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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2603.16111","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-03-17T04:25:53Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"d953b144cef755035970ab205ea9bc612450b40f5b465398eb26811cfb654571","abstract_canon_sha256":"07cd6148607810e13feeaff548c82f4c3cdf6c84f61507376d50faad0fbf2f6c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:16.227337Z","signature_b64":"FzNk3dD1XW26FHJxkgfc2dFoJiw9hV7jlAkVOFFgdePTnQVhvIpb8+L6J9KbCAg+YZRB2AQiRJwhEmtMQMiiAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9497d9d6e7dee8fd065fd9b770724d5c834cd20f73d8ed0c7898d4332e14f01","last_reissued_at":"2026-06-02T02:04:16.226812Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:16.226812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2603.16111","created_at":"2026-06-02T02:04:16.226878+00:00"},{"alias_kind":"arxiv_version","alias_value":"2603.16111v2","created_at":"2026-06-02T02:04:16.226878+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2603.16111","created_at":"2026-06-02T02:04:16.226878+00:00"},{"alias_kind":"pith_short_12","alias_value":"7FEX3HLOPXXI","created_at":"2026-06-02T02:04:16.226878+00:00"},{"alias_kind":"pith_short_16","alias_value":"7FEX3HLOPXXI7UDF","created_at":"2026-06-02T02:04:16.226878+00:00"},{"alias_kind":"pith_short_8","alias_value":"7FEX3HLO","created_at":"2026-06-02T02:04:16.226878+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2604.06237","citing_title":"On a perturbed Hofstadter $Q$-recursion","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X","json":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X.json","graph_json":"https://pith.science/api/pith-number/7FEX3HLOPXXI7UDF7WNXOBZE2X/graph.json","events_json":"https://pith.science/api/pith-number/7FEX3HLOPXXI7UDF7WNXOBZE2X/events.json","paper":"https://pith.science/paper/7FEX3HLO"},"agent_actions":{"view_html":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X","download_json":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X.json","view_paper":"https://pith.science/paper/7FEX3HLO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2603.16111&json=true","fetch_graph":"https://pith.science/api/pith-number/7FEX3HLOPXXI7UDF7WNXOBZE2X/graph.json","fetch_events":"https://pith.science/api/pith-number/7FEX3HLOPXXI7UDF7WNXOBZE2X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X/action/storage_attestation","attest_author":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X/action/author_attestation","sign_citation":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X/action/citation_signature","submit_replication":"https://pith.science/pith/7FEX3HLOPXXI7UDF7WNXOBZE2X/action/replication_record"}},"created_at":"2026-06-02T02:04:16.226878+00:00","updated_at":"2026-06-02T02:04:16.226878+00:00"}