IDOS Energy, Coherence, and Supporting Dynamics — Part 34

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components in $H^1$ except vector support $\zeta_Z$, for which $W\nabla\zeta_Z\in L^2$ suffices. Bounded lower-order coefficients make the graph norm equivalent to these derivative norms plus $\|u\|^2$. Compact cutoffs followed by local smoothing prove compact smooth fields are a form core. The associated nonnegative self-adjoint operator $A$ is the Friedrichs realization, with $D(A^{1/2})=\mathcal V$. This does not prove essential self-adjointness of the full minimal differential operator.

On $\mathcal V\times L^2$, the free wave group uses $\cos(t\sqrt A)$ and $\sin(t\sqrt A)/\sqrt A$, with the latter multiplier equal to $t$ at zero. The velocity perturbation $(u,p)\mapsto(0,-Gp)$ is bounded there. The convergent variation-of-constants construction therefore gives a unique global energy solution, without a spectral gap. Skewness and approximation from operator-domain data yield

$$E_2(t)=\tfrac12(\|\dot u(t)\|^2+q[u(t)])=E_2(0),\qquad \|u(t)\|\le\|u(0)\|+|t|\sqrt{2E_2(0)}.$$

Density reconstructs as $hr=2\sqrt{n_{\rm peak}}s+\dot u_\theta/\sqrt\beta$ and remains in weighted $L^2$. Compact curvature-bearing preparations thus have global coupled linear continuations with support, frame and flow allowed to respond. No pointwise smallness, nonlinear chart preservation or nonlinear vacuum continuation follows. The separately reviewed finite empty-label blocks are decoupled at this order; adjoining them is a finite direct sum of their own global energy evolutions. It supplies no full-system gap. The global occupied-block proof is in coupled-energy-evolution/report.md, with 74 independent checks in coupled-evolution-review/report.md.

Positive directional quadratic energy also has an explicit nonlinear limitation. Within the same finite carrier, choose positive cutoffs $1\le N_L<N_H\le M$ at one depth and let $\Delta=\mu_d[\binom{N_H+d}{d}-\binom{N_L+d}{d}]>0$. Write $h_{\rm clock}=m+(v_*^2/\kappa)(\sqrt{1+\kappa^2|a|^2/v_*^2}-1)$ for the constitutive clock function, distinct from $h=\sqrt\rho$ above. Keep the rational support, frame and flow fixed initially, set $n=\rho$ and choose the exact initial velocity $e_t=(h_{\rm clock}+\gamma V)Je$. Then the density equation holds and the complete initial energy and phase charge are

$$E=\int\rho\left[h_{\rm clock}+\gamma V+\frac\epsilon8\sum_i|R_i|^2\right]dx, \qquad Q=\int\rho\,dx.$$

A ball of fixed large radius $a$ centered at a remote $X$ can carry the lower prefix, joined through a fixed-thickness smooth wall to the high prefix. Interpolate real carrier vectors, not their phase angles. Their nested overlap has absolute value at most $c=\sqrt{D_L/D_H}<1$; the normalized chord has denominator squared at least $(1-c)/2$. Thus wall derivatives and its energy density have finite bounds independent of $X$ and $a$. This construction works even when the original finite carrier has no empty label.

If the wall thickness is $w$, its density-weighted energy satisfies

$$E_X-E_H\le-\Delta\rho_-V_{10}a^{10} +C\rho_+V_{10}[(a+w)^{10}-a^{10}], \quad V_{10}=\pi^5/120,$$

where $C$ is a finite wall bound and $\rho_\pm$ bound the actual rational density on the ball. First choose finite $a$ so that

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