{"paper":{"title":"Threshold dynamics for the 4$d$ mass-energy double critical NLS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alex H. Ardila, Jason Murphy, Jiqiang Zheng, Zuyu Ma","submitted_at":"2026-05-20T05:15:01Z","abstract_excerpt":"We consider the 4$d$ mass-energy double critical NLS \\[ (i\\partial_t+\\Delta)u = -|u|^2 u + |u| u. \\] In Luo (2024) and Cheng--Miao--Zhao (2016), the authors established a scattering/blowup dichotomy for solutions satisfying the energy constraint $E(u_0)< E^c(W)$, where $W$ is the energy-critical NLS ground state and $E^c$ is the energy for the underlying cubic NLS. We prove that the scattering/blowup dichotomy persists even at the energy threshold $E(u_0)=E^c(W)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.20715","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/2605.20715/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"}