Zero-Crossing Reset Integral Optimizer

1. Integration point: add one integral buffer per parameter tensor or, for a cheaper first test, one scalar buffer per layer. Let G_k be the minibatch gradient. The simplest version uses g_k = G_k directly and stores a tensor z_k with the same shape. To make the zero-crossing rule less sensitive to individual coordinates, use a block residual g_k = inner_product(G_k, U_(k-1)) divided by the norm of U_(k-1) plus epsilon, where U_(k-1) is the previous update direction. 2. Pseudocode: compute G; form the residual g; update z = z + g; if the elementwise or blockwise product g times g_old is nonpositive and the reset counter is at least d, set z = 0 and record the reset; form U = k_P times g plus k_I times z; update theta = theta - eta times U; store g_old = g and increment the dwell counter. Clip the integral buffer only as a secondary safety measure, not in the first ablation. 3. Taken from the paper: PI accumulation, reset at a proportional-error zero crossing, and a minimum dwell time. Estimated empirically: Hessian eigenvalue ranges, minibatch sign-flip probability, eta, k_P, k_I, and d. 4. First cheap experiment: train a two-layer MLP on MNIST and a small quadratic objective with known Hessian, comparing SGD, momentum, Adam, non-reset PI, and reset PI at equal update counts. Sweep the ratio k_P squared times lambda divided by 4 k_I. The predicted signature is a transition near ratio one: below the critical-damping boundary, the quadratic error should oscillate and generate repeated zero crossings; above it, the dominant mode should become non-oscillatory. Reset PI should reduce the post-crossing error peak and settling steps, whereas too-small d should create reset chatter and degrade convergence. On the known quadratic, the measured dominant decay rate should match the real part of the predicted characteristic root within about 20 percent before stochastic-gradient noise dominates.