PAAMcm — a complex-cell layer over the PAAM core

We develop a mathematical framework for agency in active media, in which localized patterns arise as symmetry-breaking states of a stochastic dynamical medium. The framework begins with a nondegenerate diffusion on a finite-dimensional manifold and defines a medium as a local, homogeneous system equipped with a symmetry group compatible with its drift and noise. Symmetric states provide the background, while patterns are localized states that break the background symmetry. The linearization at a pattern separates naturally into neutral modes associated with motion along the symmetry orbit, localized internal modes, and extended modes inherited from the background. This structure provides a basis for defining intrinsic pattern properties, including transverse monodromy, response functions, localization, orientation, excess production, stability margins, birth and death, and interactions between patterns. The framework further develops notions of lineages and collectives and gives procedures for identifying pattern properties from recorded observables. The mathematical core explicitly distinguishes results taken from the literature, adapted results, elementary consequences, and new statements, with provenance recorded for each formal statement. Empirical measurements, models, and predictions are maintained separately from the mathematical core. PAAM 7.1 therefore provides a formal, reproducible framework for describing localized agency in stochastic active media and for connecting its mathematical structure with observable data.