CVaR-tail active residual correction

(1) Exact integration point: place this module around a costly simulator, human labeler, safety checker, or high-resolution world model. The neural low-fidelity predictor receives candidate states or designs $x$ and produces $K$ ensemble predictions. The high-fidelity oracle is queried only for selected candidates. Use the corrected predictor in the CVaR loss, safety constraint, or downstream design optimizer; it need not replace the ordinary forward pass everywhere. (2) Pseudocode: initialize a low-fidelity model $f_L$ and a small high-fidelity set $H$; train $K$ independently initialized or bootstrapped copies; on a large cheap candidate pool compute $\mu_L(x)=K^{-1}\sum_k f_L^{(k)}(x)$ and $s_L^2(x)=(K-1)^{-1}\sum_k(f_L^{(k)}(x)-\mu_L(x))^2$; compute the empirical quantile $q_\beta^L$; retain candidates satisfying $|\mu_L(x)-q_\beta^L|\leq\epsilon s_L(x)$; select one batch fraction by largest $s_L$ and the remainder by largest $s_r+\rho|\phi(x)^T\widehat w|$; query the expensive oracle at those points; append $(x,y_H)$ to $H$; fit $\widehat w$ using the ridge objective above; optionally fine-tune $f_L$ and repeat. Estimate CVaR on a fresh Monte Carlo pool by evaluating corrected predictions and minimizing the scalar empirical objective over $\eta$. (3) Computed from the paper's mathematics: the empirical quantile, confidence tail region, ridge residual coefficients, corrected predictions, and CVaR objective. Estimated empirically: ensemble uncertainty $s_L$, residual uncertainty $s_r$ from bootstrap residual fits, and the candidate-pool distribution. Use standardized scores or rank-based selection so uncertainty and correction magnitude are comparable. (4) First cheap experiment: construct a 2D or 5D synthetic simulator with a cheap biased function and an expensive function containing a narrow failure region; compare random sampling, global ensemble-uncertainty sampling, and this two-stage policy at $\beta=0.99$. Use a five-member 3-layer MLP ensemble, 32--128 initial high-fidelity labels, and batches of 16 oracle queries. Measure absolute CVaR error, high-fidelity calls required to certify a CVaR constraint, and global MSE. The method succeeds if it reaches lower CVaR error or reliable constraint satisfaction with fewer expensive evaluations than both baselines; a useful target is at least 2x fewer high-fidelity labels than random or global uncertainty sampling, even if central-region MSE is unchanged.