IDOS Energy, Coherence, and Supporting Dynamics — Part 55

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flow derivative terms use $q_0$, while $qF$ uses $q$. Every coefficient is varied. Keep separately $(b,\kappa)=(\omega_{\min}/128,8/\omega_{\min})$ and $(m/128,8/m)$, with smooth positive algebraic $m$ and the selected flat first and second mass jets. A derivative-defined mass descriptor is not covered. Both phase-edge choices $\eta=0$ and $\eta>0$ remain available. Elimination gives $q=q_0e^{F/b}$ and the reduced entropy term $bq_0(e^{F/b}-1)$. For variable coefficients the first variation includes

$$-D(q\Vert q_0)\,\delta b,\qquad \delta F\supset-\delta m-\frac{v_*^2}{\kappa^2}(1-r_A^{-1})\delta\kappa, \quad r_A=\sqrt{1+\kappa^2 A^2/v_*^2},$$

with $\delta b=\delta m/128$, $\delta\kappa=-8\delta m/m^2$. The reduced explicit $b$ term is $q_0[e^{F/b}(1-F/b)-1]\delta b$. Metric variation also retains the clock contraction $\kappa A^\alpha A^\beta\delta g_{\alpha\beta}/(2r_A)$, the variation of $A$ itself, every other contraction and the volume measure. These terms vanish appropriately on the exact branch, but generally change encounter response. A change of charge weight is a change of action.

On that branch $j=q_0U$, $T=q_0mUU$, $E_{\rm rest}=mQ$ and complete boost inertia is $mQ$; off branch $j=q(U-h_A)$ instead. Define $M_j=\int_0^\infty s^j f(s)\,ds$ and $J=\varepsilon_p\sum_{j=0}^4\binom4j\varepsilon_p^jM_j$. Exact integration gives

$$Q_{\rm core}=\frac{\sigma\pi^5L^{10}}{120},\quad Q_{\rm out}=\frac{\sigma\pi^5L^{10}}{24}J,\quad \frac{Q_{\rm out}}{Q_{\rm core}}=5J,\quad L=\left[\frac{120\mathcal A}{\sigma\pi^5(1+5J)}\right]^{1/10} \quad(Q=\mathcal A).$$

The bounds $e^{-s}\le f(s)\le e^{-\max(0,s-1)}$ imply $j!\le M_j\le 1/(j+1)+j!\sum_{k=0}^j1/k!$ and $5J=O(\varepsilon_p)$. At $\varepsilon_p=.01$, diagnostic quadrature gives core fraction $.9185156014$. Nevertheless $h/\sigma=mf(s)<m$ at every exterior point; the relative defect tends to one and has infinite ordinary-volume integral. Finite charge-weighted diagnostics do not establish ordinary-volume uniformity. No occupied exterior has been removed.

Finite Nonlinear Data And Global Linear Evolution Have Different Domains

For this changed action, the branch drift completion retains

$$S_{\rm grad}/q_0=C_sD_s^TD_s/W^2+\epsilon_c H^TH \le\frac{37}{106}\omega_{\min}I,\qquad q_0(m-1/\kappa)I-S_{\rm grad}\ge\frac{223}{424}q_0mI.$$

Here $D_s=\nabla(W,Z)$, $H$ is the centered carrier-gradient map and $\epsilon_cH^TH\le\omega_{\min}I/4$. The absolute margin vanishes in the tail. The nonlinear statement needs more than this branch inequality. In

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