Admissibility of Height Functions under Infinitesimal Arithmetic Deformations

We study logarithmic height functions under infinitesimal arithmetic deformations treated purely at the level of first and second variations. No assumption is made on the existence of global deformation families, smooth dependence on external parameters, or convexity of local height components. We introduce a class of admissible infinitesimal directions designed to exclude trivial affine shifts and smooth second-order curvature contributions, isolating the genuinely arithmetic effects arising from the max-structure of local heights. For such admissible deformations, we prove that the second variation of the height function is non-negative. The result is local and second-order in nature, relies only on the intrinsic definition of height, and requires no auxiliary conjectures or structural interpretations. The analysis is intentionally minimal and technical. Its purpose is to provide a clean infinitesimal input usable in subsequent arithmetic arguments. The presentation is logically closed within the stated framework.