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Here, we present an arborescence graph constructed from iterations of $g(x) = (2^{e(x)}x - 1)/3$, which is the inverse of $f(x)$ and where $x \\not \\equiv [0]_3$ and $e(x)$ is any positive integer satisfying $2^{e(x)}x - 1 \\equiv [0]_3$, with $[0]_3$ denoting $0\\pmod{3}$. The integer patterns inferred from the resulting arborescence provide new insights into proving the validity"},"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":"1907.07088","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2019-07-14T21:37:14Z","cross_cats_sorted":[],"title_canon_sha256":"5781c829b69bdf5bd30035294741265afe6b4ebae47c904eb79540b45a6ff089","abstract_canon_sha256":"5028549c8423813084ed4db288830343ac3b076f978b48bcc09fbabf7e3b810e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:22.925567Z","signature_b64":"qS/+pB5UogHx5TZbt+WZbpECgGDDENfskLcMmUdwAQs1MwI4yW5whPWQERAEyUON0N3vK6Qn8wcc7+LKnhh8Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"640f036a2ef09636249bbfeff47545b6df46600a4b0af4601788882995481170","last_reissued_at":"2026-05-17T23:40:22.924932Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:22.924932Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integer patterns in Collatz sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Zenon B. 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