passiveCoupling_face

The module builds excitation ordering from the CW-filtration of the 3-cube $Q_3$. Subcells are graded by dimension: 0-skeleton (8 vertices), 1-skeleton (12 edges, of which 11 are passive), 2-skeleton (6 faces). Generation excitations are assumed to couple in that dimensional order, producing the cumulative torsion schedule ${0,11,17}$.

Passive coupling counts how many subcells at each CW level can absorb torsion without driving the active transition. By definition it is zero on vertices, the passive-edge count on edges, and the full face count on faces. Spatial dimension $D$ is the constant $3$ forced by the T8 step of the forcing chain (three spatial dimensions).

The face case therefore reduces to the elementary count of 2-faces of the unit 3-cube, which is $2D=6$.

One-line computational proof. Unfolding the passive-coupling definition on the face constructor yields the cube-face count at dimension $D$. With $D=3$ that count is the closed natural $6$, discharged by native_decide (and marked @[simp] for downstream rewriting).

This evaluation is the face contribution in the torsion schedule of the excitation-ordering argument. Generation 1 couples only to the 0-skeleton ($\tau=0$); generation 2 adds passive edges ($\tau=11$); generation 3 adds all faces, so $\tau=11+6=17$. The module then invokes J-cost monotonicity on $\varphi$-power ratios to obtain the strict cost order $J(\varphi^0)<J(\varphi^{11})<J(\varphi^{17})$.

Geometrically it records that the 2-skeleton of $Q_3$ has six faces, the structural reason faces sit after edges in CubeAdmissibleTorsion. No downstream theorems currently depend on this simp lemma in the graph, but it is the face half of the sibling triple (vertex/edge/face) that makes the cumulative schedule fully computational. The remaining open premise is the filtration principle itself: that excitations couple in CW-dimension order rather than by ad-hoc mode labels.