{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:U3MEX5HP33DRLBX76TL75UNR5W","short_pith_number":"pith:U3MEX5HP","canonical_record":{"source":{"id":"2607.28660","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-21T13:23:40Z","cross_cats_sorted":["math.DG","math.FA","math.MP"],"title_canon_sha256":"bff39c3ee2b5af4b66147b19281625d2fad4ec54dbd7ec31c8d3f1c693eec90f","abstract_canon_sha256":"bc81f4db7771ffa7fd7fb0842e364e4e0f2c38b455e8107df2a35a1721f44053"},"schema_version":"1.0"},"canonical_sha256":"a6d84bf4efdec71586fff4d7fed1b1ed92b0989460f2d29f6faea1b49db37040","source":{"kind":"arxiv","id":"2607.28660","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.28660","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"arxiv_version","alias_value":"2607.28660v1","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28660","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"pith_short_12","alias_value":"U3MEX5HP33DR","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"pith_short_16","alias_value":"U3MEX5HP33DRLBX7","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"pith_short_8","alias_value":"U3MEX5HP","created_at":"2026-08-03T00:11:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:U3MEX5HP33DRLBX76TL75UNR5W","target":"record","payload":{"canonical_record":{"source":{"id":"2607.28660","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-21T13:23:40Z","cross_cats_sorted":["math.DG","math.FA","math.MP"],"title_canon_sha256":"bff39c3ee2b5af4b66147b19281625d2fad4ec54dbd7ec31c8d3f1c693eec90f","abstract_canon_sha256":"bc81f4db7771ffa7fd7fb0842e364e4e0f2c38b455e8107df2a35a1721f44053"},"schema_version":"1.0"},"canonical_sha256":"a6d84bf4efdec71586fff4d7fed1b1ed92b0989460f2d29f6faea1b49db37040","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-03T00:11:56.486314Z","signature_b64":"oyaSJz6+k0R9ILvov/y9g7Fd5NLZquStALiSPFnhIW8vWBSQY5GxBa+VYvgc/eH+iRPYofk7FyR6/3dGeOKuCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a6d84bf4efdec71586fff4d7fed1b1ed92b0989460f2d29f6faea1b49db37040","last_reissued_at":"2026-08-03T00:11:56.483639Z","signature_status":"signed_v1","first_computed_at":"2026-08-03T00:11:56.483639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.28660","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":"2026-08-03T00:11:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qZbG7YRXqbLmWhg13s4nlhnKqJLXICZeBkD9lSoBGeZS33Y9DcwzYMpNGu7W7/DSzN1RKMaBsLiBX/JKJ+PFBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T10:12:11.875540Z"},"content_sha256":"a9b843ba20f32bf3a7e409c48e94e165497f74ba6872d52a7a0a6e51a97b48f6","schema_version":"1.0","event_id":"sha256:a9b843ba20f32bf3a7e409c48e94e165497f74ba6872d52a7a0a6e51a97b48f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:U3MEX5HP33DRLBX76TL75UNR5W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropy Geometry and Normalized Means on Infinite-Dimensional Hamiltonian Manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG","math.FA","math.MP"],"primary_cat":"math-ph","authors_text":"Jean-Pierre Magnot","submitted_at":"2026-07-21T13:23:40Z","abstract_excerpt":"We propose a geometric--analytic framework for equilibrium statistical mechanics on infinite-dimensional Hamiltonian systems. In situations where no suitable $\\sigma$-additive invariant measure is available, we use \\emph{normalized means}, which generalize probability measures and normalized integrals. This construction yields entropy and free-energy functionals on weak symplectic Fr\\'echet manifolds and gives existence and uniqueness of exponential-family equilibrium states under explicit admissibility and separation assumptions. These states are stationary under Hamiltonian flows preserving "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28660","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/2607.28660/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":"2026-08-03T00:11:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nMnubF/HdaQkhE4EurKEyh39xWV6PPfZBg/pDtb4gToS6MecNDTqs+emN2lBCAz+wxu7jUeSYdEFyxT7Q/cwDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T10:12:11.876084Z"},"content_sha256":"89c2f2c75b40682312442561ed6c2215c43b1ff2a1355087d7848e3b633c9251","schema_version":"1.0","event_id":"sha256:89c2f2c75b40682312442561ed6c2215c43b1ff2a1355087d7848e3b633c9251"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U3MEX5HP33DRLBX76TL75UNR5W/bundle.json","state_url":"https://pith.science/pith/U3