{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:XMZA2SYP6LFNUPQIWUF5DO73LX","short_pith_number":"pith:XMZA2SYP","schema_version":"1.0","canonical_sha256":"bb320d4b0ff2cada3e08b50bd1bbfb5dd000976734d7dc0e9965f85a05472ba2","source":{"kind":"arxiv","id":"2607.11069","version":1},"attestation_state":"computed","paper":{"title":"Sharp Ternary Martingale Isoperimetry and $n$-adic Takagi-Type Lower Bounds","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Natanael Alpay","submitted_at":"2026-07-13T04:17:27Z","abstract_excerpt":"Let $S_1$ be the one-variation associated with the regular $n$-adic martingale filtration on $[0,1)$. We study the martingale isoperimetric profile \\[ V_n(x):= \\inf_{\\substack{A\\subset[0,1)\\ {\\rm measurable}\\\\ |A|=x}} \\|S_1(\\mathbbm 1_A)\\|_1 . \\] For the ternary filtration we determine this profile exactly. Namely, \\[ V_3(x)=T_3(x):= \\sum_{j=0}^{\\infty}3^{-j}\\psi_3(\\{3^j x\\}), \\] where \\[ \\psi_3(t)= \\min\\left\\{ \\frac{1+2\\left|t-\\frac12\\right|}{3}, \\frac{2-4\\left|t-\\frac12\\right|}{3} \\right\\}, \\qquad 0\\le t\\le1 . \\] Thus the sharp ternary profile is a Takagi-type Bellman function. It is, howeve"},"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":"2607.11069","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.PR","submitted_at":"2026-07-13T04:17:27Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"94b9153a966b8e661d1aedd2db40d9aedbcb0a5656d789a98e46e4fcd54a1236","abstract_canon_sha256":"76092ab0f7e12977d7c74289563388c22d3494140978836be7c13a05f5b93133"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:22:14.547002Z","signature_b64":"kvuw6f55HV5JuwJJEqard4jN9mGQtEQjpxjnZ6KKutS3Uh/vKUSo4GAA/kX3giG6c4izhoRfCsid4HESan08CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb320d4b0ff2cada3e08b50bd1bbfb5dd000976734d7dc0e9965f85a05472ba2","last_reissued_at":"2026-07-14T01:22:14.546109Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:22:14.546109Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Ternary Martingale Isoperimetry and $n$-adic Takagi-Type Lower Bounds","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Natanael Alpay","submitted_at":"2026-07-13T04:17:27Z","abstract_excerpt":"Let $S_1$ be the one-variation associated with the regular $n$-adic martingale filtration on $[0,1)$. We study the martingale isoperimetric profile \\[ V_n(x):= \\inf_{\\substack{A\\subset[0,1)\\ {\\rm measurable}\\\\ |A|=x}} \\|S_1(\\mathbbm 1_A)\\|_1 . \\] For the ternary filtration we determine this profile exactly. Namely, \\[ V_3(x)=T_3(x):= \\sum_{j=0}^{\\infty}3^{-j}\\psi_3(\\{3^j x\\}), \\] where \\[ \\psi_3(t)= \\min\\left\\{ \\frac{1+2\\left|t-\\frac12\\right|}{3}, \\frac{2-4\\left|t-\\frac12\\right|}{3} \\right\\}, \\qquad 0\\le t\\le1 . \\] Thus the sharp ternary profile is a Takagi-type Bellman function. 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