{"paper":{"title":"A Gauss-Bonnet-Type Dichotomy for Unimodular Random Infinite Trivalent Hyperbolic Polyhedra","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.DG","math.GT"],"primary_cat":"math.PR","authors_text":"Chuwen Wang, Huabin Ge, Tian Zhou, Yangxiang Lu","submitted_at":"2026-08-04T12:32:41Z","abstract_excerpt":"We develop a unified geometric and probabilistic theory of conformal type for unimodular random infinite trivalent hyperbolic polyhedra in $\\mathbb{H}^3$. By corresponding these with dual angled disk triangulations and regular circle patterns, we associate to each face an intrinsic geometric characteristic number $L_f(P)$, determined entirely by local dihedral geometry. For the root face $f$, we establish the unimodular Gauss-Bonnet formula $\\mathbb{E}[L_f(P)] = 2\\pi - (\\pi/3)\\mathbb{E}[deg(f)]$. Under natural tameness and admissibility assumptions, this yields a sharp dichotomy: a unimodular "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.03575","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/2608.03575/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"}