edge_direction_count
plain-language theorem explainer
Counts edges of the D-cube along one spatial axis as the natural number 2^{D-1}. At the forced dimension D=3 this is 4, the quark color offset in the topological Z-map. Mass-layer and Z-map forcing theorems cite it as the first-principles source of that constant. The body is a one-line power-of-two definition in D.
Claim. Let $D$ be the spatial dimension. The number of edges of the $D$-cube along one fixed spatial direction equals $2^{D-1}$.
background
This module derives the charge-to-band map $Z(\tilde Q)$ from recognition boundaries on the 3-cube, without anchors or empirical masses. Stage 1 integerizes SM charges by the face count $F=2D$; Stage 2 forces the even polynomial $Z=a\tilde Q^2+b\tilde Q^4$ with $a=b=1$; Stage 3 adds a sector-dependent color offset for quarks.
The offset is identified with the number of recognition channels along cube edge directions. Upstream, $D$ is the spatial dimension forced by the T8/T9 chain and is defined as the constant $3$ in AlphaDerivation, GapDerivation, and FermionDOFGapBridge. The hypercube has $2^D$ vertices; fixing one axis leaves $2^{D-1}$ parallel edges.
proof idea
Definitional abbreviation only: the count is written as the natural-number power $2^{D-1}$. There is no tactic proof. The sibling theorem that evaluates it at $D=3$ is a one-line native_decide showing the value equals 4.
why it matters
Stage 3 of the topological Z-map derivation: quarks pick up an additive offset equal to this edge-direction count, so $Z_{\mathrm{quark}}=2^{D-1}+\tilde Q^2+\tilde Q^4$ while leptons keep $Z=\tilde Q^2+\tilde Q^4$. Downstream, edge_direction_eq_four and canonical_color_offset pin the offset at 4; full_Z_values and Z_full insert it into the numerical band labels (1332, 276, 24); the Masses.ZMapForcing pair forces and verifies the canonical tuple $(k,a,b,c)=(6,1,1,4)$ from first principles. Ties directly to the T8 landmark $D=3$ and the cube topology underlying the eight-tick octave.
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