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A question raised by Sarnak and formulated as a conjecture in Chiu: whether, among unit-volume flat tori, height is minimized by a lattice maximizing the length of its shortest nonzero vector.\n  The first counterexample is exact and also disproves the natu"},"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":"2607.11695","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2026-07-13T15:28:34Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"5e3d1b4737bd57db967a4cc0bf894599b0d72e5850932b9f6d6579122f859f58","abstract_canon_sha256":"fa8c2e0f8a6f30beeda32c468e2d0a62a4636524b9d2bec51cd71dbcf27cbc0f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T02:22:19.437063Z","signature_b64":"kr16NoGFuq7PBRjGmSVa4jBuPmHEUwAGwhxM1XcNH536OrlkqGqSLGklyNAEAxSJA60KCHDaiiH8l3CV5VY/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21674e4d10c6366b8ed679b73c0230d36eb8de5688501cfb3a388b989937d785","last_reissued_at":"2026-07-14T02:22:19.436233Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T02:22:19.436233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On two counterexamples in the geometry of numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.MG","authors_text":"Nihar Gargava","submitted_at":"2026-07-13T15:28:34Z","abstract_excerpt":"We give counterexamples to two optimization problems in dimensions eight and nine.\n  1. 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