The companion papers gave the isotropic projection  --- the map that carries the anisotropic $6\mathrm{D}$ volume of the Panvitalistic Theory (PVT) to the isotropic $3\mathrm{D}$ space of standard physics --- at the level of dimensions and geometry, and located the constants on a single $\pi$-ladder $c=\pi^{-1},G=\pi,e=\pi^2,k_B=\pi^3,h=\pi^4$ with $\pi\equiv T/L$. This paper supplies the missing operator content. On the canonical link-space algebra of the twelve volume operators~\cite{CPS} --- six coordinates (three lengths, three angles) and six conjugate momenta, subject to the Dirac constraint $\Phi_n=V(q_{n+1})-x_nV(q_n)=0$ --- we realise the projection as the maximal-orthogonality map $\theta_{jk}\to90^\circ$ and prove that it decouples the algebra: since $\partial_{\theta}\ln V=\cot\theta\to0$, the Dirac length--angle couplings vanish, the angular Fisher--Rao metric $\csc^2\theta\to1$, and the angular commutators close to zero, leaving a commuting classical length sector \derived. We then introduce the internal-measure grade operator $N$ --- the generator of the dilation $\pi\to\lambda\pi$ --- whose spectrum on the observables \emph{is} the $\pi$-ladder, prove that the volume constraint is grade-zero ($[N,\Phi_n]=0$) so that the spectral projections  are consistent with the Dirac structure \derived, and identify general relativity with the length--time sector $|n|=1$ ($c,G$) and quantum theory with the action sector $n=4$ ($h$): two orthogonal graded restrictions of one algebra. The obstruction to their coincidence is the grade gap $\Delta n=3$, and the residual areal degree of freedom $\varepsilon_0\mu_0c_{\mathrm{PVT}}^2=L^2$ sits at grade zero with residual length $m=2$ in the common kernel of both projections --- the quantum-gravitational degree of freedom that both suppress and that survives only on the unprojected volume. All operator and dimensional identities are verified symbolically; every statement is marked \derived, \posited{} or \opn.

 

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