{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:32CJA6SNP5TR5DHV2KZCCAZQRW","short_pith_number":"pith:32CJA6SN","canonical_record":{"source":{"id":"2411.17147","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-26T06:30:12Z","cross_cats_sorted":[],"title_canon_sha256":"2402228f0d998416f10f42f79d31e93620edc08ffa486fe2d8d3e1eceda1b79d","abstract_canon_sha256":"835aacee676d3a01c69a8c96ede73b3efd600af2112b4ab5c0f12a995f8e8cdf"},"schema_version":"1.0"},"canonical_sha256":"de84907a4d7f671e8cf5d2b22103308d8605580c4c556fcfe8608f627d41a98b","source":{"kind":"arxiv","id":"2411.17147","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.17147","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"arxiv_version","alias_value":"2411.17147v1","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.17147","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"pith_short_12","alias_value":"32CJA6SNP5TR","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"pith_short_16","alias_value":"32CJA6SNP5TR5DHV","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"pith_short_8","alias_value":"32CJA6SN","created_at":"2026-07-05T09:40:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:32CJA6SNP5TR5DHV2KZCCAZQRW","target":"record","payload":{"canonical_record":{"source":{"id":"2411.17147","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-26T06:30:12Z","cross_cats_sorted":[],"title_canon_sha256":"2402228f0d998416f10f42f79d31e93620edc08ffa486fe2d8d3e1eceda1b79d","abstract_canon_sha256":"835aacee676d3a01c69a8c96ede73b3efd600af2112b4ab5c0f12a995f8e8cdf"},"schema_version":"1.0"},"canonical_sha256":"de84907a4d7f671e8cf5d2b22103308d8605580c4c556fcfe8608f627d41a98b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:40:28.333337Z","signature_b64":"p/4G2+1OqFCQgbAK8qB43L0+Pbh1p2lVvH0xbEjg2509IjAsaoO9tX7IYTRhsY4x53L+J1cTKPcUacnbcRbWDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de84907a4d7f671e8cf5d2b22103308d8605580c4c556fcfe8608f627d41a98b","last_reissued_at":"2026-07-05T09:40:28.332847Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:40:28.332847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.17147","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":"2026-07-05T09:40:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LizOHZr10Xim6S7IlInx9P4+VcoetuAPHOULMYnFST83BokkbIcizBB4HHXNj+icCTK8zDJZNQ/tQU6DR+gpDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T06:39:57.334191Z"},"content_sha256":"9745991664fc06e730858ef40623a7c3147e122b5dd103b599ab88ada76dee57","schema_version":"1.0","event_id":"sha256:9745991664fc06e730858ef40623a7c3147e122b5dd103b599ab88ada76dee57"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:32CJA6SNP5TR5DHV2KZCCAZQRW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical local well-posedness of the nonlinear Schr\\\"odinger equation on the torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Beomjong Kwak, Soonsik Kwon","submitted_at":"2024-11-26T06:30:12Z","abstract_excerpt":"In this paper, we study the local well-posedness of nonlinear Schr\\\"odinger equations on tori $\\mathbb{T}^{d}$ at the critical regularity. We focus on cases where the nonlinearity $|u|^{a}u$ is non-algebraic with small $a>0$. We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering $a$, we prove a bilinear estimate for the Schr\\\"odinger operator on tori $\\mathbb{T}^{d}$, which enhances previously known multilinear estimates. We design a function space adapted to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17147","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/2411.17147/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":"2026-07-05T09:40:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xz7XR+NzG2KrEp+/JjH90nM5ItG6D8CMBX6ofhiSPhvFvh32EUS5Rzz8LQ3w9yBnt6/al+FEzIEnAMqEL7QcAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T06:39:57.334869Z"},"content_sha256":"4d9ba587cb4623051cc6a6a873a7912705282dd595d0190e9d15bfd6afbbb1e3","schema_version":"1.0","event_id":"sha256:4d9ba587cb4623051cc6a6a873a7912705282dd595d0190e9d15bfd6afbbb1e3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/32CJA6SNP5TR5DHV2KZCCAZQRW/bundle.json","state_url":"https