CRR-Net: Complex Resonant Refinement Network

This work operationalizes the notion of “resonance” as a falsifiable engineering hypothesis: representing neural states in the complex domain Cd\mathbb{C}^dCd may induce a geometry distinct from real-valued models, potentially advantageous for relational structure. We implement Hermitian real-logit attention with s=Re(q\*k)/dks=\mathrm{Re}(q^\*k)/\sqrt{d_k}s=Re(q\*k)/dk and evaluate it under controlled conditions. To test the hypothesis fairly, we designed a strict iso-capacity “Bridge Protocol” (matched parameter and compute budgets) comparing CRR-Net against a standard real-valued Transformer across S=10S=10S=10 seeds. In strict logic tasks (Parity N=12N=12N=12 under OOD generalization), both models remain near chance, indicating that—under this configuration and soft aggregation (softmax)—complex-valued geometry alone does not yield robust discrete logical generalization. In a controlled toy language-modeling micro-corpus, CRR-Net trains stably under energy-preserving initialization, demonstrating numerical viability. Main contributions: (i) a reproducible operationalization of “resonance” into an implementable and falsifiable model; (ii) an iso-capacity evaluation protocol that reduces confounds between inductive bias and brute-force capacity; (iii) an empirical delimitation showing that moving from Rd\mathbb{R}^dRd to Cd\mathbb{C}^dCd changes representational geometry (phase rotations and a different optimization landscape) but does not, by itself, introduce the algorithmic bias needed for global discrete OOD generalization (parity); in short, phase is geometry, not magic; and (iv) a negative but informative result clarifying the limits of complex-valued representations for symbolic/OOD logic in this setup.