The Exact Exchange Rate of Gradient-Free Credit

Gradient-free training is back at LLM scale, and its cost is now the central question. Folklore holds that on a layer of $d$ parameters, scalar-broadcast evolution strategies need $\sim d$ times more samples than backpropagation — directionally right, but the constant has never been measured. We derive it exactly: for antithetic rank-1 perturbations of an $m\times n$ layer, the estimate ${\hat{g}}_N$ from $N$ forward evaluations against the true gradient $G$ obeys ${\cos }^2=N/(N+N_0)$ with $N_0=2(m+2)(n+2)$, parameter-free, exact in the linear regime at frozen weights; dynamic training adds landscape-dependent excess variance atop this floor. We verify it on every layer of the reference MLP (ratios 0.93–1.12) and of a frozen extreme-aspect-ratio chain (0.976–1.006), killing all four alternatives in that pre-registered experiment (maximum deviation 0.006 across a $340\times$ span of $N_0$). The law then prices, by extrapolation, the largest gradient-free training system to date: it operates at per-step $\cos \approx 0.02$ and finances that signal with $\sim 10^5$ coherently integrated steps at $\sim 180\times$ the GPU-hours of backpropagation. Channel by channel, our measurements yield a design-space map (two primitives, four laws, six measured rows) and three impossibility statements, among them: no scalar-broadcast channel escapes the exchange rate, and approximate backpropagation is the wrong objective. This is a measurement study, not a new method.