{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:ACVHGPZYUWE6OKWHUMGL4I7IFP","short_pith_number":"pith:ACVHGPZY","canonical_record":{"source":{"id":"2606.11193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-03-27T18:19:12Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"3ebfcc8d6bf2938db495d10c1cfb81ba6329a4c44bd3a1c17a6af922809c5cf7","abstract_canon_sha256":"faedfd8090b556ce9f67a3d8f9e0bc239eb41b8d956c738490d3c58bf88fab42"},"schema_version":"1.0"},"canonical_sha256":"00aa733f38a589e72ac7a30cbe23e82bc9a6ca5535152138f5fbcc41b64fbfa0","source":{"kind":"arxiv","id":"2606.11193","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11193","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11193v1","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11193","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"pith_short_12","alias_value":"ACVHGPZYUWE6","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"pith_short_16","alias_value":"ACVHGPZYUWE6OKWH","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"pith_short_8","alias_value":"ACVHGPZY","created_at":"2026-06-11T00:08:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:ACVHGPZYUWE6OKWHUMGL4I7IFP","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-03-27T18:19:12Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"3ebfcc8d6bf2938db495d10c1cfb81ba6329a4c44bd3a1c17a6af922809c5cf7","abstract_canon_sha256":"faedfd8090b556ce9f67a3d8f9e0bc239eb41b8d956c738490d3c58bf88fab42"},"schema_version":"1.0"},"canonical_sha256":"00aa733f38a589e72ac7a30cbe23e82bc9a6ca5535152138f5fbcc41b64fbfa0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-11T00:08:05.680743Z","signature_b64":"ET4AHhx0fcaIH+hetAEB29t9aoHcViF0HIwpgyJc+75Pv6xCGMIgOx5YID+HbEcaDVP6GmeZaYvlX6TblmPtCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00aa733f38a589e72ac7a30cbe23e82bc9a6ca5535152138f5fbcc41b64fbfa0","last_reissued_at":"2026-06-11T00:08:05.679663Z","signature_status":"signed_v1","first_computed_at":"2026-06-11T00:08:05.679663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11193","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":"2026-06-11T00:08:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+dQfiVZm4vYmt75dHBMGaJ7iTegx5QBucRkDRgumqqhC3PBncyE/puYqJthkTRIRagoqPwyxoMA6eOTpMymXDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T12:13:10.143429Z"},"content_sha256":"1946ccd4d3839323fde7f7400a733a3322397919147e08e833eacb748f19b51a","schema_version":"1.0","event_id":"sha256:1946ccd4d3839323fde7f7400a733a3322397919147e08e833eacb748f19b51a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:ACVHGPZYUWE6OKWHUMGL4I7IFP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Approximation Properties of Evolutionary Dynamics in Continuous-Time Finite State Space Games","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cs.GT","authors_text":"Pietro Grassi","submitted_at":"2026-03-27T18:19:12Z","abstract_excerpt":"This thesis studies the convergence of finite-population stochastic evolutionary dynamics to their deterministic mean-field limit in continuous-time finite state space games. We first develop refined ergodic theorems for Markov chains with a single positive-recurrent class, guaranteeing the existence of a unique invariant distribution and almost-sure convergence of time averages. Next, we prove that the mean-field model, described by a system of Lipschitz-continuous ordinary differential equations, admits a unique solution that depends continuously on its initial condition and that constitutes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11193","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/2606.11193/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":"2026-06-11T00:08:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oySEJSXQCW0Ve1sBAT51Xig95K1tmKBRI3X6r5L8f8jp/5vSuSwYjqp3fExnZhvz/iGdKoV9828HW+apK39vBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T12:13:10.143804Z"},"content_sha256":"eac8c2774e3f72bde9665a273c179315bc03b1318d358c2f895efd1e880998fc","schema_version":"1.0","event_id":"sha256:eac8c2774e3f72bde9665a273c179315bc03b1318d358c2f895efd1e880998fc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ACVHGPZYUWE6OKWHUMGL4I7IFP/bundle.json","state_url":"https://pith.science/pith/ACVHGPZYUWE6OKWHUMGL4I7IFP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ACVHGPZYUWE6OKWHUMGL4I7IFP/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":"2026-06-11T12:13:10Z","links":{"resolver":"https://pith.science/pith/ACVHGPZYUWE6OKWHUMGL4I7IFP","bundle":"https://pith.science/pith/ACVHGPZYUWE6OKWHUMGL4I7IFP/bundle.json","state":"https://pith.science/pith/ACVHGPZYUWE6OKWHUMGL4I7IFP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ACVHGPZYUWE6OKWHUMGL4I7IFP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:ACVHGPZYUWE6OKWHUMGL4I7IFP","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":"faedfd8090b556ce9f67a3d8f9e0bc239eb41b8d956c738490d3c58bf88fab42","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-03-27T18:19:12Z","title_canon_sha256":"3ebfcc8d6bf2938db495d10c1cfb81ba6329a4c44bd3a1c17a6af922809c5cf7"},"schema_version":"1.0","source":{"id":"2606.11193","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11193","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11193v1","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11193","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"pith_short_12","alias_value":"ACVHGPZYUWE6","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"pith_short_16","alias_value":"ACVHGPZYUWE6OKWH","created_at":"2026-06-11T00:08:05Z"},{"alias_kind":"pith_short_8","alias_value":"ACVHGPZY","created_at":"2026-06-11T00:08:05Z"}],"graph_snapshots":[{"event_id":"sha256:eac8c2774e3f72bde9665a273c179315bc03b1318d358c2f895efd1e880998fc","target":"graph","created_at":"2026-06-11T00:08:05Z","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/2606.11193/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This thesis studies the convergence of finite-population stochastic evolutionary dynamics to their deterministic mean-field limit in continuous-time finite state space games. We first develop refined ergodic theorems for Markov chains with a single positive-recurrent class, guaranteeing the existence of a unique invariant distribution and almost-sure convergence of time averages. Next, we prove that the mean-field model, described by a system of Lipschitz-continuous ordinary differential equations, admits a unique solution that depends continuously on its initial condition and that constitutes","authors_text":"Pietro Grassi","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-03-27T18:19:12Z","title":"Approximation Properties of Evolutionary Dynamics in Continuous-Time Finite State Space Games"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11193","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":"sha256:1946ccd4d3839323fde7f7400a733a3322397919147e08e833eacb748f19b51a","target":"record","created_at":"2026-06-11T00:08:05Z","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":"faedfd8090b556ce9f67a3d8f9e0bc239eb41b8d956c738490d3c58bf88fab42","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2026-03-27T18:19:12Z","title_canon_sha256":"3ebfcc8d6bf2938db495d10c1cfb81ba6329a4c44bd3a1c17a6af922809c5cf7"},"schema_version":"1.0","source":{"id":"2606.11193","kind":"arxiv","version":1}},"canonical_sha256":"00aa733f38a589e72ac7a30cbe23e82bc9a6ca5535152138f5fbcc41b64fbfa0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"00aa733f38a589e72ac7a30cbe23e82bc9a6ca5535152138f5fbcc41b64fbfa0","first_computed_at":"2026-06-11T00:08:05.679663Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T00:08:05.679663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ET4AHhx0fcaIH+hetAEB29t9aoHcViF0HIwpgyJc+75Pv6xCGMIgOx5YID+HbEcaDVP6GmeZaYvlX6TblmPtCQ==","signature_status":"signed_v1","signed_at":"2026-06-11T00:08:05.680743Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11193","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1946ccd4d3839323fde7f7400a733a3322397919147e08e833eacb748f19b51a","sha256:eac8c2774e3f72bde9665a273c179315bc03b1318d358c2f895efd1e880998fc"],"state_sha256":"ae8bf4859abd65eacf17c285ec9659291559646afa01ecf4dd5f6d20fe5294a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xe5AxvzfkNw4e74ku40DnDnE9RDOcPhH0SYkWki0kRpOJzwyECkrHFZWjz+2p4Adfo3x9Wrus0vpSD7qW5sEDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T12:13:10.145845Z","bundle_sha256":"d837188bb58a29ebd2ff679cfe6696d5415d380c95e97d29c66c193d302d5241"}}