IDOS Energy, Coherence, and Supporting Dynamics — Part 44

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$K^{ij}=-\mathcal L_{q_iq_j}$. At the branch, in phase, horizontal, scalar-support, vector-support, frame and flow order,

$$P/\rho_0=\operatorname{diag}(1/b,\epsilon I,M_W/n_{\rm peak}, M_ZI/n_{\rm peak},\kappa_F I,\kappa_U I),\qquad B^i=0,$$ $$K^{ij}/\rho_0=\delta_{ij}\operatorname{diag} \left(\kappa,\epsilon I/4,\gamma_A/n_{\rm peak}, \frac{\gamma_Bw_0^2 I}{n_{\rm peak}(1+w_0^2)}, \kappa_Fc_F^2I,\kappa_Uc_U^2I\right).$$

Both full matrices are positive on every fixed finite spatial collar. Smoothness preserves this positivity for sufficiently small full first jets there. With $V=q_t,D_i=\partial_iq$, the first-order principal system has

$$A^0=\operatorname{diag}(P,K),\qquad A^i=\begin{pmatrix}B^i&-K^{ij}\\-K^{ki}&0\end{pmatrix}.$$

These matrices are symmetric, and $A^0>0$. The exact gradient equations give $\partial_t(D_i-\partial_iq)=0$, so compatible initial data recover the full second-order Euler system. Lower-order support, flow and label connections remain in its source.

For each finite $R$ and integer $\sigma\ge7$, sufficiently small data supported in $|x|\le R$, with $q_0\in H^{\sigma+1}$ and $q_1\in H^\sigma$, therefore have a unique short-time solution with $q\in C H^{\sigma+1}$, $q_t\in C H^\sigma$ and $q_{tt}\in C H^{\sigma-1}$. The first-order state depends locally Lipschitz-continuously on data in $H^{\sigma-1}$ and continuously in $H^{\sigma'}$ for $\sigma'<\sigma$. The external local symmetric-hyperbolic theorem is applied only after establishing these matrices and constraints.

For spatial localization, extend the positive normalized $P,K$ by convex blending with constant positive matrices, and extend $B$ and the zero-source branch smoothly outside a larger collar. First restrict time using the extended system's finite speed, then use the physical unit cone on the unchanged collar. Uniqueness gives support inside $R+|t|$ and permits gluing to the exact exterior branch. This does not assume an arbitrary extension retains unit speed.

The admissible size and lifespan depend on $R$. Although the relative support algebra contains no negative powers of $w_0$, the vector speed $c_Z=\sqrt{\gamma_B/M_Z}\,w_0/\sqrt{1+w_0^2}$ vanishes at infinity. Thus this argument gives no uniformly coercive unrestricted-tail theorem. A failed weighted first-order estimate is not an ill-posedness proof. No nonlinear stability, trajectory toward a lower-energy preparation or particle lifetime follows. The amended nonlinear-relative-chart/report.md and its independent review give the complete action, proof, explicit constraint charts and external theorem attribution. Both phase-edge and entropy-coefficient choices and all six maintained depths are covered.

Actual-Count Affine Disturbance Inertia

The affine inertia can be bounded at the actual maintained counts without assembling their very large occupation matrices. All inverses below act on the finite sum-zero occupied tangent where appropriate. Let $S=H_a^TH_a$, $a_o=2\gamma/\epsilon$, $\delta=\omega-1/\kappa$, and retain $\delta I>\epsilon S$. Define

$$C=H_a^T(a_oA_o)^{-1}H_a,\qquad T=\frac{4\kappa}{\epsilon}(\delta I-\epsilon S) (\kappa\omega\delta I-\epsilon S)^{-1}.$$

Eliminating the full independent-flow block of $M_0$ gives the occupation block $a_oA_o+H_aTH_a^T$ and affine source $(\omega/2)H_aT$. Woodbury reduction, including the separate positive eliminated-flow term, then gives exactly

$$M_A=\frac\omega\epsilon I-\frac{\omega^2}{4}(C+T^{-1})^{-1}, \qquad S<M_A<\frac\omega\epsilon I.$$

$C$ is a compression of the full inverse; it is generally not the inverse of the affine compression. Active-axis permutation symmetry implies two channels, parallel and perpendicular to the all-ones vector. It does not make the affine range invariant under $A_o$. The covariance eigenvalues are

$$\lambda_\perp=\frac{N(N+d+1)}{(d+1)(d+2)\ell^2},\qquad \lambda_\parallel=\frac{\lambda_\perp}{d+1}.$$

Set $D=\binom{N+d}{d}$ and $q=N/(d+1)$. The actual directed occupation graph sums give the following exact affine Hessian contraction:

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