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We prove this sharp inequality for every integral unimodular lattice $L$ of rank $n\\leq 32$, with equality only at $L\\cong\\mathbb{Z}^n$, and furthermore obtain the strict inequality for every even unimodular lattice of rank $40$. The proof does not use the classification of unimodular lattices in these ranks; rather, it parametrizes integral unimodular theta series as polynomials in the modular function $u=\\"},"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":"2606.01347","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-31T16:55:52Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"35ce14a891ec3ce4cc82068920e372bdc507aa75339349a5c61e42f42ee4949c","abstract_canon_sha256":"acccf0a6aa3e2892d0686ba842a416eba4ca1fe982323001bf9e260698b85d89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:30.826105Z","signature_b64":"2WELoZqtkDv7Iv057YWePXhGDgGsPEEdK1/D/IcWSZY5xcab1lJy0U7nF4gAvhQ3JZS1QXaGaeg8QKTnEy1eAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac4eb3b05f72adfd55dbdffb6fb0d6493ff9b464a30ff3d44e6af66379f3613f","last_reissued_at":"2026-06-02T02:04:30.825691Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:30.825691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"math.NT","authors_text":"Scott Duke Kominers","submitted_at":"2026-05-31T16:55:52Z","abstract_excerpt":"The integer lattice $\\mathbb{Z}^n$ is conjectured to maximize the Gaussian mass $\\Theta_L(t)=\\sum_{x\\in L}e^{-t\\|x\\|^2}$ over the set of stable lattices in $\\mathbb{R}^n$, for every $t>0$. 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