IDOS Energy, Coherence, and Supporting Dynamics — Part 13

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with the unit-carrier and orthogonality constraints enforced in variation. The derivative order is one. All constants are common across the finite positive target set. The fractional powers are $C^2$, not $C^3$ at zero.

For $t_r=|x-X|^2/L^2$, take

$$v=(1+t_r)^{-4},\quad Z_0=0,\quad Z_i=(x_i-X_i)(1+t_r)^{-5}/L,\quad n=n_{\rm peak}v^2, \quad e=e^{J\omega t}e_0(x).$$

The prepared $e_0$ has zero spatial phase current and constant selected Gram descriptor. Every support residual and proper-time support derivative vanishes pointwise. Hence the full support variation, including its metric and projector variations, vanishes. The clock equations reduce to $s=\omega$, $\partial_\mu(nu^\mu)=0$ and $u\cdot\partial e=\omega Je$. These are satisfied by the full preparation. On this branch the complete current and stress are

$$j^\mu=nu^\mu,\qquad T^{\mu\nu}=n\omega u^\mu u^\nu.$$

They are branch identities: off-branch mass and support variations contribute to both. A full Lorentz transformation, including contraction of the profile and transformation of $Z_\mu$, gives

$$Q=S,\quad E=\gamma S\omega,\quad P=\gamma S\omega V,\quad E^2-|P|^2=(S\omega)^2, \qquad S=\frac{n_{\rm peak}\pi^5L^{10}}{2520}.$$

The density has $\langle r^2\rangle=5L^2/2$ and all retained support components have finite unweighted squared norm. The support zero set on its positive smooth branch consists of translations of this profile. It has no positive global unweighted Hessian gap, since low-cost broad perturbations can move into the tail. Common $n_{\rm peak},L$ fix one common $S$, as supplied constitutive parameters rather than individual particle amplitude fits. The laboratory charge density is $\gamma n$, not the proper density $n$.

Coupled Disturbance Cone

Covariance alone does not establish causal disturbances. On flat mass jets, let $Y=(v,Z_i)$, $\mathsf G_{\alpha i}=\partial_iY_\alpha/\omega$, $b_0=n/\omega$, $a=1/\lambda$, $\mathsf M=\operatorname{diag}(M_v,M_Z,\ldots,M_Z)$ and $\mathsf C=\operatorname{diag}(\gamma_A,\gamma_Bv^2,\ldots,\gamma_Bv^2)$. The full quadratic variation includes the phase-induced support velocities $y_t-\mathsf G\nabla\theta$. Its derivative-bearing Lagrangian is

$$\mathcal L_2^{\rm pr}=\frac a2\theta_t^2 +\frac12(y_t-\mathsf G\nabla\theta)^T\mathsf M (y_t-\mathsf G\nabla\theta) -\frac{b_0}{2}|\nabla\theta|^2 -\frac12\sum_i(\partial_i y)^T\mathsf C\partial_i y.$$

Horizontal carrier perturbations $h$ also have the nondegenerate first-order term $n\langle Jh,h_t\rangle$. Using $b_i=\partial_i\theta-2\langle J\partial_i e_0,h\rangle$ in the first-order reduction retains their coupling and removes derivatives of $h$ from the principal wave block. Their principal speed is zero; they are not discarded.

Positive energy follows when $\mathsf H=b_0I-\mathsf G^T\mathsf M\mathsf G>0$. For $\mathsf M-\mathsf C>0$, the coupled cone is inside the unit light cone if

$$a\ge b_0+\lambda_{\max} \left[\mathsf G^T\mathsf M\mathsf C (\mathsf M-\mathsf C)^{-1}\mathsf G\right].$$

The positive energy symmetrizer establishes a complete characteristic reduction, rather than only real roots of a determinant. The profile bound $|\nabla Y|^2/n\le106/(n_{\rm peak}L^2)$ gives one sufficient common choice:

$$M_v=M_Z=\frac{n_{\rm peak}L^2\omega_{\min}}{212},\qquad \gamma_A=\gamma_B=M_v/4,\qquad \lambda=\frac{\omega_{\min}}{2n_{\rm peak}},\qquad\kappa>0.$$

This proves local strong hyperbolicity of the linearized constrained system on compact positive-density regions of the exact branch. Lorentz boosts preserve its causal rays. Positive coefficients alone do not suffice: explicit controls have a positive-energy superluminal root and, separately, an imaginary characteristic pair. Vacuum, uniform tail estimates, generic nonlinear evolution and reciprocal scattering remain open. A changed density-weighted kinetic version improves normalized principal tail bounds but introduces coefficients singular at generic vacuum.

The same law applies to every active depth in the common ambient space; there is no exceptional depth-five result. The zero-frequency photon branch is outside this timelike massive construction. Occupied nested bands and constant participating energy are still separate requirements.

Test Against The Uniform Local Clock

The exact branch has $h=\omega n(x)$, so its local action density varies. It does not obey a constant-$\sigma_0$ law $\Omega_{\rm local}=h/\sigma_0$ with one spatially uniform frequency. This is a substantial difference, not a negligible tail correction. Under its normalized energy measure, $z=(1+r^2/L^2)^{-1}$ has density $105z^2(1-z)^4$; the proposed local rate is proportional to $z^8$. Optimizing its normalization, the largest energy fraction within relative tolerance $\delta$ of any local rate is

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