W_eq_17

Fedorov (1891) classified the distinct discrete groups of isometries of the Euclidean plane that act cocompactly: there are exactly 17 wallpaper groups. Recognition Science seeks an endogenous account of that count from the 3-cube rather than an external geometric census.

In this module the classical 17 are split structurally into edge-generated groups (defining symmetry along 1D edge directions: translations, reflections, glides) and face-generated groups (defining symmetry from 2D face rotations, including the triangular decomposition that yields the hexagonal families). The intended split is 11 + 6.

Spatial dimension is fixed at $D = 3$ by the forcing chain (T8). The companion endogenous identity is $W_{\mathrm{endogenous}}(D) := E_{\mathrm{passive}}(D) + F(D) = 17$ if and only if $D = 3$, with $F = 6$ cube faces and eleven passive-field edges.

Both summands are closed natural-number definitions in the same module (the edge-generated and face-generated counts). The goal is a pure numeral equality on $\mathbb{N}$. The proof is the single tactic native_decide, which evaluates the concrete definitions and checks $11 + 6 = 17$ by kernel computation. No intermediate lemmas are invoked.

This pins the structural decomposition advertised in the module: $W = $ (edge-generated) $+$ (face-generated) $= E_{\mathrm{passive}} + F = 11 + 6 = 17$. It is the arithmetic witness that the cube-face bridge lands on the classical wallpaper count, strengthening the endogenous story already proved as $W_{\mathrm{endogenous}}(D) = 17 \Leftrightarrow D = 3$.

Framework landmarks in play are T8 ($D = 3$ spatial dimensions) and the cube's face count $F = 6$ with dihedral $D_4$ face stabilizers, plus the triangular face decomposition that accounts for the five hexagonal groups. No downstream theorems currently depend on this declaration; it is a verification checkpoint inside the wallpaper classification bridge rather than a lemma consumed elsewhere.