Recursive Polymodal Synthesis for Real-Time Embodied Interaction: A Contraction-Based Framework with Provable Convergence

We present a mathematically rigorous framework for multi-modal sensor fusion in real-time embodied interaction systems. Our approach, termed Recursive Polymodal Synthesis (RPS), addresses the fundamental challenge of fusing heterogeneous sensor modalities with different noise characteristics, sampling rates, and semantic meanings into a coherent internal representation suitable for generative control. The key innovation is a proximal fixed-point iteration scheme that enforces cross-modal coherence through spectral-norm-constrained relational operators, providing theoretical guarantees of convergence to a unique fixed point. We establish conditions under which the update operator is a contraction mapping on the latent representation space and prove convergence in at most mathcal(O)( log(1/ epsilon) iterations to achieve epsilon-accuracy. The framework processes sensor inputs through modality-specific encoders (E_m )_(m=1)^M, learns cross-modal predictors (T_m )_(m=1)^M with spectral norm |T_m |_2 leq sigma_( max) < 1, and iteratively refines representations via the proximal operator mathcal(P)_ alpha(z^(t)) = (1- alpha)E(x) + alpha T(z^(t)). Experimental validation on synthetic multi-modal data demonstrates 99.94% cross-modal coherence (measured via normalized mutual consistency), validation loss of 1.93 times 10^(-4), and inference latency of 15-40ms on commodity CPUs. We provide comprehensive ablation studies demonstrating the necessity of spectral constraints, analyze the learned representations through spectral analysis and information-theoretic measures, and establish performance bounds for deployment with real sensor data. Our framework achieves state-of-the-art performance on multi-modal fusion while maintaining mathematical rigor and computational efficiency, making it suitable for latency-critical applications including live performance, human-robot interaction, and adaptive interfaces.