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We exhibit an explicit integer $M < 1.78 \\times 10^{32}$ such that every integer $m \\ge M$ with gcd$(m,9)=1$ occurs as $s(p)$ for at least one prime $p$. We also prove an explicit lower bound $A_m(10^{2m/9}) \\ge C_q(m) 10^{2m/9}/m^{3/2}$, where $C_q(m)$ is explicit, positive above the sufficient threshold, and bounded a"},"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":"2606.04677","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-03T10:01:39Z","cross_cats_sorted":[],"title_canon_sha256":"328df018c647a79bd0b9f652e481c2169a01b02f6ccba2c0c247c57de10fc5d6","abstract_canon_sha256":"cf422379284bc95df21a1164d7a48eeaadd553cada3fa62c17d6a000d7938ff0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-04T01:09:24.523311Z","signature_b64":"0DeK0R51FMk9hc/PsIu1TUoT3C47kXXRPVipMz+S3aWjwZzjIpwZogMuAIPpK05yURXwlsNHYeq8K14YPQstAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bfaa826ced31b66fc94c35caa7f5a48de171edcbf8950fe8c296374b63ee0d2","last_reissued_at":"2026-06-04T01:09:24.522186Z","signature_status":"signed_v1","first_computed_at":"2026-06-04T01:09:24.522186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Explicit Surjectivity Threshold for Digit Sums of Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jens Lehmann","submitted_at":"2026-06-03T10:01:39Z","abstract_excerpt":"Let $s(n)=s_{10}(n)$ be the decimal sum-of-digits map. 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