Probabilistic Hard-Constrained Learning via Structural Latent Parameterization

Many probabilistic predictors must satisfy exact structure in every stochastic realization, yet common hard-constraint approaches form predictions in ambient coordinates and subsequently complete, correct, or project them. We introduce SLP-ProbHard, a cross-family representation-centered framework for probabilistic hard-constrained learning when suitable explicit structural parameterizations are available. Its core modelling object is a Structural Feasible Latent Parameterization (SFLP), consisting of a structural latent law $Z\sim P_{\theta }^Z(\cdot \mid x)$ and an explicit feasible map $Y=h_{\varphi }(x,Z)$, with $h_{\varphi }(x,z)\in \mathcal{C}(x)$ for every $z\in \mathcal{Z}$. Together, the structural latent law and feasible map define the predictive model itself rather than a final feasibility wrapper. The latent law and map jointly determine predictive support and boundary probability; the chosen coordinates and map specify the stochastic representation and shape attainable dependence, calibration, expressiveness, and computation. The formulation does not prescribe a particular predictive backbone or latent distribution family; the present experiments instantiate it with Gaussian latent laws and fixed geometry-derived maps. We instantiate the formulation across representative affine equalities, convex structural sets (ordering, simplex, and nonnegative ordering), explicit nonlinear equalities/manifolds (circle and scale-shape), and three structural views of the same seven-basin hydrological FDC data. A verified official-source ProbHardE2E/DPPL affine comparison gives zero practical violations for both methods: SLP-ProbHard uses 8 rather than 11 stochastic coordinates and improves MSE/MAE, whereas DPPL attains better marginal CRPS and higher, closer-to-nominal coverage; the paired test does not detect an Energy Score difference across ten seeds. Real-world affine and nonlinear FDC representations reduce 13 to 7 and 14 ambient to 8 computational coordinates, respectively. Together, these results show that exact feasibility alone does not determine a predictive law and motivate direct structural generation when scientifically meaningful feasible coordinates are available.