IDOS Energy, Coherence, and Supporting Dynamics — Part 21

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phase current is $j=n(U-h_A)$, where $h_A=\kappa_a A/r$. Thus $U\cdot j=-n$ and $j^2=n^2(-1+|h_A|^2)<0$. The phase covector may be spacelike even though this conserved current is future timelike.

The Hessian of $h$ has eigenvalues $\kappa_a/r^3$ along $A$ and $\kappa_a/r$ transverse to it. If all other density-weighted derivative backgrounds vanish, exact density elimination gives the phase tensor

$$H_{\rm phase}=\frac nb(U-h_A)\otimes(U-h_A)-n h_{AA}.$$

Its rays in the $U$ rest frame have speed bounded by $|h_A|+\sqrt{b\lambda_{\max}(h_{AA})}<v_*+c_D\le1$. This is not yet a result for the complete original saturated action. For example, a constrained frame rotation with spatial gradient $G$ and kinetic coefficient $k$ gives the coupled squared-speed equation

$$(z-b\kappa_a)(z-c_F^2)-\kappa_a k c_F^4G^2=0.$$

At $c_F=1$, any nonzero $G$ produces a root greater than one. At $k c_F^2G^2>b$, the smaller root is negative. This is an actual block of the constrained local principal system; retaining dynamic $U$ and the other channels does not remove it.

Changing The Non-Clock Derivative Weights

A further explicit candidate, SF-DW, replaces the factors $n$ by $n_0$ in the horizontal carrier, material-frame and independent-flow derivative terms. The clock and algebraic occupation retain $n$. Keep the support definitions and temporal coefficients, but replace the vector residual term $-\gamma_B B^2/2$ by $-\gamma_B B^2/[2(1+w^2)]$. This bounds its spatial speed even for finite $|w|>1$. These are changed constitutive laws, not a field redefinition or a derivation from the original master action.

The auxiliary equation is now exactly

$$n=n_0\exp[(U\cdot a-h-\gamma V)/b].$$

No other derivative invariant enters this elimination. In a local $U$-rest chart the phase tensor above is therefore its only second-order block. The horizontal carrier derivatives are phase-orthogonal; projected support and frame derivatives remove induced temporal and boost components. For material support coordinates, the shared map $(\delta z,f)\mapsto(\delta z+fz,f)$ is invertible, so the apparent support/frame derivative mixing is a congruence of the same positive forms. These statements hold at arbitrary finite background first derivatives: each remaining residual is affine in field derivatives, and its nonzero background value contributes at most one perturbation derivative to the quadratic expansion. It does not create another principal Hessian.

The complete squared wave speeds besides the phase block are

$$\frac{\gamma_A}{M_w},\quad \frac{\gamma_Bw^2}{M_Z(1+w^2)},\quad \frac14,\quad c_F^2,\quad c_U^2.$$

The earlier common $\gamma_A=\gamma_B=M_w/4=M_Z/4$, with $0<c_F,c_U\le1$, bounds all these speeds by one. The example $\kappa_a=8/\omega_{\min}$, $v_*=1/2$, $c_D=1/4$ also controls the phase rays. Positive kinetic forms give a local symmetric wave reduction on compact positive-chart regions. This establishes causal characteristics and strong hyperbolicity for the reduced independent waves with auxiliary density reconstruction. It is not uniform through vacuum. After elimination, density depends on derivatives, so a second-order normal form in every density-weighted physical coordinate does not follow automatically.

The positive phase symmetrizer uses the convected velocity $p=(\partial_t-h_A\cdot\nabla)\theta$ and the form $n[p^2/b+\nabla\theta\cdot h_{AA}\nabla\theta]/2$. It is distinct from the ordinary $U$-time canonical spatial energy matrix $n[h_{AA}-h_Ah_A^T/b]$, which can be indefinite. Causal characteristics therefore do not establish arbitrary-background Hamiltonian positivity or global stability.

What The Repair Preserves And Changes

At $A=0,n=n_0$, all added residuals and full first variations vanish. The isolated solutions, multiplier $\zeta=n\omega$, current $nU$,

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