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More precisely, for every integer \\(t\\ge 1\\) we show that \\[ c_t:=\\lim_{N\\to\\infty}\\frac{1}{N} \\#\\{0\\le n<N:\\ s_2(n+t)\\ge s_2(n)\\}>\\frac{1}{2}, \\] and in fact obtain the explicit bound \\[ c_t\\ge \\frac{1}{2}+2^{-2s_2(t)-1}, \\] where \\(s_2(m)\\) denotes the number of ones in the binary expansion of \\(m\\). The proof is based on an exact deconvolution which replaces the distribution of \\(s_2(n+t)-s_2(n)\\) by a finite stopped random-walk law. The required bias is then proved through first-exit medians for principal subsequence ideals"},"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":"2606.23398","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-22T14:21:11Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e0288c981c8c599baf40ef98fa72dd235f1e7572db406bb148b39efbc93311ec","abstract_canon_sha256":"29ed1096cd4202a660793a7e1072e093e4bc6136eea673acbaf3988eb468e7a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T03:14:19.000516Z","signature_b64":"JYKPIkwmrztLjHLzcx0cIOZF0GxmXKf/p9Bd7CYxX6rzYUj+YJnLNvS02UTQdLuU3GzhED2aloWKT9IBBdrMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"777f1c0afb4eeae7e34132eb9ee6a2b7b5c2e0190672b984d67dfb84e9641e83","last_reissued_at":"2026-06-23T03:14:19.000097Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T03:14:19.000097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A first-exit proof of Cusick's sum-of-digits conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kaimin Cheng","submitted_at":"2026-06-22T14:21:11Z","abstract_excerpt":"We prove Cusick's conjecture on the binary sum-of-digits function. More precisely, for every integer \\(t\\ge 1\\) we show that \\[ c_t:=\\lim_{N\\to\\infty}\\frac{1}{N} \\#\\{0\\le n<N:\\ s_2(n+t)\\ge s_2(n)\\}>\\frac{1}{2}, \\] and in fact obtain the explicit bound \\[ c_t\\ge \\frac{1}{2}+2^{-2s_2(t)-1}, \\] where \\(s_2(m)\\) denotes the number of ones in the binary expansion of \\(m\\). The proof is based on an exact deconvolution which replaces the distribution of \\(s_2(n+t)-s_2(n)\\) by a finite stopped random-walk law. 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