IDOS Energy, Coherence, and Supporting Dynamics — Part 56

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the smooth constrained target coframe $T_c$, the full derivative Hessian is $T_c^T\operatorname{diag}(G_a^{\alpha\beta})T_c$. Its phase block is $(q/b)(U-h_A)(U-h_A)-qh_{AA}$; the other blocks are positive multiples of $UU-c_a^2P$, with $c_a^2=1/4,1/4,W^2/[4(1+W^2)],c_F^2,c_U^2$. The saturation eigenvalues are $\kappa/r_A^3,\kappa/r_A$; both laws have $b\kappa=1/16$, giving phase ray speed below $3/4$ in the $U$ rest frame and other speeds at most one. The $3/4$ number is not invariant under a laboratory boost; the physical unit cone remains valid in every inertial frame.

Set $\beta_a^i=G_a^{0i}/G_a^{00}$ and $g_a^{ij}=\beta_a^i\beta_a^j-G_a^{ij}/G_a^{00}>0$. The simultaneous first-order symmetrizer on increments is

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

Here $V=X_t$, $D_i=\partial_iX$; the curl constraints propagate under the associated homogeneous transport system. Thus each fixed finite carrier, fixed $\varepsilon_p>0$ and compact compatible datum in regular positive charts has a data-dependent nonlinear lifespan, with relative $X\in C H^{s+1}$, $X_t\in C H^s$, $X_{tt}\in C H^{s-1}$ for integer $s\ge7$. Smooth positive coefficients must exist on a neighborhood of the whole first-jet image. Finite large data are allowed; no uniform lifespan is asserted. Symmetrized finite- collar extension and uniqueness give support inside $R+|t|$. Reconstructed $q$ and $\sqrt{2q}e$ have only the guaranteed $H^s$ relative regularity because $F$ contains first derivatives.

A separate result gives global linear evolution about one fixed branch. Write $h=\sqrt{q_0}$, $k=\sqrt\mu$ and normalize displacement as $u=(h\theta/\sqrt b,h\sqrt{\epsilon_c}X,h\sqrt{\epsilon_c}Y, k\delta(W,Z),h\sqrt{\kappa_F}f,h\sqrt{\kappa_U}v)$. Its $2D+65$ real components retain the sum-zero carrier tangents and all 45 skew-frame components. Let $T_0$ be the finite first-order residual stack of the exact nonnegative static quadratic form after this substitution, with every weighted derivative retained as $D_h=\nabla-\nabla\log h$ or $D_k=\nabla-\nabla\log k$. In particular its support squares are

$$\frac14|D_k\zeta_W+8\zeta_Z/L|^2+ \frac{\sum_{ij}|W D_{k,i}\zeta_{Z,j}+\zeta_W\partial_iZ_j -[(9/4)W^{5/4}\zeta_W\delta_{ij}-10(\zeta_{Z,i}Z_j+Z_i\zeta_{Z,j})]/L|^2}{4(1+W^2)} +\frac{\kappa_G}{\mu}|2Z\cdot\zeta_Z-g_W\zeta_W|^2, \quad g_W=\tfrac94W^{5/4}-\tfrac52W^{3/2}.$$

Denote the other normalized coordinates by $(\vartheta,\xi,\upsilon,\zeta,\varphi,\nu)$ in the order above. The remaining static density is

$$\tfrac14|D_h\xi|^2+\tfrac14\sum_i|D_{h,i}\upsilon-\sqrt{\epsilon_c/\kappa_F}\sum_jH_j\varphi_{ij}|^2 +\tfrac{2\gamma}{\epsilon_c}(\xi^TA_o\xi+\upsilon^TB_o\upsilon) +c_F^2|D_h\varphi|^2+c_U^2|D_h\nu|^2 +\kappa|\sqrt bD_h\vartheta+2H^T\xi/\sqrt{\epsilon_c} +(m-1/\kappa)\nu/\sqrt{\kappa_U}|^2+\nu^T\mathcal R\nu,$$

where $A_o,B_o$ are the retained nonnegative occupation Hessians and $\mathcal R=[(m-1/\kappa)I-\epsilon_cH^TH-C_sD_s^TD_s/W^2]/\kappa_U>0$. The finite stack $T_0$ is defined by these

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