Engineering teams increasingly ask of an optimization algorithm more than a good design: They ask for a justification of it. This paper presents Interpretable Population Dynamics (IPD), a metaheuristic for optimization over Riemannian manifolds in which population evolution is governed by an explicitly modeled causal system, so that the algorithm can audit, and explain, its own search behavior. Within IPD, geodesic variation operators act as interventions on a causal graph of search dynamics, scheduling selects the action with maximal predicted counterfactual progress, and inverse-probability-weighted estimation on logged trajectories quantifies the Average Treatment Effect (ATE) of every algorithmic factor. The experimental program evaluates the method on ten problems spanning four manifold species, the sphere, the Stiefel manifold, the Symmetric Positive Definite (SPD) cone, and SO(3), against twelve racing competitors, including newly admitted Riemannian trust-region, adaptive stochastic, and quasi-Newton rivals, over 21 independent runs under a 12,000-evaluation budget. IPD attains the smallest median optimality gap on seven of the ten problems, with a median reduction of 26.2% against the strongest rival on those problems; exact paired permutation tests with Benjamini-Hochberg control confirm 94 of 100 contrasts, while Riemannian CMA-ES retains the two smooth SPD instances and a Riemannian trust-region method retains the best-conditioned Stiefel instance. In the audit leg, an honest Causal Forest reproduces the audit's effect ordering with Spearman correlation 0.90, whereas naive correlational attribution overstates the dominant operator's effect by 43.7%. Grouping by curvature regime locates the scheduling gain chiefly on positive-curvature multimodal instances, two one-sided equivalence tests certify wall-clock parity with the fastest baselines wherever objective evaluation dominates, and a regime surrogate with 85.9% cross-validated accuracy converts logged trajectories into decision-ready evidence.