Theorem: Coupled longitudinal-state comparison

For common input and Gaussian draws,

All displayed first moments are finite for finite parameters and the stated Gaussian input law.

Under common forward controls, only weight and bias residuals remain. For different protection levels, the offsets do not cancel.

Proof. Subtract the hidden updates and insert the primed preactivation under the unprimed activation and multiplier. The Lipschitz property gives

Iterate and apply the head bound. Both activations have norm at most their preactivation norm, so finite input and Gaussian moments propagate recursively.