IDOS Energy, Coherence, and Supporting Dynamics — Part 46

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The remaining blocks are $\epsilon\rho(UU-P/4)$, $M_WUU-\gamma_AP$, $M_ZUU-\gamma_BW^2P/(1+W^2)$, $\kappa_F\rho(UU-c_F^2P)$ and $\kappa_U\rho(UU-c_U^2P)$, repeated for their components. Nonzero residuals add lower-order forces but no omitted derivative Hessian: each nonclock residual is affine in derivatives with algebraic coefficients.

For finite $W>0$, positive smooth constitutive coefficients and regular future flow/frame coordinates, each $G_a$ is Lorentzian and its ray cone lies in the ordinary unit cone. Therefore any inertial laboratory time has $G_a^{00}>0$, and

$$\beta_a^i=G_a^{0i}/G_a^{00},\qquad g_a^{ij}=\beta_a^i\beta_a^j-G_a^{ij}/G_a^{00}>0.$$

This remains true when the canonical spatial energy matrix is indefinite. In holonomic variables $V=q_t,D_i=\partial_iq$, define $B^i=T^{-1}\operatorname{diag}(\beta_a^i)T$ and $C^{ij}=T^{-1}\operatorname{diag}(\beta_a^i\beta_a^j-g_a^{ij})T$. The exact principal equations can be augmented by curls:

$$V_t+2B^i\partial_iV+C^{ij}\partial_iD_j=f(q,V,D;t,x), \qquad D_{i,t}-\partial_iV+B^j(\partial_jD_i-\partial_iD_j)=0.$$

A smooth simultaneous symmetrizer is the following quadratic form on tangent increments $(\delta V,\delta D)$:

$$\sum_a\left[ \left((T\delta V)_a+\beta_a^i(T\delta D_i)_a\right)^2 +g_a^{ij}(T\delta D_i)_a(T\delta D_j)_a\right].$$

It is not the Jacobian of a nonlinear change to convected velocities. That distinction matters because $\beta$ depends on the first jet. At a frozen state the corresponding wave blocks have symmetric matrices $\left(\begin{smallmatrix}\beta^i&-g^{ij}\\-g^{ki}&\beta^ig^{kj}\end{smallmatrix}\right)$ and positive time form $\operatorname{diag}(1,g)$. Pullback gives the displayed symmetrizer for every direction at once.

For $\mathcal C_{ij}=\partial_iD_j-\partial_jD_i$, exact differentiation gives

$$\mathcal C_{ij,t}+B^k\partial_k\mathcal C_{ij} =-(\partial_iB^k)\mathcal C_{kj} +(\partial_jB^k)\mathcal C_{ki}.$$

$T^TT$ symmetrizes every $B^k$, so this homogeneous constraint system preserves zero curl. Then $D_i-\partial_iq$ remains zero as well. The regular algebraic charts and auxiliary reconstruction preserve the remaining constraints. No global phase coordinate or vanishing carrier curvature was assumed.

Every fixed compatible compact datum with $q_0\in H^{s+1}$, $q_1\in H^s$, integer $s\ge7$, lying in this regular positive region therefore has a positive data-dependent local existence time. Its norm and finite gradients need not be small. On a finite collar the compact initial first-jet image provides positive bounds for the symmetrizer. Apply the local symmetric-hyperbolic theorem there. Alternatively extend the matrices after symmetrization, blending $A^0$ to the identity and $A^i$ to zero; positivity, symmetry and the cone inequalities survive. Local uniqueness glues to the exact rational exterior. Overlapping regular target charts may be treated locally and glued on compatible Euler solutions.

The solution has $q\in C H^{s+1}$, $q_t\in C H^s$, $q_{tt}\in C H^{s-1}$, with the usual lower-Sobolev-norm continuous dependence. Reconstructed $n$ and $\sqrt{2n}e$ generally have only $H^s$ perturbation regularity because $n$ depends on first derivatives. There are no independently prescribed density velocity data.

This theorem includes each fixed regular compact tail competitor, but does not supply evolution toward it. No uniform lifespan in tail radius, vacuum crossing, infinite-carrier limit, instability, decay or nonlinear stability follows. Both phase-edge choices and both smooth positive coefficient branches obey the proof at every maintained depth. The source compact-large-data-symbol/report.md and independent 25-check review give the complete action, constraint proof and external PDE theorem attribution. The 431 author checks are supplementary small-matrix controls, not numerical solutions of all full-carrier actions.

Smooth Density Relaxation With All Retained Charges

The two-region calculation has a smooth spatial counterpart that retains the actual joining layer. Fix rational support positions, rest independent

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