<aside> 🧊
Linked synthesis credential: Professor’s Cube Exhibit — Shared-Carrier S₃ Symmetry Bridge.
</aside>
The plate’s constrained continuum density is
$$ H_\perp=\frac12\rho_I|\nabla S_I|^2+U(\rho_I)+\kappa|\nabla\rho_I|^2+\alpha E_{\mathrm{Sk}}+\cdots, $$
with momentum constraint
$$ H_i=\rho_I\nabla_iS_I=0. $$
The Madelung kinetic term, Casimir potential, Fisher-information term, and Skyrme stabilizer are therefore registered as one typed Hamiltonian density. The finite computational branches are $L_{C_5^3}$ for lattice diffusion and $K^2$ for E47 spectral folding.
$\mathcal Q[\mathsf{Bohm\text{-}Madelung\ Kernel\ Identity}\mid \Psi=\sqrt\rho e^{iS/\hbar};\ \mathbf{Hilb};\ \text{Madelung split with }V_K\propto K^2;\ E0;\ \text{interpretive physical bridge}]$
$i\hbar\partial_t\Psi=\left(-\frac{\hbar^2}{2m}\Delta+V_K\right)\Psi$
implies
$\partial_t\rho+\nabla\cdot\left(\rho\frac{\nabla S}{m}\right)=0$
and
$\partial_tS+\frac{\|\nabla S\|^2}{2m}+V_K+Q_B=0,\qquad Q_B=-\frac{\hbar^2}{2m}\frac{\Delta\sqrt\rho}{\sqrt\rho}.$
With $V_K\propto\langle\Psi,K^2\Psi\rangle$, the zero-potential sector is $E_{47}=\ker K^2$.
Boundary: $V_K$ and $Q_B$ coexist in the Hamilton–Jacobi equation but are not the same operator.