cabibbo_coefficient_eq_3_2
plain-language theorem explainer
The Cabibbo coefficient in the PMNS radiative corrections equals 3/2 by the cube geometry ratio. Neutrino physicists checking mixing angle predictions from 3D voxel topology would cite this to fix the vertex-edge contribution in the quark sector. The proof reduces to a single reflexivity step on the explicit rational value assigned by the preceding definition.
Claim. The Cabibbo coefficient, obtained as the ratio of the six faces to four in the 3-cube geometry, equals $3/2$.
background
The PMNS Radiative Correction Derivation module extracts integer coefficients for neutrino mixing angles from the topology of the 3-cube. Atmospheric mixing draws on the six faces, solar mixing on twelve edges minus two, and the Cabibbo term on the six-to-four ratio arising from vertex-edge duality in the quark sector. The upstream definition records: The Cabibbo correction coefficient is faces / 4. Physical interpretation: The quark sector's 3-generation torsion (φ^{-3} weight) couples via face-diagonal paths. The 6 faces divided by 4 (the number of vertices per face, or equivalently the vertex-edge weight in the dual lattice) gives 3/2. This construction operates inside the Recognition Science setting with D = 3 spatial dimensions fixed by the eight-tick octave.
proof idea
The proof is a one-line reflexivity that matches the Cabibbo coefficient definition to the rational 3/2. It invokes no external lemmas beyond the built-in equality judgment on the explicit assignment.
why it matters
This equality supplies the concrete value required by the parent theorem correction_derivation_verified, which certifies that atmospheric_from_faces, solar_from_edges_minus_2, and cabibbo_from_faces_over_4 all match their geometric sources. It realizes the module's account of the 3/2 coefficient from vertex-edge duality in the 3D voxel ledger, consistent with the Recognition Science forcing chain at T8 where D = 3. The result closes the integer-coefficient derivation for the predicted angles sin²θ₁₃ = φ^{-8}, sin²θ₁₂ = φ^{-2} − 10α, and sin²θ₂₃ = ½ + 6α.
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