The Triangular Register: Fermion Mixing Angles from Discrete Spacetime Geometry

Abstract

The Problem. In 1963, Nicola Cabibbo discovered that quarks mix between generations with a specific angle: sin θ_C ≈ 0.225. This single number governs the rates of countless particle decays, from kaon oscillations to beta decay. Yet for over sixty years, no theory has explained WHY this particular value. The Cabibbo angle appears in the Standard Model as a free parameter—measured, not predicted. The same mystery extends to all six fermion mixing angles: three in the CKM matrix (quarks) and three in the PMNS matrix (leptons). Why these values? Why do quarks mix weakly while leptons mix strongly? The Standard Model offers no answer. The Resolutionist Approach. This paper applies the framework of Resolutionism, which posits that our universe is characterized by finite information capacity, discrete structure at the Planck scale, and anthropic selection for parameters permitting observer emergence. Rather than treating physical constants as arbitrary, Resolutionism asks: which values are NECESSARY for complexity and observers to exist? The framework combines information-theoretic constraints with evolutionary selection pressure—universes (or regions) that cannot support observers are never observed, leaving only those with specific parameter values. Here we show that this approach, applied to discrete spacetime geometry, can explain all six fermion mixing angles. Our Solution. We propose that spacetime possesses a discrete triangular microstructure characterized by three integers: 3 (edges per triangle), 5 (triangles at a curved defect), and 6 (triangles at a flat vertex). From these three numbers alone—with no free parameters—we derive all six mixing angles. The Cabibbo angle emerges as ε = (ln 2/3) × (41/42), where ln 2/3 represents one bit of information shared across three generations, and 41/42 is the 'Dougness factor' arising from the Douglas number D(6) = 42 of string landscape counting. This yields sin θ_C = 0.2255, matching the measured value 0.2257 ± 0.0010 to 0.07%—well within experimental uncertainty. The Cabibbo angle is no longer arbitrary; it is a geometric inevitability. The Quark-Lepton Puzzle Resolved. Why do leptons mix with large angles (~33°, ~45°) while quarks mix with small angles (~13°, ~2°, ~0.2°)? Our framework provides a geometric answer: quarks propagate through the flat hexagonal bulk of the lattice, where information spreads evenly and mixing is suppressed. Leptons inhabit the curved interface between pentagon defects and hexagona

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