We propose a new principle of physics---MATHICCS---requiring that every mathematical axiom used in a physical law must correspond to an operation persistently realizable by a physical subsystem. Applying this criterion forces reality's geometry to be a discrete, self-similar, singularity-free structure: the Apollonian Soddy Sphere Packing (ASSP). We conjecture that 3D dynamics on this packing is a emph{self-computing fractal}: a dynamical computer whose registers are fractal congruent objects that store angular and linear momentum as cross-scale registers. This fractal quine---the first self-describing, MATHICCS-valid physical theory---derives the fine-structure constant $alpha^{-1} = 4pi^3+pi^2+pi = 137.036$ from the minimal Presburger computational cost of a Soddy sphere state update, derives the hydrogen binding energy from a single IO API equation $E = alpha^2 m_e c^2/2$, and derives the gravitational constant $G$ from atomic structure by two independent first-principles routes. We present two validated predictions (25-arcminute CMB seam; Karlsson quasar log periodicity) and one pending falsifiable prediction ($alpha^{-1}_{mathrm{Yb}} = 137.035998945 pm 20,mathrm{ppb}$). The framework resolves many major tensions in observational physics with zero free parameters.