Continuum Scaling Limits of PMIR Rivalry Dynamics in Networked Oscillator Systems

This preprint investigates the continuum scaling limit of PMIR (Phase-Modulated Information Rivalry) dynamics on discrete network-coupled oscillator systems. Extending prior fixed-size analyses, we study how spectral structure and rivalry dynamics evolve as system size NNN increases across multiple graph topologies, including regular lattices, small-world networks, and random regular graphs. Using linear Laplacian-driven dynamics with fixed local coupling and controlled frequency detuning, we examine the scaling behavior of key observables: algebraic connectivity, low-mode spectral aggregates, and rivalry response curves. We show that low-lying Laplacian eigenvalues exhibit topology-dependent power-law scaling with NNN, while higher low-frequency modes collapse band-by-band toward dense spectral continua. Beyond static spectral measures, we demonstrate that rivalry curves undergo a discrete-to-continuum transition under appropriate rescaling of the detuning axis. For sufficiently large NNN, rivalry responses collapse onto a universal master curve, providing a dynamical signature of continuum emergence. The rescaling exponent required for collapse is topology-dependent, defining distinct universality classes. Spectrum-matched null experiments confirm that observed topology effects are fully mediated by low-order Laplacian eigenstructure: when the spectrum is held fixed, rivalry responses collapse across realizations. Together, these results support a field-theoretic interpretation in which continuum behavior is governed by spectral density and low-mode structure rather than graph combinatorics. This work establishes PMIR rivalry dynamics as a scalable, falsifiable probe of discrete-to-continuum transitions in network-based dynamical systems, with implications for lattice field theories, discrete approximations to continuous media, and string-inspired models.