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If $\\mu$ is the cascade on $[0,1]$, then $\\widehat{\\mu}(\\xi)\\to 0$ as $|\\xi|\\to\\infty$, almost surely on non-extinction. For every fixed nondegenerate $C^2$ embedded arc $\\gamma:[0,1]\\to\\mathbb{R}^2$, the pushforward $\\gamma_\\#\\mu$ is likewise Rajchman almost surely on non-extinction. The analogous conclusion holds for the scalar cascade on the parameter circle pushed forward by any fixed nondegenerate $C^2$ Jordan curve. No moment condition of order strictl"},"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":"2607.15966","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-17T13:53:14Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"77c26ea6be9d1775b9587a65b86071f81ba6f4470df1f4d85f6b51412f8955e0","abstract_canon_sha256":"9592b38e7a18f8c3c69f4b9a57c91fb61df04ff4fe61c786c94d9f62a6068f7e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-20T02:19:23.067459Z","signature_b64":"KlmOHYzPh4Busu0EW1vbtq3W080L62J+fLpLdGUkIqBdSDu4QCgkp6VdodELSRBQ5Q2FI4iSyrgWwZmT6fubCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc165da6a6de8550006e6a84ef480a3182c92befcf805bd9187aba1fbbe40595","last_reissued_at":"2026-07-20T02:19:23.066549Z","signature_status":"signed_v1","first_computed_at":"2026-07-20T02:19:23.066549Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Canonical Mandelbrot Cascades on Curves Are Rajchman","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Chengbo Xiao, Guozheng Cheng, Hongdou Qu, Menghan Li, Xiang Fang, Yin Cai","submitted_at":"2026-07-17T13:53:14Z","abstract_excerpt":"We settle the Rajchman problem for canonical scalar dyadic Mandelbrot cascades at the minimal Kahane--Peyri\\`ere integrability threshold. 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