gauge_rank_match
plain-language theorem explainer
The theorem asserts that the fundamental representation dimensions of the three gauge layers extracted from the hyperoctahedral group B₃ are 3 for color, 2 for weak, and 1 for hypercharge. Researchers deriving the Standard Model gauge group from geometric symmetries would cite this when confirming the rank match to SU(3) × SU(2) × U(1). The proof is a one-line term that applies reflexivity directly to the explicit definitions of each layer.
Claim. The fundamental representation dimensions of the three layers extracted from the automorphism group B₃ of the 3-cube satisfy $d_{fund}(color) = 3$, $d_{fund}(weak) = 2$, and $d_{fund}(hypercharge) = 1$.
background
Module P-014 derives the Standard Model gauge group SU(3) × SU(2) × U(1) from the automorphism group B₃ of the 3-cube Q₃, where B₃ = (ℤ/2ℤ)³ ⋊ S₃ has order 48 and acts by signed permutations on ℤ³. The module decomposes B₃ into three layers: axis permutations (S₃, order 6) yielding the color structure, even sign flips ((ℤ/2ℤ)², order 4) yielding the weak structure, and the parity homomorphism yielding hypercharge. A GaugeLayer is a record carrying a name, fundamental representation dimension, and discrete order; the three sibling definitions color_layer, weak_layer, and hypercharge_layer hard-code the dimensions 3, 2, and 1 respectively.
proof idea
The proof is a one-line term that applies reflexivity to the three layer definitions, which already set fund_rep_dim to 3, 2, and 1 by construction.
why it matters
This theorem completes the rank extraction step in the P-014 derivation of the Standard Model gauge group from cube symmetry. It directly supplies the (3,2,1) dimensions that match SU(3) × SU(2) × U(1) and thereby closes the gap left by earlier modules on particle generations and quark colors. The result sits inside the Recognition Science forcing chain at the point where T8 fixes D = 3 spatial dimensions and the eight-tick octave supplies the discrete symmetry structure; the full dynamical embedding of these ranks into the phi-ladder remains open.
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