Positive diagonal concavity does not imply concavity for elementary-symmetric sums

The four-variable polynomial f=1+2*sigma_1+sigma_2+2*sigma_3+sigma_4 has strictly positive coefficients and a diagonal fourth root concave on the entire positive axis, while f^(1/4) is not concave on the positive orthant. An exact identity and a strictly interior rational midpoint certificate refute Lachaume Conjecture 1. The literal elementary symmetric functions and regularity are proved in Lean. The older real-rootedness-converse failure is explicitly uncredited. 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: No earlier Conjecture1refutation found in inspected primary/citing/companion sources. Actual2019citingSection5/Theorem4.17read byparent and finalreviewer; typesetJMP2018body and subscription-index/worldwide coverage remain unavailable. Original preliminary translation/positivity-range errors corrected and independently rechecked. No absolute priority guarantee.