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For $n\\ge 4$, we prove a saturation theorem for the natural scalar Poisson-summation certificates of this conjecture: any such certificate that is sharp at $\\mathbb{Z}^n$ must interpolate the Gaussian, and have vanishing Fourier transform, at every nonzero point of integer squared norm. Applied to the lattice $E_8 \\oplus \\mathbb{Z}^{n-8}$, this rigidity is incompa"},"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":"2605.26803","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-26T10:17:37Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"dd711ec6672469e813cd05b095f315cd9a5251656dbf3eb6a9d5c02dadb080ec","abstract_canon_sha256":"541509fc0572138278afcab5d6eb81226f021dba9ac7b64bcf8de717fc8e53f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-27T01:06:13.422540Z","signature_b64":"SrxRTAWnShH5UK/uP15/n2DbdQ16/6nefB6g8ph9IrvnSG/nRZDT5e+vNiHVrOis6Zopi/ZWhM4YD9Jc1jjoCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40d183342afec9d438dbcb630712b61f75c17472b1ec2d5d8507d88979da71c6","last_reissued_at":"2026-05-27T01:06:13.421804Z","signature_status":"signed_v1","first_computed_at":"2026-05-27T01:06:13.421804Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Saturation and No-Go Theorems for Scalar Poisson Certificates of Gaussian Mass Maximality","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":"2026-05-26T10:17:37Z","abstract_excerpt":"Regev and Stephens-Davidowitz conjectured that the Gaussian mass $\\Theta_\\Lambda(t) = \\sum_{x \\in \\Lambda} e^{-t\\lVert x\\rVert^2}$ of any integral lattice $\\Lambda \\subset \\mathbb{R}^n$ is bounded above by $\\Theta_{\\mathbb{Z}^n}(t)$. For $n\\ge 4$, we prove a saturation theorem for the natural scalar Poisson-summation certificates of this conjecture: any such certificate that is sharp at $\\mathbb{Z}^n$ must interpolate the Gaussian, and have vanishing Fourier transform, at every nonzero point of integer squared norm. 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