{"paper":{"title":"The Complex Spectral Flow: Spectral Conditions for Two-Parameter Equivariant Bifurcation Guarantees","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ziad Ghanem","submitted_at":"2026-06-03T20:49:08Z","abstract_excerpt":"We introduce the \\emph{complex equivariant spectral flow} -- a virtual $G$-representation assembling the eigenvalue winding numbers of the linearization at an isolated two-parameter critical point for $G = S^1 \\times \\Gamma$ bifurcation problems -- and prove that, for maximal twisted orbit types, the coefficient of the local bifurcation invariant in $A_1^t(G)$ reduces to a closed-form dimension formula, bypassing the standard pipeline of basic degree factorization and Burnside ring multiplication entirely. When the eigenvalue dependence is holomorphic, topological cancellation among winding nu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.05428","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/2606.05428/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"}