Q3_aut_order
plain-language theorem explainer
Q3_aut_order is defined as the integer 48, representing the order of the automorphism group of the three-dimensional cube. This constant enters symmetry reduction formulas for counting n-fold face-wallpaper configurations in higher-order corrections to alpha inverse. The assignment follows immediately from the semidirect product structure Aut(Q_D) = S_D ⋊ Z_2^D whose order is D! times 2^D.
Claim. The automorphism group order of the three-dimensional cube satisfies $|Aut(Q_3)| = 48$.
background
The module AlphaHigherOrder develops higher-order voxel-seam corrections to the fine-structure constant alpha inverse. It starts from the geometric seed alpha_seed = 4 pi times 11, applies the gap weight f_gap = w_8 ln phi, and adds curvature terms delta_n to approach the CODATA value. Each delta_n is a combinatorial sum over n-fold face-wallpaper configurations on Q3, weighted by the Z_2^5 half-period measure, with symmetry reduction applied via the automorphism group order.
proof idea
The declaration is a direct definition assigning the constant 48. It replicates the value established by the theorem in SpectralEmergence that applies norm_num to the factorial expression for aut_order 3. The CubeSpectrum sibling provides an equivalent definition justified by native_decide on the semidirect product order.
why it matters
This supplies the group order used by reduced_configs to bound the number of symmetry-reduced configurations for computing delta_n terms in the alpha series. It feeds directly into the spectral_emergence master theorem, which assembles the certificate that the entire Standard Model structure emerges from D equals 3. The value also completes the numerological summary listing 48 as the automorphism count alongside vertices 8 and edges 12.
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