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Holevo and Utkin \\cite{HU26} conjectured that for $0<p\\le 1$, \\[ \\min \\left\\{\\frac{\\|x\\|_p}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\} =2^{1/p-1/2}; \\] for $1<p<2$, \\[ \\min \\left\\{\\frac{\\|x\\|_p}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\} = \\min\\left\\{2^{1/p-1/2},\\left(\\frac{(d-1)^{p/2}+(d-1)^{1-p/2}}{d^{p/2}}\\right)^{1/p}\\right\\}; \\] and for $2<q<\\infty$ \\[ \\max\\left\\{\\frac{\\|x\\|_q}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\"},"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":"2605.05243","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-04T03:11:54Z","cross_cats_sorted":["math-ph","math.FA","math.MP"],"title_canon_sha256":"9cf4a152028ac9498483138188955fadc2bed8a15e0f04fe952d6c2d0f561efd","abstract_canon_sha256":"0ad62238a58ea42fcf9f43ba600934df88735f8a5b6c1a57b08aca32d767e405"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:04.511186Z","signature_b64":"CYgCCO+aRvORg33JcG0UqXu7lvvbQCD5ZpjL7TfuM3PCeD+m6596tqgK7uYygbffr4FR8nXmN+IhdAmrz/ZsBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a4e684367ccd9179c6390e5f58452b09f6d9b02ad4c972350d04dbd0e6c6cae","last_reissued_at":"2026-05-22T01:04:04.510376Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:04.510376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of the Holevo--Utkin conjecture on sharp $\\ell_p$ norms for zero-sum vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The conjectured sharp bounds on p-norm to 2-norm ratios for zero-sum vectors hold for all dimensions four and higher.","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.CA","authors_text":"Haonan Zhang","submitted_at":"2026-05-04T03:11:54Z","abstract_excerpt":"Let $d\\ge 3$ and $p>0$. Let $\\|x\\|_p$ denote the $\\ell_p$ (quasi-)norm of a $d$-dimensional vector $x$. Holevo and Utkin \\cite{HU26} conjectured that for $0<p\\le 1$, \\[ \\min \\left\\{\\frac{\\|x\\|_p}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\} =2^{1/p-1/2}; \\] for $1<p<2$, \\[ \\min \\left\\{\\frac{\\|x\\|_p}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\} = \\min\\left\\{2^{1/p-1/2},\\left(\\frac{(d-1)^{p/2}+(d-1)^{1-p/2}}{d^{p/2}}\\right)^{1/p}\\right\\}; \\] and for $2<q<\\infty$ \\[ \\max\\left\\{\\frac{\\|x\\|_q}{\\|x\\|_2}:\\vec{0}\\neq x\\in\\mathbb R^d,\\ \\sum_{i=1}^d x_i=0\\right\\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For d ≥ 4 the minimum of ||x||_p / ||x||_2 over non-zero zero-sum x equals 2^{1/p-1/2} when 0 < p ≤ 1, equals the min of that quantity and ((d-1)^{p/2} + (d-1)^{1-p/2})/d^{p/2})^{1/p} when 1 < p < 2, and the analogous maximum statement holds for q > 2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The proof must correctly identify and compare the two candidate extremal configurations (the two-support vector and the equitable (d-1)-support vector) and show no other zero-sum vector yields a smaller or larger ratio; 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