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In this paper, we consider the set $\\mathscr{D} := \\{\\delta(n): n \\ge 1 \\}$ of all defects. We show that as a subset of the real numbers, the set $\\mathscr{D}$ is well-ordered, of order type $\\omega^\\omega$. More specifically, for $k\\ge 1$ an integer, $\\mathscr{D}\\cap[0,k)$ has order type $\\omega^k$. We also"},"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":"1310.2894","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-10T17:33:14Z","cross_cats_sorted":[],"title_canon_sha256":"f8c5a3844834740676242eed37530ff5b187aaf8f251ac38f694fb852637d742","abstract_canon_sha256":"374750579c77aa3cc2aae7ce524315c937c24709740d51a9892fd9ba607f6923"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:24.688527Z","signature_b64":"ZghpzwyMGWuH2GMKP8P2oOi9SJchyX3A+lwB8DdgdSWeLPyMSelQqocPUmi/iPQxaPlRDCxNCbtvJ+qysSLSDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"973652740c65a15a698026d0e16550d8397c69ea3fe1c6a9ef3c94d3ca2a2772","last_reissued_at":"2026-05-18T00:40:24.687963Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:24.687963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integer Complexity and Well-Ordering","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Harry Altman","submitted_at":"2013-10-10T17:33:14Z","abstract_excerpt":"Define $\\|n\\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. 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