Q3_aut_order
plain-language theorem explainer
The definition assigns the integer 48 to the order of the automorphism group of the three-dimensional hypercube. Researchers counting symmetry-reduced lattice configurations or verifying Standard Model numerology from D = 3 cite this constant when applying group order reductions. The assignment is a direct constant binding with no further computation.
Claim. The order of the automorphism group of the three-dimensional hypercube equals 48.
background
The CubeSpectrum module treats the three-dimensional hypercube Q₃ as the unit cell of the lattice ℤ³. This graph has eight vertices, twelve edges, and six faces; its automorphism group is the semidirect product S₃ ⋊ ℤ₂³ whose order is 3! × 2³ = 48. The module uses these counts to derive spectral properties that enter critical exponent corrections in Recognition Science.
proof idea
The declaration is a direct definition that binds the name to the literal integer 48. It is consistent with the upstream theorem that evaluates the general automorphism order function at dimension 3 and obtains 48 by normalization with the factorial.
why it matters
This supplies the automorphism order required by the master theorem spectral_emergence, which certifies that the full Standard Model structure follows from D = 3 with zero free parameters. It also feeds the numerological summary that equates 48 to 2³ × 3! and counts total fermion states. In the Recognition framework the value anchors the eight-tick octave and the spatial dimension D = 3, enabling symmetry reductions for higher-order alpha terms.
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