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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","authors_text":"Marco Mantovanelli","cross_cats":["math.NT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-03-17T04:25:53Z","title":"Dyadic Frequency Laws, Clock Dynamics, and Defect Scaling in a Perturbed Hofstadter $Q$-Recursion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2603.16111","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ea95621917086bc5e6bfa4e4f4b487f15140c311ba6f9a09d707cb66a651dae4","target":"record","created_at":"2026-06-02T02:04:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"07cd6148607810e13feeaff548c82f4c3cdf6c84f61507376d50faad0fbf2f6c","cross_cats_sorted":["math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-03-17T04:25:53Z","title_canon_sha256":"d953b144cef755035970ab205ea9bc612450b40f5b465398eb26811cfb654571"},"schema_version":"1.0","source":{"id":"2603.16111","kind":"arxiv","version":2}},"canonical_sha256":"f9497d9d6e7dee8fd065fd9b770724d5c834cd20f73d8ed0c7898d4332e14f01","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f9497d9d6e7dee8fd065fd9b770724d5c834cd20f73d8ed0c7898d4332e14f01","first_computed_at":"2026-06-02T02:04:16.226812Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:16.226812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FzNk3dD1XW26FHJxkgfc2dFoJiw9hV7jlAkVOFFgdePTnQVhvIpb8+L6J9KbCAg+YZRB2AQiRJwhEmtMQMiiAQ==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:16.227337Z","signed_message":"canonical_sha256_bytes"},"source_id":"2603.16111","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea95621917086bc5e6bfa4e4f4b487f15140c311ba6f9a09d707cb66a651dae4","sha256:ff05b5d2229ec91b4ebf81d27e9fbe73a608ed24a3e4c4bf8c3caacb19bcf41a"],"state_sha256":"51c03d6d13a971968ebc80600ae659757cfaef8a60a25109802d59aa13957c7c"}