Sigmoid Contact Smoothing Recovers a Local Boundary Term in First-Order Pathwise Gradients

First-order pathwise gradients through rigid-contact simulation can be biased: the return of a rollout may jump when a contact event switches on or off, and differentiating a fixed rollout never sees that jump. We analyze one deterministic remedy used in recent differentiable locomotion pipelines, in which hard contact-mode selection is replaced by a sigmoid gate on the contact impulse, and identify precisely which part of the true gradient the gate supplies. Near a single regular switching surface in the space of rollout randomness, the gradient of the expected return equals the expectation of the pathwise gradient plus a boundary integral over that surface, weighted by the jump of the return. This decomposition is classical in form, a transport (Reynolds) identity; we give a rigorous local proof of it under mild hypotheses, from which it follows that hard pathwise differentiation misses exactly the boundary term. Our main result concerns the gate: whenever the smoothed return depends on the contact activation through a differentiable interpolation whose endpoints reproduce the two hard branches, the gate part of the smoothed pathwise gradient converges in expectation to the boundary term as the gate hardens. In this limit, analytic gate smoothing supplies exactly the term that hard differentiation discards. We complement this with a finite-sharpness theory: the second moment of the gate term grows at most linearly in the sharpness, and under one extra smoothness hypothesis its bias and second moment have explicit expansions whose orders are set by the readout and the noise density at the surface, giving a bias–variance scaling law in the batch size; we illustrate the results numerically on a single-contact model, including a two-dimensional-noise configuration.