The Golden Ratio as Minimal Spectral Complexity in Discrete Compensatory Dynamics

We formulate and solve a structural variational problem for discrete compensatory bidimensional dynamics under positivity, primitivity, unimodularity, and integrality constraints. Within this class of operators, we prove that the golden ratio = 1+52 is the unique minimal strictly positive spectral radius. Equivalently, is the minimal topological entropy, and is the minimal Mahler measure in the quadratic unimodular positive setting. We further show that this minimality is dimensionally rigid: in dimensions n 3, no strictly positive lower bound for hyperbolic expansion exists under analogous unimodular constraints. Thus the golden ratio arises as the minimal hyperbolic unit compatible with discrete compensatory bidimensional structure. The associated projective dynamics admits a unique positive attractor ratio equal to, with reciprocal contraction factor 1/. Perturbation analysis establishes stability of the ratio attractor under small structural deviations. The results provide a structural synthesis linking spectral minimality, entropy, Mahler measure, hyperbolic translation length, and projective dynamics. No universality is claimed: the emergence of is strictly conditional on explicitly stated structural assumptions. This deposit is part of an independent mathematical line of work. Related conceptual framework: EROI (see Related identifiers).