{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:LRXQO7LVR2QBFHS7DWJL42YISM","short_pith_number":"pith:LRXQO7LV","canonical_record":{"source":{"id":"2607.20400","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-22T17:35:47Z","cross_cats_sorted":["math.AP","math.MP","math.PR"],"title_canon_sha256":"6d685c850d683e82b2674e0485f28d1ffbb1c19bc9d90ca3d63944434df05093","abstract_canon_sha256":"5503d1b7aed8d3fe7bc5022f5018e91760b69a032881f0a1e94d0d98c291f152"},"schema_version":"1.0"},"canonical_sha256":"5c6f077d758ea0129e5f1d92be6b08932d5d501e40e58a5716549dcd59e43f22","source":{"kind":"arxiv","id":"2607.20400","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.20400","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"arxiv_version","alias_value":"2607.20400v1","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20400","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"pith_short_12","alias_value":"LRXQO7LVR2QB","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"pith_short_16","alias_value":"LRXQO7LVR2QBFHS7","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"pith_short_8","alias_value":"LRXQO7LV","created_at":"2026-07-23T01:25:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:LRXQO7LVR2QBFHS7DWJL42YISM","target":"record","payload":{"canonical_record":{"source":{"id":"2607.20400","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-22T17:35:47Z","cross_cats_sorted":["math.AP","math.MP","math.PR"],"title_canon_sha256":"6d685c850d683e82b2674e0485f28d1ffbb1c19bc9d90ca3d63944434df05093","abstract_canon_sha256":"5503d1b7aed8d3fe7bc5022f5018e91760b69a032881f0a1e94d0d98c291f152"},"schema_version":"1.0"},"canonical_sha256":"5c6f077d758ea0129e5f1d92be6b08932d5d501e40e58a5716549dcd59e43f22","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-23T01:25:19.107261Z","signature_b64":"Qlgyrc06fTkCUHural5qzrV7pkHZjMJtiBHNUrEDQ0Apc5J5K1JrQh2+ZfmpeLw0da0i0NQgpUXt/TUXcqxCBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c6f077d758ea0129e5f1d92be6b08932d5d501e40e58a5716549dcd59e43f22","last_reissued_at":"2026-07-23T01:25:19.106403Z","signature_status":"signed_v1","first_computed_at":"2026-07-23T01:25:19.106403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.20400","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":"2026-07-23T01:25:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wA9S6Pa6QwQHyIrk2s2/CxXzhxNgdn5PizC9xu9kYZ8o99HA6f0R5EpBLRxGjdVuRUBjO70FRpby83JRBJpzAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T22:14:22.243903Z"},"content_sha256":"6d6ed9f91b4cd6728daf770ee4269550f9e5c33e11d06c03480e941ea9e58a03","schema_version":"1.0","event_id":"sha256:6d6ed9f91b4cd6728daf770ee4269550f9e5c33e11d06c03480e941ea9e58a03"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:LRXQO7LVR2QBFHS7DWJL42YISM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A perturbative microscopic derivation of the focusing $\\Phi^6_1$ measure with rough cut-off","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Andrew Rout, Shahnaz Farhat","submitted_at":"2026-07-22T17:35:47Z","abstract_excerpt":"We give a derivation of the Gibbs measure for the focusing nonlinear Schr\\\"odinger equation (NLS) on the circle with rough cut-off. This extends earlier work by Sohinger and the second author, which proved analogous results for smooth cut-offs. Our proof is based on the perturbative expansion developed by Fr\\\"ohlich, Knowles, Schlein, and Sohinger (2017), and provides an alternative proof of the recent derivation given in L\\\"u, Nam, and Zhu (2026). To prove convergence of the explicit terms, we employ a Wigner measure approach and an inductive argument to overcome the lack of smoothness for th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20400","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/2607.20400/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":"2026-07-23T01:25:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QNH/Sx8ZzzCtRfKLbhSktfaXm2sEGTTKRO+Lm4D+uCRCnNxC8ZdERj1BedyOvvFWR/nS+VJIWOVk3uZ9ROcGBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-10T22:14:22.244424Z"},"content_sha256":"ff25db317799e4e298386b9dff09b98b1de2aab073084c90591c4ae7430f8406","schema_version":"1.0","event_id":"sha256:ff25db317799e4e298386b9dff09b98b1de2aab073084c90591c4ae7430f8406"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LRXQO7LVR2QBFHS7DWJL42YISM/bundle.json","state_url":"https://pith.science/pith