Midpoint Consistent-Gain Cancellation

Use a small one-dimensional RNN or diagonal state-space model as the first integration point. Decompose each state transition as z[t+1,i] = a[i] z[t,i] + q[t,i] + e[t,i], where z is the hidden state, q is the learned input and nonlinear contribution, a[i] is an unknown persistent gain, and the residual e has a known or conservatively estimated bound epsilon[i]. Initialize an interval A[i] = [-Delta[i], Delta[i]]. At each step compute mu[i] = (left[i] + right[i]) / 2 and use the transition q[t,i] = learned_input[t,i] - mu[i] z[t,i], thereby canceling the estimated recurrent drift. After z[t+1] is available, and whenever absolute z[t,i] exceeds a threshold tau, construct the candidate interval with endpoints (z[t+1,i] - learned_input[t,i] - epsilon[i]) / z[t,i] and (z[t+1,i] - learned_input[t,i] + epsilon[i]) / z[t,i]. Sort the endpoints, intersect with A[i], and skip the update if the intersection is empty. The paper supplies the interval logic, midpoint policy, and exact peak law; the implementation must estimate Delta and epsilon from calibration data. First test a scalar recurrence with fixed random gain a, unit-bounded adversarial noise, and Delta swept over 0.1, 0.5, 1, 2, and 5. Compare midpoint selection with zero-gain cancellation, random exploration, and endpoint selection. Then test a tanh RNN on noisy sine-wave prediction. The predicted signature is a worst hidden-state peak approximately equal to 1 plus Delta in the isolated scalar experiment, with slope one versus Delta and a midpoint peak lower than endpoint selection. Agreement within 20 percent is the initial acceptance criterion.