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We show that for all but finitely many pairs $(m,n)\\in \\mathbb{Z}_{\\ge 0}\\times \\mathbb{N}$ there exists a place $\\mathfrak{p}$ of $K$ such that the point $\\alpha$ has preperiod $m$ and minimum period $n$ under the action of $\\varphi$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under s"},"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":"1407.1573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-07T03:54:13Z","cross_cats_sorted":["math.AG","math.DS"],"title_canon_sha256":"3dba44009e11140afd224606443a19fbb5f8542e9668fcdac7d7923fab017407","abstract_canon_sha256":"52fd15e7501106c36cca821f1b2eb1b2bc472ab6c0384525d5f8251ebafb3157"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:37.562812Z","signature_b64":"XYV7kKvk1qGQI/fMUORryjlq1ulBaj84i/QMRA1dZTuYIicJQ8MTUbABikpuo7XdRrz8YIFrTNKl7/mBKw9ZDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d769da7672506943ae0f8397dd9714590d81012173b3d55d1e4c5ac6f020b12","last_reissued_at":"2026-05-18T02:03:37.562110Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:37.562110Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Portraits of preperiodic points for rational maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.DS"],"primary_cat":"math.NT","authors_text":"Dragos Ghioca, Khoa Nguyen, Thomas J. 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