Killed-Brownian diffusion score

Integrate this at the forward-noising and score-target construction of a continuous-time diffusion model whose state lies in H={x:x_n>0}; do not modify the U-Net or transformer initially. For each clean training example y and sampled time t, sample a perturbed x from the killed heat transition or use an analytically tractable positive proposal, then train a score network s_theta(x,t) with denoising score matching. The per-pair analytic target is g(x,y,t)=-(x-y)/(2t) for coordinates 1,...,n-1. For the normal coordinate use g_n=-(x_n-y_n)/(2t)+y_n/[t(expm1(x_n*y_n/t))]. Compute the second term stably with a small-z series y_n/t[1/z-1/2+z/12] when z=x_n*y_n/t is small, and use a reciprocal exponential form for large z. At sampling time, use the learned reverse SDE or probability-flow ODE, but parameterize x_n=softplus(h_n)+epsilon and add the analytic boundary score correction as a control variate: s_used=s_theta-g_model_estimate+g_boundary, where g_boundary is evaluated with the current predicted clean point or a one-step Tweedie estimate. Alternatively, start with direct target prediction and compare against clipping. The exact mathematics supplies K_t, g_n, and the inverse-square curvature bound; empirical components are the proposal sampler, time weighting, and any estimate of y from the network. First test on a 2D toy distribution supported on x_2>0, such as a Gaussian mixture at distances 0.01, 0.1, and 1 from the boundary, using a small MLP score model. Compare ordinary Gaussian DSM plus clipping, a log-coordinate baseline, and the killed-kernel target. Measure boundary violation rate, sliced Wasserstein distance, score error near x_2=0, and reverse-process stability at equal network evaluations. Success would be substantially lower boundary mass leakage and improved likelihood or Wasserstein error, especially for modes close to the absorbing boundary.