Negative-order Vendi–LCR comparison without a full-rank assumption

For finite positive semidefinite similarity matrices with diagonal one and entries between zero and one, we prove the uniform Vendi–LCR comparison at every negative order using the positive spectral support evidenced in the source implementation. A spectral Jensen argument also proves the directly defined minus-infinity endpoint. Classical Jensen and Holder–McCarthy inequalities and known nonnegative/full-rank cases are credited. No analytic order-limit theorem or weighted-distribution formal extension is claimed. AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review. The accompanying Lean certificate covers the precise scoped main statement; inspect the formal endpoint, assumptions and provenance in the supporting package. Prior-art investigations are documented bounded searches, not worldwide-priority certification. Original prose is offered under CC BY 4.0 and original code/Lean under MIT; third-party rights and existing notices are preserved. The public supporting ZIP contains original proof sources and verification evidence, with bibliographic links and hashes instead of redistributed third-party source manuscripts. Recorded prior-art/source-access limits: PRE final full body unread; current primary abstract retains fullranknegative restriction. Later Bioengineering article broadly asserts comparison citing PRE without separate singular-negative proof. Classical Jensen/Hölder–McCarthy and all known source cases credited; no earlier demonstrated full support comparison found.