IDOS Energy, Coherence, and Supporting Dynamics — Part 12

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The zero-energy density squares preserve the exact translating Gaussian, but their perturbation dynamics need a separate check. For the logarithmic square at $\rho_G$, put $v=\delta\rho/\rho_G$ and

$$\mathcal A=-\rho_G^{-1}\nabla\cdot(\rho_G\nabla),\qquad \mathcal B=\rho_G^{-1}\partial_i\partial_j (\rho_G\partial_i\partial_j)=\mathcal A(\mathcal A-\alpha).$$

On the common flat mass jets, the full orientation perturbation is driven by the induced transport and must not be held fixed. Nevertheless the density-current subsystem closes: $a_0=0$ implies zero phase curvature, so $\delta a_t=\nabla\delta\mu$, with $\delta\mu=\beta\mathcal Bv$. Gaussian Hermite degree $k$ therefore has

$$\Omega_{{\rm shape},k}^2 =\frac{\beta c^2\alpha^3}{\omega}k^2(k-1).$$

Degrees zero and one are action and translation directions; higher density shape modes oscillate in this linear subsystem. These disturbance rates are distinct from the particle's actual clock $\omega$. A nonflat second mass derivative can couple additional orientation stiffness, so this formula does not transfer to every stationary mass function.

At a smooth positive point the local high-frequency relation is $\Omega_{\rm shape}^2\sim(\beta c^2/\omega)|\xi|^6$. Thus the local logarithmic support has unbounded high-frequency group speed. The polynomial square has the corresponding coefficient $\gamma\rho_G^3c^2/\omega$; its weighted operator is not diagonal in the same Hermite basis. The action-current bound remains valid, but bounds transport into vacuum rather than the propagation of differences between solutions on a nonzero background. The everywhere-positive Gaussian has no vacuum boundary.

The independent homogeneous controls also distinguish prepared solutions from arbitrary backgrounds. For the log square, $\Omega_{\rm shape}^2=(\beta c^2/\omega)(|\xi|^6-2\alpha|\xi|^4)$, so a uniform positive-density background has a long-wave instability. For the polynomial square at constant density $R$,

$$\Omega_{\rm shape}^2=\frac{\gamma R^3c^2}{\omega}|\xi|^2 \left[|\xi|^4-8\alpha|\xi|^2+6n\alpha^2\right].$$

Its reduced $n=2$ control has an unstable interval $2\alpha<|\xi|^2<6\alpha$, while the other maintained reduced dimensions have a positive bracket. Full ambient localization uses $n=10$ at every active depth. There is no exceptional depth-five effect.

One changed positive support smooths the residual spatially:

$$K_{\rm smooth}=\frac\beta2\langle\sqrt\rho Q, (1-\ell^2\Delta)^{-2}\sqrt\rho Q\rangle.$$

It and its first variation still vanish on every selected translated Gaussian. Its frozen principal dispersion is $\Omega=c\sqrt{\beta/\omega}|\xi|^3/(1+\ell^2|\xi|^2)$, with group speed bounded by $(9/8)c\sqrt{\beta/\omega}/\ell^2$. This controls the high-frequency growth at the cost of explicit spatial nonlocality. It is not a finite-front-speed theorem or a derived elimination of specified field components. The generic local scalar saturation of a nondegenerate square leaves the same high-frequency linearization. A local dynamical completion must therefore retain and test its own supporting degrees of freedom and characteristic equations. The independent derivation and 73 exact checks are preserved in support-dispersion-control/report.md.

Finite Support With A Covariant Count Clock

A local inverse construction can retain explicit supporting components while preserving the complete energy of the count clock. This changes the kinetic law; it is not an off-branch equivalence to the canonical current action. Use ordinary ten-dimensional physical space, signature $(-,+,\ldots,+)$, and $c=1$. The real unit carrier $e$ is a Lorentz-scalar multiplet here. Identifying it with a fundamental Lorentz spinor would need a separate bridge. Its components and the following support components belong to the proposed single field, not additional physical dimensions or mediator particles.

Put $a_\mu=\langle Je,\partial_\mu e\rangle$, $s=\sqrt{-a_\mu a^\mu}>0$, $u^\mu=-a^\mu/s$, and $P_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu$ on the future timelike branch. The proper density is $n=|\psi|^2/2>0$. Horizontal derivatives $D_\mu e=\partial_\mu e-a_\mu Je$ give a projected spatial Gram tensor. Its first ten power sums form covariant descriptors $C$ of the nonzero rest-space eigenvalues. One positive common function $m(C)$ can take the supplied count values on distinct prepared descriptors, with flat jets in their neighborhoods. This is an inverse fit of a finite ladder, not a proof of modal occupation. Identical descriptors cannot receive different frequencies; in particular the zero-cutoff descriptor does not identify depth.

Let $v$ be real and $Z_\mu$ be a real covector with $u\cdot Z=0$. For $q=4$, define the spacelike residuals

$$A_\mu=P_\mu{}^\nu\partial_\nu v+2qZ_\mu/L,$$

$$B_{\mu\nu}=vP_\mu{}^\alpha P_\nu{}^\beta\nabla_\alpha Z_\beta -[|v|^{2+1/q}P_{\mu\nu}-2(q+1)Z_\mu Z_\nu]/L,$$

$$G=Z^2-|v|^{2+1/q}+|v|^{2+2/q},\qquad R=n-n_{\rm peak}v^2.$$

With $W_\mu=P_\mu{}^\nu u^\alpha\nabla_\alpha Z_\nu$, the action density is

$$\mathcal L=n[s-m(C)]+\frac{M_v}{2}(u\cdot\partial v)^2 +\frac{M_Z}{2}W^2-\frac{\gamma_A}{2}A^2-\frac{\gamma_B}{2}B^2 -\frac\kappa2G^2-\frac\lambda2R^2,$$

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