quark_lepton_ratio
plain-language theorem explainer
The theorem establishes that the color sector dimension times face-pair count times two equals three times the hypercharge sector dimension times the same factors, yielding the 3:1 quark-to-lepton multiplicity ratio from Q3 geometry. Particle physicists reconstructing Standard Model fermion counts from forced D=3 would cite this when enumerating chiral states. The proof is a direct numerical check via native_decide on the explicit sector dimensions and combinatorial face pairs.
Claim. Let $d_c=3$ be the dimension of the color spectral sector and $d_h=1$ the dimension of the hypercharge sector. Let $f$ be the number of face pairs in the 3-cube. Then $d_c f 2 = 3 (d_h f 2)$.
background
The Spectral Emergence module starts from T8 (D=3) forcing the binary cube Q3 with 8 vertices and automorphism group B3 of order 48. The SpectralSector inductive type partitions this symmetry into four layers: color (from S3, dimension 3), weak (dimension 2), hypercharge (dimension 1 from single-axis parity), and conjugate (dimension 2). The face_pairs function on dimension 3 counts the three independent face-pair orbits, which the module identifies with the three fermion generations.
proof idea
The proof is a one-line wrapper that applies native_decide to evaluate the arithmetic equality after substituting the explicit values dim color = 3, dim hypercharge = 1, and the face-pair count for the 3-cube.
why it matters
This equality feeds directly into the master theorem spectral_emergence, which certifies that the full gauge content SU(3)×SU(2)×U(1), 24 chiral fermions, and |Aut(Q3)|=48 all follow from D=3 with zero free parameters. It closes the step from color dimension to the 3:1 quark-lepton ratio inside the Recognition framework's eight-tick octave and phi-ladder mass structure.
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