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We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime $p$ the signature rank of the units of ${\\mathbb Q}( \\zeta_{p^n})^+$ differs from $\\varphi(p^n)/2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for"},"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":"1809.02185","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-06T19:32:55Z","cross_cats_sorted":[],"title_canon_sha256":"ac0d1669ee8376e3502c751409a36fe3ce4fb5b589692d55d7a87048594dc958","abstract_canon_sha256":"f62f2b92e3cc1a127775e0ed178a40a7dc9ef08fcb0630cd3b04e3b514af0bc8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:17.712057Z","signature_b64":"C194iD8Fz4v6bwgELxsFY3OkheBLqQMFzxFU29y8lmNPLh4Pgc+wNjSKoKYUPkIST/HHZ9dlcI+t42/EuWU6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74b478cbb790d4f51bad18540da3e60466d14ad699af067634d56a3b81b11f6d","last_reissued_at":"2026-05-18T00:06:17.711606Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:17.711606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Signature Ranks of Units in Cyclotomic Extensions of Abelian Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David S. 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