Shifted-Hermite Gaussian-tail primitive

Integrate this at the Gaussian-tail operation in a probit, censored-regression, diffusion-guidance, or rare-event classification loss, specifically where the network produces a standardized threshold y=(a-\mu)/\sigma and the code evaluates log\bar\Phi(y) or the Mills ratio. During each batch, choose a positive reference x and shift t so that y=x+t; for example set x=max(y-T,x_min), t=y-x, with x_min=8 and T=2. For K=4 or K=6, compute Hermite values by the recurrence and accumulate S=sum_{n=0}^{K}(-1)^n He_n(t)x^{-n-1}. Pseudocode: y=(a-mu)/sigma; x=max(y-T,xmin); t=y-x; h0=1; h1=t; S=h0/x-h1/x**2; for n in 1..K-1: h2=t*h1-n*h0; S += (-1)**(n+1)*h2/x**(n+2); h0,h1=h1,h2; use Hermite result if y>=yswitch and abs(t)<=T else exact_log_ndtr(y). Evaluate log M as log(max(S,epsilon)) and combine it with log phi(y)=-y^2/2-log(2pi)/2. The mathematical quantities are the integral representation, Hermite recurrence, and asymptotic coefficients; x_min, T, K, y_switch, and epsilon are empirical safeguards. First test on a small MLP trained with a probit binary loss on synthetic imbalanced Gaussian data, comparing exact log_ndtr and a rational approximation at equal batch size. Measure maximum absolute tail error, derivative error by finite differences or autodiff, kernel latency, and end-to-end throughput. A successful result is at least 1.5x faster tail evaluation, relative probability error below 1e-5 for y>=8, stable gradients, and unchanged validation likelihood. If S becomes negative or gradients become unstable, reduce K or immediately use the exact fallback.