Information Propagation via Sign-Flip Dynamics

Classical initialization theory analyzes networks through second-order observables such as variance, correlation, and Jacobian conditioning that are natural when weights are symmetric around zero. Identity-plus-noise initialization, which initializes the network to behave locally like a residual block, sits outside this picture: variance can be preserved while coordinate-wise signal identity is destroyed, making such second-order quantities indirect indicators of trainability. We identify the terminal sign-flip rate as the natural forward-pass observable for this regime, and derive its critical scale $\sigma \sim L^{-1/2}$ from a dimensionless margin--sensitivity ratio, rather than postulating it. The retention curve emerges as a first-passage limit of a signed-margin process, yielding a calibration procedure that targets a chosen sign-flip rate at any depth. Experiments on synthetic and real benchmarks suggest that this calibrated initial rate behaves as a training-time parameter rather than an inert calibration detail.