face_generated_groups

This module bridges the classical count of 17 planar symmetry groups (Fedorov, 1891) to cube-intrinsic geometry. The endogenous program writes the wallpaper count as a sum of edge-generated and face-generated contributions, aiming to show the total is forced once spatial dimension is fixed.

Spatial dimension is the constant $D = 3$ forced by the Recognition forcing chain (T8). The face count of the $D$-cube is the standard combinatorial formula $\mathrm{cube_faces}(d) = 2d$, so at $D = 3$ one has $F = 6$. Each face is a square 2D domain whose stabilizer is the dihedral group $D_4$, compatible with oblique, rectangular, and square lattice types in the classical wallpaper list.

Face-generated groups are those arising from 2D rotational symmetries within face planes, as opposed to 1D symmetries along passive edges. The module doc states the structural target $W = E_{\mathrm{passive}} + F = 11 + 6 = 17$.

Pure definitional abbreviation: the value is exactly cube_faces D, with $D = 3$ and $\mathrm{cube_faces}(d) = 2d$ from AlphaDerivation (and matching copies elsewhere). No tactic proof; evaluation is immediate arithmetic once those constants unfold.

This constant is the face half of the structural decomposition used by W_decomposition (edge-generated plus face-generated equals the wallpaper total) and by W_eq_17 (that sum equals 17). Together they strengthen the endogenous bridge: the classical 17 is rewritten as passive-edge count plus cube face count, which equals 17 precisely when $D = 3$.

In the module narrative, the six face-type generators cover planar rotations on the six square faces; the remaining classical groups are attributed to edge directions and to triangular decompositions of faces (linking hexagonal types via dual polyhedra). The declaration therefore pins the $F = 6$ term in $W = E_{\mathrm{passive}} + F$ against the T8 landmark $D = 3$, rather than treating 17 as an external combinatorial accident.