{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:PJBCD7PARY2I4N4Y6F7C7QT5KH","short_pith_number":"pith:PJBCD7PA","canonical_record":{"source":{"id":"2104.14348","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-04-29T13:57:00Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"9530aed20926f70c9c93c32596cc39b69a0e6e54e278997b523623012ca73af3","abstract_canon_sha256":"b4e19ecaee92404e38b418e8ea8a65f27e385c10d799f0c1cb55028a9956f1b7"},"schema_version":"1.0"},"canonical_sha256":"7a4221fde08e348e3798f17e2fc27d51ee871d115da428625f1c32cf4db32555","source":{"kind":"arxiv","id":"2104.14348","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2104.14348","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"arxiv_version","alias_value":"2104.14348v1","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.14348","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"pith_short_12","alias_value":"PJBCD7PARY2I","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"pith_short_16","alias_value":"PJBCD7PARY2I4N4Y","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"pith_short_8","alias_value":"PJBCD7PA","created_at":"2026-07-05T02:36:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:PJBCD7PARY2I4N4Y6F7C7QT5KH","target":"record","payload":{"canonical_record":{"source":{"id":"2104.14348","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-04-29T13:57:00Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"9530aed20926f70c9c93c32596cc39b69a0e6e54e278997b523623012ca73af3","abstract_canon_sha256":"b4e19ecaee92404e38b418e8ea8a65f27e385c10d799f0c1cb55028a9956f1b7"},"schema_version":"1.0"},"canonical_sha256":"7a4221fde08e348e3798f17e2fc27d51ee871d115da428625f1c32cf4db32555","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:36:15.605278Z","signature_b64":"Bnj+avRh5q5EfrIDqt7oAlr69vQEKlod1n8GOkQCW/YeXDjnFoJeWF8AiJ8C9+QZK08I+oRKMFWCJ8XEHi7yDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a4221fde08e348e3798f17e2fc27d51ee871d115da428625f1c32cf4db32555","last_reissued_at":"2026-07-05T02:36:15.604883Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:36:15.604883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2104.14348","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":"2026-07-05T02:36:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"h8qY+JWfbGKlhs77VFPzXvI7okC+/YEfmgdgqxQmOU550rzbTULehW4MVaOfdjpfd2HbU45k0/AEbVJnILWkDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:48:25.313612Z"},"content_sha256":"088bfb83599743f645e700789d80cdf1a253e204e77bacf00e3969e550582a45","schema_version":"1.0","event_id":"sha256:088bfb83599743f645e700789d80cdf1a253e204e77bacf00e3969e550582a45"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:PJBCD7PARY2I4N4Y6F7C7QT5KH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant Gibbs measure for a Schrodinger equation with exponential nonlinearity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.AP","authors_text":"Tristan Robert","submitted_at":"2021-04-29T13:57:00Z","abstract_excerpt":"We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $i\\partial_t u + (-\\Delta)^{\\frac{\\alpha}2} u = 2\\gamma\\beta e^{\\beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $\\mathcal{M}$, for a dispersion parameter $\\alpha>d$, some coupling constant $\\beta>0$, and $\\gamma\\neq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $\\gamma>0$, the measure is well-defined in the whole regime $\\alpha>d$ and $\\beta>0$ (Theorem 1.1 (i)), while in the focusing case $\\gamma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.14348","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/2104.14348/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":"2026-07-05T02:36:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R+pigsmaAefBAvzYW1Le7ZctAg77yCMMuKdJzJoHYobb8Z+XSOZeFLO7vpxodq5iexq9GkNTy10sRgVb9qhEDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:48:25.314542Z"},"content_sha256":"7a7d3b001eeef9aad7e8ed8c0eeb1b3f6ce6c22dcb76eb520a6c9e3210a1b290","schema_version":"1.0","event_id":"sha256:7a7d3b001eeef9aad7e8ed8c0eeb1b3f6ce6c22dcb76eb520a6c9e3210a1b290"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PJBCD7PARY2I4N4Y6F7C7QT5KH/bundle.json","state_url":"https://pith.science