R↔L HARMONICS AS A MINIMAL TESTBED FOR DUAL-DOMAIN CONSISTENCY

We introduce R↔L Harmonics, a minimal yet expressive testbed designed to probe dual- domain consistency in learned representations. The core idea is to maintain two coupled latent domains, RRR and LLL, connected by learned linear maps and regularized by a harmony functional Φ̃ that combines cross-domain coherence, information-theoretic spread, and a penalty on cycle inconsistency. On top of this dual-domain core, we place a simple 2-bit delayed match-to-sample (DMS) decision head. Despite its simplicity, the testbed exposes a rich space of behaviors through four structurally motivated regimes: Full (harmony + cycle + temporal kernel), No Temporal, No Cycle, and No Harmony. We run a systematic multi-run ablation study (10 runs × 1000 episodes per regime) and show: • The No Temporal regime achieves the best task performance (89.0% ± 3.7% accuracy, 0.3952 ± 0.0242 cross-entropy), significantly outperforming the Full regime, which reaches 71.2% ± 4.2% accuracy. • Removing cycle consistency (No Cycle) leaves task-level performance comparable to Full, but qualitatively alters the geometry of the R↔L dynamics. • Completely disabling the harmony functional (No Harmony, so Φ̃ = 0 by design) still yields a reasonably good solution (68.6% ± 5.1% accuracy), demonstrating that Φ̃ is not required for solvability but instead shapes how the dual-domain solution is formed. • Across all three regimes with harmony enabled (Full, No Temporal, No Cycle), the average Φ̃ remains tightly clustered around ≈ 0.04, suggesting that Φ̃ captures a robust harmonic mode of the R↔L dynamics that is only indirectly reflected in task-level metrics. We argue that this testbed provides a clean, reproducible and computationally cheap environment for studying structural priors such as dual-domain harmony, temporal kernels and cycle consistency, and that it can serve as a bridge between abstract theoretical frameworks (e.g. dual-domain plasma dynamics) and concrete, measurable learning behavior.