IDOS Energy, Coherence, and Supporting Dynamics — Part 31

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ordinary $\mathbb R^{10}$, irrespective of the pattern's active depth. Write $W=(1+|x|^2/L^2)^{-4}$, $Z_i=(x_i/L)(1+|x|^2/L^2)^{-5}$, $\rho=n_{\rm peak}W^2$, $y=(s,z)$, $D_s=\nabla(W,Z)$ and $M_s=\operatorname{diag}(M_W,M_Z,\ldots,M_Z)$. Let $f_{ij}=-f_{ji}$ be frame rotation, $v$ the ten-component flow variation and $H=\sqrt w L=C/2$. Put

$$b_* = \nabla\theta+2H^TX,\qquad F_i=\partial_iY-\sum_jH_jf_{ij},\qquad \alpha=2s/W.$$

The density equation gives $r=\alpha+\dot\theta/\beta$, where $\beta=b$ at the reference branch. Its elimination retains $\rho\alpha\dot\theta$ as well as $\rho\dot\theta^2/(2\beta)$. The other temporal squares are $\rho\epsilon(|\dot X|^2+|\dot Y+Hv|^2)/2$ and $(\dot y+D_sv)^TM_s(\dot y+D_sv)/2$, together with the frame and flow kinetic terms. The clock also retains $\rho(2X\cdot\dot Y+\omega|v|^2/2+v\cdot b_*)$; its spatial alignment term is $-\rho\kappa|b_*+\omega v|^2/2$. Thus setting support, frame and flow to zero is generally not invariant. Their omitted equations would additionally require $\dot\theta=0$, $H_b\cdot\partial_aY-H_a\cdot\partial_bY=0$ and $(\kappa\omega-1)b_*-\epsilon H^T\dot Y=0$.

In the conserved quadratic energy, velocity-linear terms cancel. The phase-flow potential completes to

$$\frac{\rho\kappa}{2}|b_*+(\omega-1/\kappa)v|^2+ \frac12v^T\left[\rho(\omega-1/\kappa)I-S_{\rm grad}\right]v, \qquad S_{\rm grad}=D_s^TM_sD_s+\rho\epsilon H^TH.$$

The negative drift contributions in this expression are essential. For the existing common choices $M_W=M_Z=n_{\rm peak}L^2\omega_{\min}/212$, $\epsilon=\omega_{\min}\ell^2/(4M^2)$ and $\kappa=8/\omega_{\min}$, the actual rational support obeys

$$\frac{\|D_s\|^2}{W^2}\le\frac{32161}{1600L^2}<\frac{21}{L^2}, \qquad S_{\rm grad}\le\rho\omega_{\min}\frac{37}{106}I,$$ $$\rho(\omega-1/\kappa)I-S_{\rm grad} \ge\rho\omega\frac{223}{424}I>0.$$

The alternative branch-flat $\kappa=8/\omega$ retains this sufficient margin. All other energy terms are the nonnegative carrier, support, frame and flow derivative or occupation squares. This is the second variation of $E-\omega Q$ at the branch, not an arbitrary-background laboratory-energy theorem. Its no-growing-mode conclusion requires a conserved finite-energy realization; positivity alone supplies neither a spectral gap nor nonlinear persistence. The full equations and support residuals are in coupled-spatial-support/report.md, independently checked in coupled-support-review/review.md with 506 exact assertions.

A useful exact spatial operator is $A_\rho=-\rho^{-1}\operatorname{div}(\rho\nabla)$ on $L^2(\rho\,dx)$. The unitary transformation $Uf=\sqrt\rho f$ gives

$$UA_\rho U^{-1}=-\Delta-\frac{80L^2}{(L^2+|x|^2)^2}.$$

The potential is bounded. The transformed operator has domain $H^2$, compact smooth core, and closed form

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