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Equivalently, if $p(z)=\\mathbb{E}[(z-X)^n]$, then $p$ encodes a truncated moment version of the classical perpetuity equation $X\\stackrel{d}{=}AX+B$ with $X$ and $(A,B)$ independent. This places finite free perpetui"},"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":"2606.19115","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2026-06-17T14:28:50Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"df5b49a1003142aed35764822d70f20644886b3b740ace36af7e0e63291fd1be","abstract_canon_sha256":"fb92fced53470da7ca4a01896b5a6186d728372596e2e2be220349aa2bafc431"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:11:56.944544Z","signature_b64":"KJ8Fatx6WMLkZzpZAiFrQEiPhtWM/bw04YF/jfi0dLGmCweWz7zRXJ5hEtugOHvIf06K7JvCeR7NdAchXN+GBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f78e7f7f85926c4007814280a6f0d519fb7b4f13fbd0d684f936f0986b4c7a7","last_reissued_at":"2026-06-19T16:11:56.944150Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:11:56.944150Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite free perpetuities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.PR","authors_text":"Bartosz Ko{\\l}odziejek, Julia Le Bihan","submitted_at":"2026-06-17T14:28:50Z","abstract_excerpt":"We introduce and study finite free perpetuities, defined as monic polynomial solutions of degree $n$ to the affine fixed-point equation \\[ p(z) = \\mathbb{E}\\!\\left[ A^{n}\\,p\\!\\left(\\frac{z-B}{A}\\right)\\mathbf{1}_{\\{A\\neq0\\}} \\right] + \\mathbb{E}\\!\\left[ (z-B)^n\\mathbf{1}_{\\{A=0\\}} \\right], \\] where $A$ and $B$ are complex-valued random variables with finite moments up to order $n$. Equivalently, if $p(z)=\\mathbb{E}[(z-X)^n]$, then $p$ encodes a truncated moment version of the classical perpetuity equation $X\\stackrel{d}{=}AX+B$ with $X$ and $(A,B)$ independent. 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