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 motivates reality's geometry as a discrete, self-similar, singularity-free structure: the Apollonian–Soddy Sphere Packing (ASSP).We conjecture that 3D dynamics on this packing is a self-computing fractal: a dynamical computer whose registers are fractal-congruent objects that process angular and linear momentum as cross-scale registers. This fractal quine is proposed as a self-consistent, MATHICCS-valid physical theory.It gives zero-free-parameter derivations of the baseline fine-structure constant α⁻¹_TFOFT = 4π³ + π² + π = 137.036, the electron–star logarithmic mass hierarchy, the electron mass, the proton mass, the proton/electron mass ratio, the 77-layer self-encoding register, the hydrogen binding energy, the hydrogen spectral ladder, and the gravitational constant G from atomic structure by two independent first-principles routes.We present empirical cross-checks involving the observed electron–star hierarchy, dark/visible channel fractions 23/27 and 4/27, Fermi-bubble morphology, the 25-arcminute CMB seam, and the near-match between Karlsson quasar logarithmic periodicity and the derived Info-channel leakage. The Karlsson complement also gives a 77-layer mass-channel spot check inside the same 137–139 register interval.We present one pending falsifiable prediction, α⁻¹_Yb = 137.035998945 ± 0.000000045. The framework is presented as a nearly zero free-parameter ontology whose derived consequences are compared against major tensions in observational physics.