IDOS Energy, Coherence, and Supporting Dynamics — Part 69

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complete stress and common action normalization therefore survive. At rest the linearized $\mathscr B$ has only spatial-spatial components. The extra quadratic term is a nonnegative static square with no new kinetic or gyro term, so the positive flow margins survive too.

Set $u=\log(W/W_0)$, $z=Z/W_0$, $a_0=d\log W_0$ and $R_w=w(W)/w_0$. The exact relative residuals include

$$\mathscr A_{\rm rel}=e^uP(du+a_0)+8z/L,$$ $$\mathscr B_{\rm rel} =PP(\nabla z+a_0z)-W_0^{1/4}e^{5u/4}P/L +10e^{-u}zz/L.$$

Their common static coefficient in $\mathcal L_s/w_0$ is $-C_sR_we^{-2u}/8$. No inverse power of $W$ multiplies a relative derivative. At $p=8$, $d\log W_0$, $W_0^{1/4}$, $W_0^4/w_0$ and all needed fixed derivatives are bounded. The material support coframe obeys

$$\frac{dW}{W_0}=e^u(du+a_0),\qquad \frac{\langle E_i,dZ\rangle}{W_0} =dz_i+a_0z_i+\Omega_{ij}z_j.$$

The independent boost and rotation forms make the complete coframe triangular and uniformly invertible on a small fixed relative chart. It need not be a coordinate differential. Dividing the physical Euler equations by $w_0$ retains the resulting $d\log w_0$ terms.

The derivative Hessian has the phase block $(\rho/b)(U-h_A)(U-h_A)-\rho h_{AA}$ and the horizontal, scalar-support, vector-support, frame and flow blocks with rest speeds $1/2,1/2,1/2,c_E,c_U$. The phase rays remain below $3/4$ in the material rest frame; the complete laboratory cones are inside the unit cone. The time and spatial Schur margins are uniformly positive in this chart. A compatible symmetric-hyperbolic reduction propagates its zero curl constraints. The local existence, continuation and domain tools are standard mathematical input; the action-specific margins and coframe verification above supply their hypotheses. See Métivier, sections 3.3, 3.4 and 3.6.

For integer $s\ge7$ let $\|Y\|_s=\|X\|_{H_{\rm uloc}^{s+1}}+\|X_t\|_{H_{\rm uloc}^{s}}$, where the uniformly local norm is the supremum over translations of one fixed smooth compact cutoff. Use smooth bounded data or their strong completion with the required spare derivatives. The full local theorem and continuation estimates have constants independent of the spatial location of data. Within its small positive tube, $W/W_0$ and $\rho/w_0$ are bounded above and below, and all physical energy and charge densities are integrable. This is a local theorem on all of space, not global stability or continuation through vacuum.

A Verified Fixed-Size Nonlinear Departure

The following more specific conclusion uses constant $\mu=\omega_*=1$, $\lambda=2$, $q=2$, $\eta=0$ and the singleton $k=e_1$, in a fixed finite carrier $M\ge1$. Retain either entropy law. Use the new horizontal coefficient or $\epsilon=1/4$ with $\ell\ge M$. Its branch values and complete empty coefficients are

$$\Omega=13,\qquad Q_0=-26,\qquad \epsilon_0=\frac14,\qquad \Gamma^2=-\epsilon_0^{-2}-2Q_0/\epsilon_0=192,$$ $$Q_l=14+10\,\mathbf1_{\{l=e_1+e_j\text{ for some }j\}} +20\sum_{j\ne1}l_j\ge14\qquad(l\ne0,e_1).$$

The full common operator is now nonnegative and uniformly elliptic, with form domain $H^1$ and operator domain $H^2$. Time-commuted conserved energies and elliptic estimates give its physical group bound $C_s(1+|t|)$. The negative origin block has rate $\Gamma$, while every other empty block has positive energy. Hence the complete physical group is bounded by $C_s(1+|t|)e^{\Gamma|t|}$. Finite propagation, the weight ratio and a ten-dimensional collar cover imply the sufficient relative bound

$$\|S(t)\|_{Y_s\to Y_s}\le C_s(1+|t|)^{14}e^{\Gamma|t|}.$$

The exponent is an upper bound, not an asserted optimal loss. The full linear mode with origin component

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