A Non-Invasive Population Profiling Framework: Problem Well-Posedness, Attractors, and Convergence Rates

We present a mathematically well-posed framework for population-level profiling based on anonymous headcount dynamics, nonlinear feedback, and symbolic compression. The framework admits rigorous analysis at three layers: population state evolution, attractor formation with controlled collapse, and discrete symbolic dynamics. We establish existence, uniqueness, exponential forgetting of initial conditions, attractor convergence rates, and symbolic convergence in distribution. Crucially, we prove that all inverse problems related to individual reconstruction or identity inference are ill-posed by construction. Numerical examples illustrate convergence behavior and emergent basins of attraction.