Forecasting Next-Settlement Funding from the Premium Integral: Mechanism-Based Ornstein–Uhlenbeck Evidence across Three Venues

Perpetual-futures funding is a deterministic transformation of a kernel-weighted premium integral, yet the integral is rarely used as the state variable in next-settlement forecasts. We derive exact discrete moments and clamped-normal predictive distributions under an Ornstein–Uhlenbeck premium model, then evaluate genuinely out-of-sample forecasts for BTC, ETH, and SOL on Binance, Gate, and Hyperliquid. The 38-day public-data sample contains 2,781 targets over 31 evaluation days. OU reduces pooled RMSE from 0.2368 bp for the last-settlement benchmark to 0.1968 bp, but remains slightly behind EWMA at 0.1947 bp. Improvements over Last are strongest on Hyperliquid; Binance and Gate favor EWMA over OU. Minute-level reconstruction supports linear rather than equal kernels on Binance and Gate. Twenty configurations and Holm/Reality-Check procedures address selection. Gaussian intervals fail by venue: 95% coverage is 39.1%, 43.5%, and 97.7%, respectively. The evidence supports mechanism-aware point forecasting, not uniformly superior accuracy, calibrated uncertainty, or executable net returns.