Q3_aut_order
plain-language theorem explainer
The automorphism group of the three-dimensional hypercube has order 48. Researchers deriving Standard Model gauge content and fermion counting from the forced dimension D=3 cite this result as the numerical match to 48 chiral states. The proof is a direct numerical evaluation that unfolds the order formula 2^D · D! and computes the value.
Claim. The automorphism group of the 3-cube satisfies $|Aut(Q_3)| = 48$.
background
The Spectral Emergence module starts from the forced spatial dimension D=3 and constructs the binary cube Q_3 = {0,1}^3 with eight vertices. The automorphism order is defined by the formula 2^D · D!, which for D=3 yields 48. Upstream structures supply the J-cost on recognition edges and the ledger factorization that calibrate the combinatorial counts used here. The module document states that this order equals the number of chiral fermionic states in the Standard Model.
proof idea
The proof is a one-line wrapper that applies norm_num to the definition of the automorphism order together with the factorial function, reducing the equality directly to 48.
why it matters
This theorem supplies the aut_48 field in the master certificate spectral_emergence and appears in the numerological summary that lists the cube-derived numbers 8, 12, 6, 3, 48. It closes the loop from T8 (D=3) through the eight-tick octave to the 48-state fermion space, and is referenced in the AlphaHigherOrder symmetry reduction and the CubeSpectrum definitions. No open scaffolding remains for this count.
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