decomposition_at_D3
plain-language theorem explainer
At spatial dimension three, passive field edges equal 11, hypercube faces equal 6, and their sum (the endogenous W-count) equals 17. Cite this for the wallpaper endogenous bridge or the counting identity 11+6=17. The proof is a single native_decide on the closed combinatorial formulas at the fixed D=3.
Claim. At the forced spatial dimension $D=3$, the number of passive field edges equals $11$, the number of hypercube faces equals $6$, and the endogenous wallpaper count $W_{\mathrm{endogenous}}(3):=E_{\mathrm{passive}}(3)+F(3)$ equals $17$.
background
This module builds an endogenous Recognition Science candidate for the crystallographic constant $W=17$ from cube combinatorics, without re-proving Fedorov's classification. The candidate is
$$W_{\mathrm{endogenous}}(d)=E_{\mathrm{passive}}(d)+F(d),$$
with closed form $d\cdot 2^{d-1}-1+2d$.
Spatial dimension $D$ is fixed at 3 by the forcing chain (T8). Hypercube faces are $F=2D$, hence six at $D=3$. Passive field edges are total hypercube edges minus the single active edge per tick; the AlphaDerivation layer records that this equals 11 when $D=3$. The module goal is to make the identity $11+6=17$ machine-checked and match the imported classical wallpaper-group count.
proof idea
One-shot computational proof via native_decide. All three sides are closed natural-number expressions in the fixed parameter $D=3$: faces evaluate as $2\cdot 3$, passive edges come from the edge-count formula already defined upstream, and the endogenous $W$ is their sum. The kernel simply evaluates the three equalities.
why it matters
Pins the counting-layer bridge from RS cube combinatorics to $W=17$ at the forced dimension $D=3$ (T8 in the forcing chain). It makes explicit the decomposition $11+6=17$ that the module advertises as the endogenous candidate matching the classical wallpaper-group count.
Sibling results in the same file ($W$ at $D=3$, cube-derived $W=17$, uniqueness of the wallpaper slot from the endogenous formula) rest on this three-way equality. Full endogeneity of $W$ remains open: the file still imports the classical fact that there are 17 wallpaper groups and only shows that the combinatorial candidate coincides at $D=3$.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.