Doob barrier consolidation

Integrate the rule into the optimizer update for parameters that should retain previous-task knowledge. At the end of task t, copy each protected parameter into an anchor tensor mu, estimate an importance tensor s using the diagonal Fisher or an online-EWC running sum, and set a positive barrier-width tensor b. A practical initialization is b_i=clip(c/(sqrt(s_i)+epsilon), b_min, b_max), with c selected on a validation stream. For each minibatch, compute the current gradient g, choose step size eta, and sample epsilon from a standard normal distribution. For every protected parameter, calculate z=clip((w-mu)/b, -1+delta, 1-delta), then calculate the Doob correction r=-(sigma^2*pi/(2*b))*tan(pi*z/2). Apply either the direct Euler-Maruyama update w=w+eta*(-g-s*(w-mu)+r)+sqrt(eta)*sigma*epsilon, or add eta*r and the stochastic term to an AdamW update while keeping the baseline optimizer identical across comparisons. The clipping delta, for example 1e-4, avoids floating-point overflow near the barriers. Track the maximum absolute normalized displacement |z|, the percentage of coordinates with |z| above 0.95, current-task accuracy, and prior-task accuracy. Reduce eta or sigma if discrete updates frequently jump across the interval. The paper-derived quantities are the cosine ground state, its logarithmic derivative, and the sigma-squared scaling; s, b, sigma, and the discrete-step stabilization policy are empirical choices. First run a small MLP on rehearsal-free Split-MNIST, comparing SGD, noisy SGD, EWC, OU-style quadratic anchoring, and the Doob rule at equal injected noise. Sweep sigma over {0, 0.005, 0.01, 0.02, 0.04, 0.08} and sweep barrier widths. The key success signal is an inverted-U prior-task retention curve with a significant interior optimum, improved retention over matched EWC at equal current-task accuracy, and fewer large parameter excursions. A failure is monotone degradation with sigma or no advantage over the matched unconditioned noisy anchor.