{"paper":{"title":"Dimers, filters, and $q$-deformed real numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"James Propp","submitted_at":"2026-07-15T19:53:30Z","abstract_excerpt":"This article associates to each positive real number $x$ an inhomogeneous dimer model with an activity parameter $q>0$ and uses it to define a positive-real-valued function $q \\mapsto [[x]]_q$ that is related to (and indeed was inspired by) the algebraic $q$-deformation $[x]_q$ introduced by Morier-Genoud and Ovsienko~\\cite{MO22}. The technical details are most transparent when the dimer model is replaced by an equivalent model involving filters in partially ordered sets. When $x$ is rational we have $[[x]]_q = q \\: [x]_q$, and it seems likely that more generally the two sides of the equation "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14332","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/2607.14332/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"}