W_from_cube_eq_wallpaper_groups
plain-language theorem explainer
The cube-derived endogenous wallpaper count at spatial dimension three equals the classical Fedorov constant of seventeen wallpaper groups. Anyone closing the counting-layer bridge from cube combinatorics to the imported crystallographic W cites this equality. The proof is a one-line unfold of the named cube constant against the already-proved match of the endogenous formula to wallpaper_groups.
Claim. The endogenous wallpaper candidate obtained from the cube formula at spatial dimension $D=3$ equals the classical number of two-dimensional wallpaper groups: $W_{\mathrm{endogenous}}(3)=17$.
background
This module is Pass 2 of the wallpaper endogenous bridge: an explicit link from cube combinatorics to the crystallographic constant $W=17$. The framework still imports the classical count (Fedorov 1891) and does not re-prove wallpaper classification. Instead it defines an RS-native candidate
$$W_{\mathrm{endogenous}}(D):=E_{\mathrm{passive}}(D)+F(D).$$
Spatial dimension is fixed at $D=3$ by the forcing chain (T8/T9). At that value one has $E_{\mathrm{passive}}=11$ and $F=6$, so the endogenous count is $17$. The named constant in the signature is simply that evaluation: the cube-derived candidate at $D=3$. Upstream, the sibling match already records that this endogenous value equals the imported wallpaper-group constant used in the alpha-derivation denominator (faces times wallpaper groups).
proof idea
One-line wrapper. Unfold the definition of the cube-derived constant as the endogenous formula evaluated at $D$, then apply the sibling lemma that already equates $W_{\mathrm{endogenous}}$ at that $D$ with the imported wallpaper-group count. The simpa step discharges the resulting definitional equality; no new arithmetic is performed here.
why it matters
Closes the named-constant form of the counting-layer identity so three parents can quote a single equality. Downstream, the full bridge package packages this with the numerical claim "equals 17" and the generator-level iff that the wallpaper slot is exactly $E_{\mathrm{passive}}+F$. The slot-closure theorem uses it to identify the imported constant with the cube formula. On the mass-topology path it lets the endogenous cube value replace the imported $W$ without changing ledger fractions.
In the Recognition framework this is a bridge step toward full endogeneity of $W$, not a replacement of Fedorov. It makes the machine-checked identity $11+6=17$ available wherever alpha derivation or mass topology needs the wallpaper slot, with $D=3$ inherited from the forcing chain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.