total_gauge_dim
plain-language theorem explainer
The sum of the Standard Model gauge ranks equals the face count of the three-cube. Researchers deriving the gauge group from hyperoctahedral symmetry cite this numerical identity. The proof evaluates both sides by direct computation to confirm they equal six.
Claim. $3 + 2 + 1 = 6$, where the left side sums the Standard Model gauge ranks and the right side counts the faces of the three-dimensional cube.
background
The module derives the Standard Model gauge group SU(3) × SU(2) × U(1) from the automorphism group of the 3-cube Q₃. The gauge ranks are defined as the tuple (3, 2, 1), with 3 from axis permutations in S₃, 2 from even sign flips in (ℤ/2ℤ)², and 1 from the remaining U(1) factor. Cube face count for dimension D is 2D, so the three-cube contributes exactly six faces. SpectralEmergence.of states that Q₃ simultaneously forces gauge content with sector dimensions summing to six, alongside three generations and 24 chiral fermions.
proof idea
This is a one-line wrapper that applies native_decide to evaluate the finite sum over the gauge rank tuple and the face count definition directly.
why it matters
This equality is invoked by gauge_group_certificate, which certifies the full Standard Model gauge group from the cube automorphism group of order 48. It completes the dimension count in P-014 by matching the rank sum to the face count, consistent with D = 3 in the forcing chain and the three-layer decomposition of B₃ = (ℤ/2ℤ)³ ⋊ S₃.
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