://pith.science/pith/32CJA6SNP5TR5DHV2KZCCAZQRW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/32CJA6SNP5TR5DHV2KZCCAZQRW/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":"2026-08-13T06:39:57Z","links":{"resolver":"https://pith.science/pith/32CJA6SNP5TR5DHV2KZCCAZQRW","bundle":"https://pith.science/pith/32CJA6SNP5TR5DHV2KZCCAZQRW/bundle.json","state":"https://pith.science/pith/32CJA6SNP5TR5DHV2KZCCAZQRW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/32CJA6SNP5TR5DHV2KZCCAZQRW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:32CJA6SNP5TR5DHV2KZCCAZQRW","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":"835aacee676d3a01c69a8c96ede73b3efd600af2112b4ab5c0f12a995f8e8cdf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-26T06:30:12Z","title_canon_sha256":"2402228f0d998416f10f42f79d31e93620edc08ffa486fe2d8d3e1eceda1b79d"},"schema_version":"1.0","source":{"id":"2411.17147","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.17147","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"arxiv_version","alias_value":"2411.17147v1","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.17147","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"pith_short_12","alias_value":"32CJA6SNP5TR","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"pith_short_16","alias_value":"32CJA6SNP5TR5DHV","created_at":"2026-07-05T09:40:28Z"},{"alias_kind":"pith_short_8","alias_value":"32CJA6SN","created_at":"2026-07-05T09:40:28Z"}],"graph_snapshots":[{"event_id":"sha256:4d9ba587cb4623051cc6a6a873a7912705282dd595d0190e9d15bfd6afbbb1e3","target":"graph","created_at":"2026-07-05T09:40:28Z","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/2411.17147/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the local well-posedness of nonlinear Schr\\\"odinger equations on tori $\\mathbb{T}^{d}$ at the critical regularity. We focus on cases where the nonlinearity $|u|^{a}u$ is non-algebraic with small $a>0$. We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering $a$, we prove a bilinear estimate for the Schr\\\"odinger operator on tori $\\mathbb{T}^{d}$, which enhances previously known multilinear estimates. We design a function space adapted to","authors_text":"Beomjong Kwak, Soonsik Kwon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-26T06:30:12Z","title":"Critical local well-posedness of the nonlinear Schr\\\"odinger equation on the torus"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17147","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":"sha256:9745991664fc06e730858ef40623a7c3147e122b5dd103b599ab88ada76dee57","target":"record","created_at":"2026-07-05T09:40:28Z","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":"835aacee676d3a01c69a8c96ede73b3efd600af2112b4ab5c0f12a995f8e8cdf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-26T06:30:12Z","title_canon_sha256":"2402228f0d998416f10f42f79d31e93620edc08ffa486fe2d8d3e1eceda1b79d"},"schema_version":"1.0","source":{"id":"2411.17147","kind":"arxiv","version":1}},"canonical_sha256":"de84907a4d7f671e8cf5d2b22103308d8605580c4c556fcfe8608f627d41a98b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de84907a4d7f671e8cf5d2b22103308d8605580c4c556fcfe8608f627d41a98b","first_computed_at":"2026-07-05T09:40:28.332847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:40:28.332847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"p/4G2+1OqFCQgbAK8qB43L0+Pbh1p2lVvH0xbEjg2509IjAsaoO9tX7IYTRhsY4x53L+J1cTKPcUacnbcRbWDw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:40:28.333337Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.17147","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9745991664fc06e730858ef40623a7c3147e122b5dd103b599ab88ada76dee57","sha256:4d9ba587cb4623051cc6a6a873a7912705282dd595d0190e9d15bfd6afbbb1e3"],"state_sha256":"71c40decff14fa0b641c5eabdab84482242909dd14af86af1b8a4e37f3449c6b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qCMs2DsPhYlO5HsNlt2V9PlNaoybfErxRQy7av/x11zTF/bOYlUJ2V3WSnufEVtNIm0Eh1m7FDUzRnQ/zwiCCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T06:39:57.393476Z","bundle_sha256":"4f8e669c5d050b7c7cb0e9ecc70605a37561275792331ed703ed3e55ad63290b"}}