MEX5HP33DRLBX76TL75UNR5W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U3MEX5HP33DRLBX76TL75UNR5W/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":"2026-08-13T10:12:11Z","links":{"resolver":"https://pith.science/pith/U3MEX5HP33DRLBX76TL75UNR5W","bundle":"https://pith.science/pith/U3MEX5HP33DRLBX76TL75UNR5W/bundle.json","state":"https://pith.science/pith/U3MEX5HP33DRLBX76TL75UNR5W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U3MEX5HP33DRLBX76TL75UNR5W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:U3MEX5HP33DRLBX76TL75UNR5W","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":"bc81f4db7771ffa7fd7fb0842e364e4e0f2c38b455e8107df2a35a1721f44053","cross_cats_sorted":["math.DG","math.FA","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-21T13:23:40Z","title_canon_sha256":"bff39c3ee2b5af4b66147b19281625d2fad4ec54dbd7ec31c8d3f1c693eec90f"},"schema_version":"1.0","source":{"id":"2607.28660","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.28660","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"arxiv_version","alias_value":"2607.28660v1","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.28660","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"pith_short_12","alias_value":"U3MEX5HP33DR","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"pith_short_16","alias_value":"U3MEX5HP33DRLBX7","created_at":"2026-08-03T00:11:56Z"},{"alias_kind":"pith_short_8","alias_value":"U3MEX5HP","created_at":"2026-08-03T00:11:56Z"}],"graph_snapshots":[{"event_id":"sha256:89c2f2c75b40682312442561ed6c2215c43b1ff2a1355087d7848e3b633c9251","target":"graph","created_at":"2026-08-03T00:11:56Z","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/2607.28660/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We propose a geometric--analytic framework for equilibrium statistical mechanics on infinite-dimensional Hamiltonian systems. In situations where no suitable $\\sigma$-additive invariant measure is available, we use \\emph{normalized means}, which generalize probability measures and normalized integrals. This construction yields entropy and free-energy functionals on weak symplectic Fr\\'echet manifolds and gives existence and uniqueness of exponential-family equilibrium states under explicit admissibility and separation assumptions. These states are stationary under Hamiltonian flows preserving ","authors_text":"Jean-Pierre Magnot","cross_cats":["math.DG","math.FA","math.MP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-21T13:23:40Z","title":"Entropy Geometry and Normalized Means on Infinite-Dimensional Hamiltonian Manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28660","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:a9b843ba20f32bf3a7e409c48e94e165497f74ba6872d52a7a0a6e51a97b48f6","target":"record","created_at":"2026-08-03T00:11:56Z","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":"bc81f4db7771ffa7fd7fb0842e364e4e0f2c38b455e8107df2a35a1721f44053","cross_cats_sorted":["math.DG","math.FA","math.MP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math-ph","submitted_at":"2026-07-21T13:23:40Z","title_canon_sha256":"bff39c3ee2b5af4b66147b19281625d2fad4ec54dbd7ec31c8d3f1c693eec90f"},"schema_version":"1.0","source":{"id":"2607.28660","kind":"arxiv","version":1}},"canonical_sha256":"a6d84bf4efdec71586fff4d7fed1b1ed92b0989460f2d29f6faea1b49db37040","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a6d84bf4efdec71586fff4d7fed1b1ed92b0989460f2d29f6faea1b49db37040","first_computed_at":"2026-08-03T00:11:56.483639Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-03T00:11:56.483639Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oyaSJz6+k0R9ILvov/y9g7Fd5NLZquStALiSPFnhIW8vWBSQY5GxBa+VYvgc/eH+iRPYofk7FyR6/3dGeOKuCA==","signature_status":"signed_v1","signed_at":"2026-08-03T00:11:56.486314Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.28660","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a9b843ba20f32bf3a7e409c48e94e165497f74ba6872d52a7a0a6e51a97b48f6","sha256:89c2f2c75b40682312442561ed6c2215c43b1ff2a1355087d7848e3b633c9251"],"state_sha256":"c650f46114741020a870a23f040c346c14f65afa03a05c8fb2ebdad9addf4fa4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q9YSKDh2NBsBzbxAFrY/Po4qGp9PUaFgrqY2exU95aXzpGzSCQY3JOeXvUhh+tlMygzfaxq5iNeq6O5AzBYuCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T10:12:11.961409Z","bundle_sha256":"e6097d8cf57f645a678597b7b260670796d5b39a086570e83716c266bac6ff47"}}