Building on the intrinsic variation principle~\cite{IVP} and the canonical link-space construction, this paper carries the Panvitalistic Theory (PVT) from the single comparison link to the closed comparison chain, and from the exact harmonic solution to the anharmonic physics. The paper is organised in two parts.

 

For the periodic chain with link constraints $\Phi_n=V(q_{n+1})-x_nV(q_n)$ we derive the holonomy condition $\prod_n x_n=1$, exact transverse momentum matching, the left null vector of the multiplier system (the volume profile itself), and a single scalar global condition $H_{\mathrm{glob}}=\sum_n\tilde c_n=0$ --- the discrete Gauss law and the classical shadow of a Wheeler--DeWitt constraint. On the quantum level we construct the transfer operator of the chain, prove that it carries the mode content of the intrinsic variation principle for every chain length simultaneously (the benchmark theorem), implement the volume constraint as a multiplication operator (superselection of the volume deficit), and solve the discrete Wheeler--DeWitt equation exactly at harmonic order, $\lambda_\ell(r)=4\pi e^{-3r^2/2}i_\ell(r^2)$, with the isotropic vacuum as ground state.

 

We prove a \emph{potential-absorption lemma} --- exact to all orders --- that fixes the on-site potential to the sector label, so that the entire anharmonic difficulty resides in the geometry of the constraint surface. Three independent methods agree on the ground-channel curvature, $\ln\lambda_0(\varepsilon)=\ln4\pi-3\varepsilon-\tfrac{9}{5}\varepsilon^2+O(\varepsilon^3)$, with quartic contribution $\delta a_2=-37/15$. The octahedral anharmonicity splits the excitation multiplets into cubic irreps; its leading structural feature is a symmetry-enforced \emph{exact annihilation} of the $\ell=2$ $E_g$ channel, while the surviving channels retain their harmonic coefficients. The

transfer operator reproduces the closed-form breathing branch exactly, including its $-7/(72N)$ and $61/(6480N^2)$ coefficients.

 

The measure ambiguity is resolved by consistency with the classical uniqueness theorem, and under the selected measure the sector selection is $\varepsilon_\ast=0$: the isotropic vacuum survives the quartic corrections, which reinforce rather than shift it. The parameter-free content of the theory therefore resides in the excitation spectrum; these numbers are $N$-independent, while $N=6$ retains its special role in the longitudinal chain-mode rationality. Every step is marked \derived, \posited{} or \opn; all numerical checks are documented in a reproducibility appendix.

 

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