Same Ratio, Wrong Derivative: MLX Division at Extreme Scales

Independent CPU audit of Apple MLX 0.32.2. For positive C, C/(C*t)=1/t, so the derivative at t=1 must equal -1. Extreme representable scales instead produce zero or negative infinity while the forward value remains 1. Intermediate numerator multiplication or denominator squaring loses a representable weighted derivative.

A research patch passes all 1328 native comparisons, versus 600 mismatches before. Another 1886 comparisons pass with the previously published inverse-hyperbolic and arctan2 patches applied together. Higher derivatives and zero gradients are checked for float32/64. Moderate complex64 controls pass before and after; extreme complex stability is not established.

The evidence archive contains source pins, exact references, actual before/after results, patch, portable runner, bundled compatibility harnesses and checksums. Fresh sequential CPU runs reproduce every historical row exactly. One numerical defect; comparison counts are not distinct findings.

Selected translation units were rebuilt against an existing CPU archive, with one numerical thread and no GPU arithmetic. GPU, performance, compile/vmap, zero denominators, exhaustive nonfinite/subnormal behavior and full-model impact remain untested. No production-readiness, absolute-priority or upstream-acceptance claim. AI-assisted research and publication preparation.

Report and code: https://github.com/kadyrbekovhamit-cyber/gero-numerical-observatory/tree/main/audits/2026-09-10-mlx-divide-scale-autodiff/

Article: https://www.gero.uz/research/articles/mlx-divide-scale-autodiff.html

Original report and deposit text are CC BY 4.0; code and copied MLX source/patches retain the included MIT notices. No blanket relicensing of third-party material. Evidence ZIP SHA-256: f40bcebce434801f5cb53dbaac06c87479898d5b47b574b54677803498b6ea47.