IDOS Energy, Coherence, and Supporting Dynamics — Part 20

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require $\zeta=n\omega$; its metric variation supplies $T^{\mu\nu}=n\omega U^\mu U^\nu$. Consequently the complete boosted energy and momentum remain $E=\Gamma S\omega$ and $\mathbf P=\Gamma S\omega\mathbf V$. The exact phase current is

$$j=n(U-\kappa_a Pa),\qquad U\cdot j=-n,\qquad j^2=n^2[-1+\kappa_a^2|Pa|^2].$$

Structural flow, phase gradient and conserved current are distinct. Spacelike or zero $a$ is no longer a singularity of the action at $n>0$; the current itself can be spacelike. Physical weighted components $\sqrt n U$ and $\sqrt n E_j$ have finite branch norms. Rewriting their derivatives requires the complete density connection; vacuum remains open.

Complete Branch Energy And Its Off-Branch Limitation

Let $v=\delta U_{\rm spatial}$, $b=\nabla\theta-2\langle J\nabla e_0,h\rangle_H$, and $S_{\rm grad}=(\nabla Y)^TM_s\nabla Y+ \epsilon n(\nabla e_0)^T\nabla e_0$. Besides positive velocity and complete residual-square terms, the constrained quadratic energy contains

$$E_{\rm static}= \frac{n\kappa_a}{2}|b+(\omega-1/\kappa_a)v|^2 +\frac12v^T[n(\omega-1/\kappa_a)I-S_{\rm grad}]v.$$

The negative supporting-drift contribution is essential. The earlier common support and jet coefficients give $S_{\rm grad}\le3n\omega I/4$. With $\kappa_a=8/\omega_{\min}$ the Schur remainder is at least $n\omega I/8$. Taking $\lambda=\omega_{\min}/(16n_{\rm peak})$ gives phase squared speed $\lambda\kappa_an=n/(2n_{\rm peak})$. Support and horizontal carrier speeds are at most $1/2$; frame and flow channels have their specified speeds $c_F,c_U\le1$. The complete branch principal system is causal on compact positive-density regions. Its quadratic energy is the constrained variation of $E-\omega Q$, not a global nonlinear laboratory-energy theorem or empirical lifetime result.

This action is not causal on arbitrary newly admitted data. In the constant-$U$ pure phase-density subsystem its rays are

$$-\kappa_a a_{\rm spatial}\cdot\hat k \ \pm\sqrt{\lambda\kappa_a n}.$$

At $n=n_0=m=1$, $\kappa_a=8$, $\lambda=1/16$ and $a=(5,1,0,\ldots)$, the auxiliary density equation holds while both collinear rays $-8\pm1/\sqrt2$ are acausal. This is a regular local constrained jet, not a constructed global pair solution. A branch causal cone, particularly one containing luminal channels, does not establish an open causal neighborhood without further bounds.

The changed action preserves uniform band occupation and its assigned count clocks, but the branch energy density remains $n(x)\omega$. It therefore still does not satisfy the separate spatially uniform energy requirement for a constant-action-density local clock. Neither the selected observed indices nor a special depth-five response follows. All six maintained depths use the same ambient law. The explicit action, first variation, complete quadratic reduction and counterexamples are in independent-flow-completion and flow-completion-audit, with respectively 2,545 and 168 exact checks. The original derived-flow theory remains a separate conditional alternative on its more restrictive phase domain.

A Saturated Clock And A Causal Positive-Density Completion

The quadratic alignment failure can be repaired on the positive-density chart, but saturating the phase drift alone is insufficient. Define

$$A=Pa,\quad r=\sqrt{1+\kappa_a^2|A|^2/v_*^2},\quad h=m+\frac{v_*^2}{\kappa_a}(r-1),\quad n_0=n_{\rm peak}w^2,$$ $$\mathcal L_{\rm clock}=n(U\cdot a-h) -b[n\log(n/n_0)-n+n_0],\qquad b=c_D^2/\kappa_a.$$

Require $n,n_0>0$, $0<v_*<1$, $c_D>0$ and $v_*+c_D\le1$. With the previous algebraic occupation term $-\gamma nV$, the exact

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