{"paper":{"title":"The Ramsey community number as a renormalization-group crossing","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.CO","physics.soc-ph"],"primary_cat":"cond-mat.stat-mech","authors_text":"Alexei Vazquez","submitted_at":"2026-07-07T07:54:39Z","abstract_excerpt":"The Ramsey community number $r_k$ is the smallest size at which a network is better described by communities than by none, under a Bayesian detection rule. On the diamond hierarchical lattice we show that $r_k$ is an exact renormalization-group crossing: the block-model sufficient statistics obey a linear map with eigenvalues $\\{bs,b\\}$, the degree-corrected evidence density flows to $\\ln K$ at a community fixed point, and $r_k$ is the generation at which the running evidence clears the detection threshold. Degree correction advances detection by two generations. We derive $r_k(b,s;q)$ in clos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05954","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/2607.05954/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"}