Unlike MLPs, Kolmogorov-Arnold Networks (KANs) expose explicit learnable edge functions on every connection, enabling mechanistic explanation in time-series forecasting. This paper introduces Temporal Functional Circuits, a framework that transforms KAN edge functions from latent visualizations into faithful, temporally grounded explanations. Built on a gated residual KAN that decomposes forecasts into a linear base and a sparsely activated KAN correction, the framework (i) maps each edge to input lags via output-aware attribution, (ii) ranks edges by learned activation range, and (iii) validates faithfulness through edge-level interventions including zeroing and spline removal. Removing the learned B-spline component while retaining the base SiLU term degrades forecasts, providing evidence that the spline shape itself carries predictive value beyond the base activation. On four synthetic regimes of increasing complexity, the learned gate opens progressively wider as signal complexity grows. On regime-switching signals, gated KAN achieves 59% lower MSE than linear-only models. Across eight benchmarks, the gated architecture is competitive with linear, attention, and MLP alternatives, while providing interpretable edge functions that MLP-based corrections cannot offer.