Multiplicative Log-Time Update Clock

Integrate the clock at the outer loop of an anytime recurrent-refinement model, state-space model, or continual-learning learner. Keep the neural transition h_{n+1}=F_theta(h_n,z_n) unchanged, but invoke it only when wall-clock time t reaches the next event time. At initialization set T=t_0, sample or estimate mu and sigma^2 from a chosen positive multiplier law p_U, and run: T=t0; n=0; h=h0; while wall_time < budget: U=sample_pU(); T=T*exp(U); wait_until(T); h=F(theta,h,input); n=n+1; output=G(theta,h). For a deterministic ablation use U=mu; for a stochastic version use the paper's exact p_U if available, or fit a positive distribution to reported multiplier samples while preserving its first two moments. The paper supplies the renewal construction and asymptotic coefficients; the implementation estimates mu, sigma^2, event-time histograms, and task-specific output changes empirically. First run a cheap experiment with a GRU or four-layer recurrent refinement MLP on sequential MNIST or a small denoising task. Compare fixed-rate updates, geometric deterministic updates, and stochastic multiplicative updates at equal wall-clock budgets. Record N(t), validation loss, and hidden-state changes. The quantitative prediction is that E[N(t)] versus ln(t/t_0) is linear with slope 1/mu, while E[N(t)] versus t is sublinear. Var[N(t)] should have slope sigma^2/mu^3 against log-time, within about 20 percent after the renewal transient. Relative count scatter should decay as 1/sqrt(ln(t/t_0)). Falsify the idea if these slopes are absent or if it provides no anytime-compute advantage over a matched deterministic geometric schedule.