{"paper":{"title":"Dyadic Frequency Laws, Clock Dynamics, and Defect Scaling in a Perturbed Hofstadter $Q$-Recursion","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Marco Mantovanelli","submitted_at":"2026-03-17T04:25:53Z","abstract_excerpt":"We study the perturbed Hofstadter $Q$-recursion \\[ Q(1)=Q(2)=1,\\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n \\quad (n\\ge3). \\] We investigate its value frequencies and dyadic fluctuation structure. Our first main result is an explicit dyadic frequency law: if $F(s)$ denotes the number of occurrences of the value $2s-1$, then for every $k\\ge0$, \\[ \\{F(s):2^k\\le s<2^{k+1}\\} = \\{3+\\nu_2(j):1\\le j\\le2^k\\} \\] as multisets. The proof uses Clo\\^itre's binary interleaving structure, dyadic hitting-time identities, and an induced rank-lifting mechanism for plateau zero-runs.\n  We also study deviations fro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.16111","kind":"arxiv","version":2},"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/2603.16111/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"}