/pith/PJBCD7PARY2I4N4Y6F7C7QT5KH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PJBCD7PARY2I4N4Y6F7C7QT5KH/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":"2026-08-16T13:48:25Z","links":{"resolver":"https://pith.science/pith/PJBCD7PARY2I4N4Y6F7C7QT5KH","bundle":"https://pith.science/pith/PJBCD7PARY2I4N4Y6F7C7QT5KH/bundle.json","state":"https://pith.science/pith/PJBCD7PARY2I4N4Y6F7C7QT5KH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PJBCD7PARY2I4N4Y6F7C7QT5KH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:PJBCD7PARY2I4N4Y6F7C7QT5KH","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":"b4e19ecaee92404e38b418e8ea8a65f27e385c10d799f0c1cb55028a9956f1b7","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-04-29T13:57:00Z","title_canon_sha256":"9530aed20926f70c9c93c32596cc39b69a0e6e54e278997b523623012ca73af3"},"schema_version":"1.0","source":{"id":"2104.14348","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2104.14348","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"arxiv_version","alias_value":"2104.14348v1","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.14348","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"pith_short_12","alias_value":"PJBCD7PARY2I","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"pith_short_16","alias_value":"PJBCD7PARY2I4N4Y","created_at":"2026-07-05T02:36:15Z"},{"alias_kind":"pith_short_8","alias_value":"PJBCD7PA","created_at":"2026-07-05T02:36:15Z"}],"graph_snapshots":[{"event_id":"sha256:7a7d3b001eeef9aad7e8ed8c0eeb1b3f6ce6c22dcb76eb520a6c9e3210a1b290","target":"graph","created_at":"2026-07-05T02:36:15Z","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/2104.14348/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $i\\partial_t u + (-\\Delta)^{\\frac{\\alpha}2} u = 2\\gamma\\beta e^{\\beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $\\mathcal{M}$, for a dispersion parameter $\\alpha>d$, some coupling constant $\\beta>0$, and $\\gamma\\neq 0$. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case $\\gamma>0$, the measure is well-defined in the whole regime $\\alpha>d$ and $\\beta>0$ (Theorem 1.1 (i)), while in the focusing case $\\gamma","authors_text":"Tristan Robert","cross_cats":["math.PR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-04-29T13:57:00Z","title":"Invariant Gibbs measure for a Schrodinger equation with exponential nonlinearity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.14348","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":"sha256:088bfb83599743f645e700789d80cdf1a253e204e77bacf00e3969e550582a45","target":"record","created_at":"2026-07-05T02:36:15Z","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":"b4e19ecaee92404e38b418e8ea8a65f27e385c10d799f0c1cb55028a9956f1b7","cross_cats_sorted":["math.PR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2021-04-29T13:57:00Z","title_canon_sha256":"9530aed20926f70c9c93c32596cc39b69a0e6e54e278997b523623012ca73af3"},"schema_version":"1.0","source":{"id":"2104.14348","kind":"arxiv","version":1}},"canonical_sha256":"7a4221fde08e348e3798f17e2fc27d51ee871d115da428625f1c32cf4db32555","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a4221fde08e348e3798f17e2fc27d51ee871d115da428625f1c32cf4db32555","first_computed_at":"2026-07-05T02:36:15.604883Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:36:15.604883Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bnj+avRh5q5EfrIDqt7oAlr69vQEKlod1n8GOkQCW/YeXDjnFoJeWF8AiJ8C9+QZK08I+oRKMFWCJ8XEHi7yDA==","signature_status":"signed_v1","signed_at":"2026-07-05T02:36:15.605278Z","signed_message":"canonical_sha256_bytes"},"source_id":"2104.14348","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:088bfb83599743f645e700789d80cdf1a253e204e77bacf00e3969e550582a45","sha256:7a7d3b001eeef9aad7e8ed8c0eeb1b3f6ce6c22dcb76eb520a6c9e3210a1b290"],"state_sha256":"808807ade42ce3ac2cc255016c961707964c55872265c0d5fe556cd3728cd473"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KHyof4diROR+cJ9y5DzzGWPJE4rxizWDzn65gUhDq/wojYeg6rWVjHUqb3/QlkHof5J2zCCUEyuppV9Fo6NVDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T13:48:25.320400Z","bundle_sha256":"12756f7dc978b52ebeb2f156369b4bb60742cdc02e97dbd713a3603d3db3ba56"}}