Dirichlet-calibrated mean-loss alarm

Integrate this at the training-monitoring layer, not inside the forward pass. Choose a nonnegative scalar $Z_i$ for each independently sampled example: per-example loss, squared error, energy, latency proxy, safety cost, or a nonnegative constraint violation. Set a target mean $\tau$ and collect a fresh validation batch $Z_1,\ldots,Z_n$ that is not reused for gradient updates. Normalize $X_i=Z_i/\tau$. To estimate $K$, sample $M$ Dirichlet vectors by drawing $G_j^{(m)}\sim\operatorname{Gamma}(1,1)$ and setting $D_j^{(m)}=G_j^{(m)}/\sum_{k=0}^{n}G_k^{(m)}$. Compute $S=\sum_m\mathbf{1}[\sum_{i=1}^nX_iD_i^{(m)}\le1]$ and use the one-sided Clopper–Pearson lower confidence bound $\operatorname{Beta}^{-1}(\delta;S,M-S+1)$ for $K$; this avoids falsely declaring a violation because of Monte Carlo noise. If the lower bound is at most alarm level $\alpha$, reduce the learning rate, increase regularization, or stop and reject the checkpoint. The theorem supplies the super-uniform guarantee; the engineer must empirically check independence, batch contamination, and Monte Carlo error. Start with a small MLP or ResNet on MNIST or CIFAR-10, using squared per-example loss and a target $\tau$ chosen from a baseline model. Compare this alarm with a moving-average mean threshold over repeated random seeds. Success means calibrated false-alarm frequency at or below $\alpha$ under a model known to satisfy the target, and earlier detection of mean-loss degradation under heavy-tailed or heteroscedastic validation losses at the same batch size.