{"paper":{"title":"Re-Rooting-Assisted Edge-Minimum Runtime Repair for Node and Link Failures in Dense Eisenstein--Jacobi Broadcast Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.NI","math.IT"],"primary_cat":"cs.DC","authors_text":"Bader Albader","submitted_at":"2026-06-19T06:17:16Z","abstract_excerpt":"One-to-all broadcasting in dense Eisenstein--Jacobi (EJ) networks relies on diameter-level spanning trees that fragment when nodes or links fail. This paper introduces the selected triple $(r,\\theta,\\Kcomp_{r,\\theta})$--a chosen root, a chosen EJ coordinate-reduction orientation, and the healthy component graph induced by that choice--as the fundamental unit of analysis for joint node/link fault recovery. The central result is a necessary and sufficient condition: hybrid repair succeeds if and only if the healthy EJ graph $G'=\\Ht-\\Fv-\\Fe$ is connected. When $G'$ is connected, a spanning tree o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.21133","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/2606.21133/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"}