Forecasts Are Paths: Curvature-Matched Hyvärinen Residuals and Overlap-Constrained Ordinal Flow

Long-horizon forecasters produce trajectories, yet they are usually trained as if the horizon were a bag of independent coordinates. Pointwise squared error measures amplitude, but it neither controls how forecast errors evolve nor distinguishes trajectories whose local peaks, valleys, and turning order differ. We introduce HyOT, a single-pass loss that scores a forecast in two complementary geometries. In the cardinal geometry, a curvature-matched Gaussian–Student Hyvärinen score acts on both residual levels and residual innovations, combining the local efficiency of squared error with controlled tail influence. In the ordinal geometry, each five-point window induces a differentiable law over all \(5! = 120\) rankings. Because adjacent windows share four observations, every hard transition has an exact source–insertion representation over a \(120 \times 5\) transition alphabet. A factorized differentiable soft descriptor and entropy-gated Hellinger score align the predicted and observed ordinal flows. The resulting objective directly optimizes trajectory geometry during training while preserving the original inference procedure. Experiments across five forecasting architectures, six datasets, and four horizons show that optimizing the path, rather than isolated points, improves aggregate MSE and MAE. A representative trajectory-structure evaluation further shows that the MSE gain can be accompanied by lower Innovation MAE and Ordinal Hellinger and higher Local Rank Agreement.