<aside> 🛰️
MC-E47-RADAR-BRIDGE-20260927-001 · PASS
Exact E47 finite algebra + E1 synthetic radar bridge. Measured-radar validation remains open.
</aside>
A calibrated local complex radar cube
$$ X(a,r,t)=I(a,r,t)+iQ(a,r,t),\qquad a,r,t\in\{-2,-1,0,1,2\} $$
has exactly $5^3=125$ complex samples. Vectorization gives the explicit carrier map
$$ \iota_R:\mathbb C^{5\times5\times5}\to V_2^{\otimes3}\cong\mathbb C^{125},\qquad x=\operatorname{vec}(X). $$
The E47 operator is unchanged:
$$ C=J_x^2+J_y^2+J_z^2, $$
$$ K=(C-6I)(C-30I), $$
$$ E_{47}=\ker K=W_2\oplus W_5, $$
$$ P_E=P_6+P_{30},\qquad \operatorname{rank}P_E=47. $$
The seven normalized Casimir-shell populations are
$$ q_j(x)=\frac{\|P_jx\|^2}{\|x\|^2},\qquad \sum_{j=0}^6q_j=1. $$
The E47 occupancy is
$$ \eta_E=q_2+q_5=\frac{\|P_Ex\|^2}{\|x\|^2}. $$
For isotropic calibrated complex noise
$$ x\sim\mathcal{CN}(0,\sigma^2I_{125}), $$
one has
$$ \mathbb E[\eta_E]=\frac{\operatorname{Tr}P_E}{125}=\frac{47}{125}=0.376. $$