Neural Flip Laws and Tangent Stabilization

A positive flip coefficient supports a local period-two branch of projected gradient descent (GD), while tangent motion can suppress oscillations in full GD. We quantify this distinction in wide additive networks. Exact empirical functionals determine the projected flip coefficient and the tangent-drift coefficient; two Softplus initialization laws share the same population tangent kernel but produce opposite flip signs. For one residual, balanced i.i.d. initialization with positive tangent drift, and a near-critical step calibrated by the initial kernel, a single GD trajectory tracks an explicit amplitude–sharpness system with $O_P(m^{-1/2})$ error over $O(\sqrt{m})$ steps, converges to its own interpolant, and becomes subcritical under either flip sign. The theorem covers unbounded initializations through a finite-eighth-moment derivative envelope. Experiments from random initialization verify the trajectory law, while endpoint interventions identify the tangent sharpness direction that produces damping.