/LRXQO7LVR2QBFHS7DWJL42YISM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LRXQO7LVR2QBFHS7DWJL42YISM/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":"2026-08-10T22:14:22Z","links":{"resolver":"https://pith.science/pith/LRXQO7LVR2QBFHS7DWJL42YISM","bundle":"https://pith.science/pith/LRXQO7LVR2QBFHS7DWJL42YISM/bundle.json","state":"https://pith.science/pith/LRXQO7LVR2QBFHS7DWJL42YISM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LRXQO7LVR2QBFHS7DWJL42YISM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:LRXQO7LVR2QBFHS7DWJL42YISM","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":"5503d1b7aed8d3fe7bc5022f5018e91760b69a032881f0a1e94d0d98c291f152","cross_cats_sorted":["math.AP","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-22T17:35:47Z","title_canon_sha256":"6d685c850d683e82b2674e0485f28d1ffbb1c19bc9d90ca3d63944434df05093"},"schema_version":"1.0","source":{"id":"2607.20400","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.20400","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"arxiv_version","alias_value":"2607.20400v1","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.20400","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"pith_short_12","alias_value":"LRXQO7LVR2QB","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"pith_short_16","alias_value":"LRXQO7LVR2QBFHS7","created_at":"2026-07-23T01:25:19Z"},{"alias_kind":"pith_short_8","alias_value":"LRXQO7LV","created_at":"2026-07-23T01:25:19Z"}],"graph_snapshots":[{"event_id":"sha256:ff25db317799e4e298386b9dff09b98b1de2aab073084c90591c4ae7430f8406","target":"graph","created_at":"2026-07-23T01:25:19Z","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/2607.20400/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a derivation of the Gibbs measure for the focusing nonlinear Schr\\\"odinger equation (NLS) on the circle with rough cut-off. This extends earlier work by Sohinger and the second author, which proved analogous results for smooth cut-offs. Our proof is based on the perturbative expansion developed by Fr\\\"ohlich, Knowles, Schlein, and Sohinger (2017), and provides an alternative proof of the recent derivation given in L\\\"u, Nam, and Zhu (2026). To prove convergence of the explicit terms, we employ a Wigner measure approach and an inductive argument to overcome the lack of smoothness for th","authors_text":"Andrew Rout, Shahnaz Farhat","cross_cats":["math.AP","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-22T17:35:47Z","title":"A perturbative microscopic derivation of the focusing $\\Phi^6_1$ measure with rough cut-off"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.20400","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":"sha256:6d6ed9f91b4cd6728daf770ee4269550f9e5c33e11d06c03480e941ea9e58a03","target":"record","created_at":"2026-07-23T01:25:19Z","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":"5503d1b7aed8d3fe7bc5022f5018e91760b69a032881f0a1e94d0d98c291f152","cross_cats_sorted":["math.AP","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2026-07-22T17:35:47Z","title_canon_sha256":"6d685c850d683e82b2674e0485f28d1ffbb1c19bc9d90ca3d63944434df05093"},"schema_version":"1.0","source":{"id":"2607.20400","kind":"arxiv","version":1}},"canonical_sha256":"5c6f077d758ea0129e5f1d92be6b08932d5d501e40e58a5716549dcd59e43f22","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5c6f077d758ea0129e5f1d92be6b08932d5d501e40e58a5716549dcd59e43f22","first_computed_at":"2026-07-23T01:25:19.106403Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-23T01:25:19.106403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Qlgyrc06fTkCUHural5qzrV7pkHZjMJtiBHNUrEDQ0Apc5J5K1JrQh2+ZfmpeLw0da0i0NQgpUXt/TUXcqxCBA==","signature_status":"signed_v1","signed_at":"2026-07-23T01:25:19.107261Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.20400","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d6ed9f91b4cd6728daf770ee4269550f9e5c33e11d06c03480e941ea9e58a03","sha256:ff25db317799e4e298386b9dff09b98b1de2aab073084c90591c4ae7430f8406"],"state_sha256":"ac99ead61bd040364d03e9c6766b40647f08c4dbeff72f71d437e5d2a6b12dab"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B00C6k9lg+oC920ve8pxLek4u15/x/2OWHuIdneeDsLeUIry7PoB74a10/eB4VVOTx6akMhbCF7FkVWDuCFXCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-10T22:14:22.249288Z","bundle_sha256":"9fc8b8067df5394f4cae856e341cadefb0888b323c6c7d6f43fb3ff19fa812e0"}}