Missing Curvature in MLX logcumsumexp at a Zero Incoming Gradient

A local CPU reproduction of a higher-order differentiation defect in MLX 0.32.2. For singleton logcumsumexp, the second derivative of f(x) = x^2/2 at zero is computed as 0 instead of 1. A research C++ prototype passes 477 comparisons across 186 scenarios; the original implementation has 189 mismatches. The evidence archive includes an independent scalar reference, finite differences, source pins, executable regression tests, a prototype patch and recorded logs. Tests were run sequentially on CPU with one computational thread; native validation used a partial rebuild. An attached 24-check GatherQMM mixed-derivative extension validates a previously published repair; it is not presented as another new defect. GPU, large-tensor performance, complete models, production readiness and novelty remain unestablished. This is an independent technical note, not a peer-reviewed certification or an accepted upstream repair. Files: the audit evidence ZIP; an English narrated short video; and the video source archive with captions. AI-assisted research, writing, narration and publication. No Apple endorsement. Report and code: https://github.com/kadyrbekovhamit-cyber/gero-numerical-observatory/tree/e061fe06214f6bdcc5c2ea05eee4bd088eefbca6/audits/2026-09-10-mlx-logcumsumexp-hessianArticle: https://www.gero.uz/research/articles/mlx-logcumsumexp-hessian-at-zero.htmlVideo: https://youtube.com/shorts/kjaxRkQXgm4 Original explanatory report text is CC BY 4.0; copied MLX source remains MIT under LICENSE-MLX.txt and other files retain their supplied notices. No blanket relicensing of third-party code is intended.