f167f41a57d5fa8121a735db2a24469ed5a00810c8ea98f3ff792f6eccea","abstract_canon_sha256":"a2bd5ec7aacbf0a8674e12446fbcbc06d3c5a3ce9dbb47cc514113740a179df8"},"schema_version":"1.0"},"canonical_sha256":"ee0a41aa4b0f70f908705e67c1848ada8ec60254af42ca5e667fd3a642b391a9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:41.750094Z","signature_b64":"sQN+j3AMf73FQbDhy0E25Kf2tcUafQxHfoShM1v0Z+Ck3D/ScYh5NgiV9zyt69/SGimesx65UIFzKda6gHlpDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee0a41aa4b0f70f908705e67c1848ada8ec60254af42ca5e667fd3a642b391a9","last_reissued_at":"2026-05-17T23:55:41.749491Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:41.749491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.07712","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":"2026-05-17T23:55:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qf9qkcjBDCQFI+hPhe66u2XsXLLNE8R2Yv6r/0gbUQnMX5it06OqPs+jnrmjAW1CLmyi4OWt8YPxNAGMiPVFDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:26:15.972832Z"},"content_sha256":"6001c903711ce0aadbff5614029101717c5ab4a46d02fc3bfdd8fb26bc2d5f22","schema_version":"1.0","event_id":"sha256:6001c903711ce0aadbff5614029101717c5ab4a46d02fc3bfdd8fb26bc2d5f22"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:5YFEDKSLB5YPSCDQLZT4DBEK3K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discounted Gottschalk-Hedlund theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.DS","authors_text":"Philippe Thieullen, Xifeng SU","submitted_at":"2019-01-23T03:54:51Z","abstract_excerpt":"Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable cases, the observable is cohomologous to its ergodic minimizing value on the Mather set, and the discrete weak KAM solution plays the role of the transfer function. One possibility of construction of such a coboundary is by using the non linear Lax-Oleinik operator. The other possibility is by using a discounted cohomological equation. It is known that the disco"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.07712","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":""},"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":"2026-05-17T23:55:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SBZbmOI/jQk3Xk8jL/dkXFcxNUoj+UxkhTUO8KvD+2K6hIMT8Lvw4i5Ol3g0SfF6aXcCzcAKogEnxdGiWg2LDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:26:15.973545Z"},"content_sha256":"56d41018ebc7114378c7018743022f39f67058d969de5c4298eb5fb9ba439aa1","schema_version":"1.0","event_id":"sha256:56d41018ebc7114378c7018743022f39f67058d969de5c4298eb5fb9ba439aa1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5YFEDKSLB5YPSCDQLZT4DBEK3K/bundle.json","state_url":"https://pith.science/pith/5YFEDKSLB5YPSCDQLZT4DBEK3K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5YFEDKSLB5YPSCDQLZT4DBEK3K/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":"2026-05-30T15:26:15Z","links":{"resolver":"https://pith.science/pith/5YFEDKSLB5YPSCDQLZT4DBEK3K","bundle":"https://pith.science/pith/5YFEDKSLB5YPSCDQLZT4DBEK3K/bundle.json","state":"https://pith.science/pith/5YFEDKSLB5YPSCDQLZT4DBEK3K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5YFEDKSLB5YPSCDQLZT4DBEK3K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:5YFEDKSLB5YPSCDQLZT4DBEK3K","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":"a2bd5ec7aacbf0a8674e12446fbcbc06d3c5a3ce9dbb47cc514113740a179df8","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-23T03:54:51Z","title_canon_sha256":"9276f167f41a57d5fa8121a735db2a24469ed5a00810c8ea98f3ff792f6eccea"},"schema_version":"1.0","source":{"id":"1901.07712","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.07712","created_at":"2026-05-17T23:55:41Z"},{"alias_kind":"arxiv_version","alias_value":"1901.07712v1","created_at":"2026-05-17T23:55:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.07712","created_at":"2026-05-17T23:55:41Z"},{"alias_kind":"pith_short_12","alias_value":"5YFEDKSLB5YP","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"5YFEDKSLB5YPSCDQ","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"5YFEDKSL","created_at":"2026-05-18T12:33:10Z"}],"graph_snapshots":[{"event_id":"sha256:56d41018ebc7114378c7018743022f39f67058d969de5c4298eb5fb9ba439aa1","target":"graph","created_at":"2026-05-17T23:55:41Z","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"},"paper":{"abstract_excerpt":"Ergodic optimization and discrete weak KAM theory are two parallel theories with several results in common. For instance, the Mather set is the locus of orbits which minimize the ergodic averages of a given observable. In the favorable cases, the observable is cohomologous to its ergodic minimizing value on the Mather set, and the discrete weak KAM solution plays the role of the transfer function. One possibility of construction of such a coboundary is by using the non linear Lax-Oleinik operator. The other possibility is by using a discounted cohomological equation. It is known that the disco","authors_text":"Philippe Thieullen, Xifeng SU","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-23T03:54:51Z","title":"Discounted Gottschalk-Hedlund theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.07712","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":"sha256:6001c903711ce0aadbff5614029101717c5ab4a46d02fc3bfdd8fb26bc2d5f22","target":"record","created_at":"2026-05-17T23:55:41Z","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":"a2bd5ec7aacbf0a8674e12446fbcbc06d3c5a3ce9dbb47cc514113740a179df8","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-01-23T03:54:51Z","title_canon_sha256":"9276f167f41a57d5fa8121a735db2a24469ed5a00810c8ea98f3ff792f6eccea"},"schema_version":"1.0","source":{"id":"1901.07712","kind":"arxiv","version":1}},"canonical_sha256":"ee0a41aa4b0f70f908705e67c1848ada8ec60254af42ca5e667fd3a642b391a9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee0a41aa4b0f70f908705e67c1848ada8ec60254af42ca5e667fd3a642b391a9","first_computed_at":"2026-05-17T23:55:41.749491Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:55:41.749491Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sQN+j3AMf73FQbDhy0E25Kf2tcUafQxHfoShM1v0Z+Ck3D/ScYh5NgiV9zyt69/SGimesx65UIFzKda6gHlpDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:55:41.750094Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.07712","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6001c903711ce0aadbff5614029101717c5ab4a46d02fc3bfdd8fb26bc2d5f22","sha256:56d41018ebc7114378c7018743022f39f67058d969de5c4298eb5fb9ba439aa1"],"state_sha256":"4dcb36ba6a63568b118ab52144c378c5afd75fc98076e1d0d6ae69eb9f6d1d88"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SbFCtjLT/IQ7kb4pIeNel8PGkMoAb+6dU9HAnpUwqeosxKMaK2Li93gH/Ik8gyPtJ6vTLJBBcUUKxayWMv52AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T15:26:15.977158Z","bundle_sha256":"299c2396c10aa144c96cc8bd1108aa6d6d5e36f54d03f02df00b805542aacda2"}}