Both gradients die together. dJ/dθ and the Fisher share the factor σ(1−σ). Park θ at 5: the policy still is not where you want it, but gradient ascent has almost nothing left to climb with.

The natural gradient is a constant. F⁻¹ dJ/dθ = r₁ − r₀ everywhere on the curve. Move the reward sliders: the purple readout tracks the gap and ignores θ entirely. The parameterization is gone from the update.

The same KL budget allows a bigger step near saturation. The shaded band is the set of steps with KL ≤ Δ, to second order. It widens exactly where a unit of parameter motion stops changing the policy.