The Impossible Trinity of Time-Series Validation: A Conservation Law among Training Sufficiency, Test Coverage, and Temporal Causality

The paper proves that in time‑series validation three desirable properties—training sufficiency, test coverage, and temporal causality—cannot all be satisfied simultaneously. It introduces quantitative bounds involving the smallest training fraction (α), test coverage (β), future training fraction (Λ), and distance to nearest future training point (δ), showing that exceeding the causal frontier α+β=1 requires training on future data that must lie within (1−α)T of a test point. The authors demonstrate that the impact of such future leakage depends on distance rather than amount, and compare different validation schemes (walk‑forward, k‑fold, purged k‑fold) in terms of their position on this Pareto frontier, illustrating the trade‑offs with empirical results on noise data.