Why Learn What You Can Compute? Closed-Form Operators for Periodic Time-Series Forecasting

Periodic patterns are a defining characteristic of many real-world time series, and exploiting them has become central to modern long-horizon forecasting. Yet most existing learned forecasters that leverage such patterns still rely on iterative gradient-based optimization. We show that period folding transforms a period-$p$ wide-sense cyclostationary process into phase-indexed sequences with cycle-invariant first- and second-order statistics, and that multi-horizon forecasting on the resulting finite delay-coordinate dictionary is a Tikhonov-regularized Galerkin projection with a unique closed-form solution. Under ergodic sampling, the empirical estimator converges to its population projected counterpart, while exact Koopman closure arises as a special case when the folded process has an affine cycle-conditional mean. This theoretical result directly yields **CAPE** (**C**losed-form **A**pproximation of **P**eriodic **E**mbeddings), a gradient-free algorithm that estimates a compact cycle-level predictor, supports direct transfer when source and target datasets have compatible projected cycle structure, and adaptively reduces to a direct multi-output ridge autoregressor when $p^{\star }=1$. To evaluate CAPE, we conduct extensive experiments across zero-shot, few-shot, in-distribution, and cross-domain tasks. CAPE ranks first in 20 of 28 in-distribution MSE settings, wins seven of eight within-family and all three cross-domain zero-shot transfers, and achieves the best few-shot result, without gradient-based or GPU training and with under one second of CPU fitting.