Path-Locked Neural Composition: Exact Evaluation-Path Invariance with Geometric Algebra

We combine learned motion calibration with exact geometric algebra (GA) composition. Paired motor products in projective GA form a mergeable 16-coordinate summary, followed by a learned gate. On development-selected, family-held-out ETH3D odometry, this gate attains $0.440$ m translation error with 577 head parameters, compared with $0.915$ m for identity motion, a $51.9\%$ reduction. Its throughput is $2.17\times$ that of a contextual gated recurrent unit (GRU) variant at batch 256 on one CPU thread, excluding odometry extraction. Feature-matched direct and residual controls support the translation benefit of motor-constrained prediction. We characterize evaluation-path invariance by associativity on reachable states and evaluate it on four public recorded-data benchmarks. Fixed-input tests identify path-locking: MovieLens-1M accuracy falls from $0.4319$ to $0.3483$ when the same bilinear-residual model uses a right rather than left fold; TUM RGB-D motion error increases from $6.83$ to $36.53$ cm. Exact GA composition has a maximum observed path discrepancy of $2.28\times 10^{-15}$ m on TUM. Intermediate-state regularization reduces empirical path sensitivity without certifying invariance. Associative motor composition supports accurate learned calibration and consistent chunk merging, while predictive improvements remain dataset- and horizon-dependent.