Coherent Routing in Decision Trees: Phase-Interference Learning for Interpretable Tabular Prediction

We introduce Phase-Interference Decision Trees (PIDT), a differentiable routing model for tabular prediction in which each node propagates a bounded complex state rather than only a scalar routing probability. The main model uses two-state norm-preserving branch maps: a gate controls how much amplitude is sent to the left and right children, while learned unitary maps rotate the latent state before subsequent splits. This formulation contains ordinary soft decision trees as the phase-free scalar case, and two-state routing creates explicit score-level cross terms among latent-state trajectories. We give a self-contained PIDT definition with exact mass conservation, prove scalar phase cancellation, establish a scoped containment relation with soft trees, derive an algebraic interference decomposition, give a companion construction for shared two-state parity routing, and derive a fixed-architecture empirical Rademacher upper bound for bounded two-state PIDT scores. We evaluate PIDT on a ten-dataset OpenML suite of low-class-count tabular tasks under matched differentiable-tree controls. Across ten seeds per dataset, the main summaries track accuracy as context and report small descriptive NLL and Brier deltas relative to the scalar PIDT-1S, phase-frozen PFB, and SDT controls. The empirical study is framed as a mechanism-level study for compact differentiable trees, not as a broad tabular-performance claim.