Exact proof monograph · Python source · JSON certificate
The previously missing Rubik representation is now explicit on the full 125-site carrier. Fifteen outer and inner quarter-turn operators act as exact permutation matrices:
$$ \mathcal P_{15}=\{P_R,P_L,P_U,P_D,P_F,P_B,P_{R2},P_M,P_{L2},P_{F2},P_S,P_{B2},P_{U2},P_E,P_{D2}\}. $$
Each satisfies $P^TP=I$, $P^4=I$, and $\|P\rho\|_2=\|\rho\|_2$. The supplied eigenmode impact table is reproduced exactly to its displayed rounding. Eight citizens RUB5-C01…RUB5-C08 now carry the operator, spectral, energy, low-band, flow, and cross-carrier results.
Canonical correction: the $C_5^3$ Laplacian has $\dim\ker L=1$. Its heat flow converges to the uniform projector $P_0=\mathbf1\mathbf1^T/125$, not to the 47-dimensional $E_{47}$ projector. The shared dimension 125 is an exact carrier correspondence; the kernel bridge remains typed and separate.
<aside> △
Status: valid graph-to-field visualization pipeline; no independent derivation of $E_{47}$ from the social graph.
</aside>
Let
$$ G=(V,E) $$
be a graph whose vertices are people and whose edges are observed connections. Assign each vertex a feature vector
$$ f_i\in\mathbb R^d. $$
Choose an embedding
$$ \Phi:V\to\mathbb R^m,\qquad x_i=\Phi(f_i). $$
A smoothed influence field may be defined by
$$ I(x)=\sum_iw_i\kappa(x,x_i), $$
and a density field by
$$ \rho(x)=\sum_i\exp\!\left(-\frac{\|x-x_i\|^2}{2\sigma^2}\right). $$
A composite scalar field can then be written