IDOS Energy, Coherence, and Supporting Dynamics — Part 73

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$$T_\epsilon=(c_*/\epsilon)^{1/12}-1,\qquad \sup_{|t|\le T_\epsilon}B_s(t) \le C_*c_*^{3/4}\epsilon^{1/4}.$$

The density is reconstructed exactly as $\rho/w_0=[w(W)/w_0]\exp[(U\cdot a-h-\Xi)/b]$ and is controlled one derivative below $X$. The physical carrier satisfies $\|\Psi-\Psi_{\rm prefix}\|_2\le C\sqrt{\mathcal A}B_s$. Complete energy, momenta and actual phase charges are conserved; angular momentum has one additional spatial weight. The exterior fluxes vanish because $w_0=O(|x|^{-16})$ in ten dimensions.

The Actual Fastest Singleton Direction

For any nonzero singleton label $k$, evaluate the complete two-label restriction $\mathcal G_l(P)=H(p_k=1-P,p_l=P)$ and define

$$Q_l=\mathcal G_l'(0),\qquad Q_{\min}=\min_{l\in\mathcal K_M\setminus\{k\}}Q_l,\qquad \Gamma_*^2=-\varepsilon_k^{-2}-2Q_{\min}/\varepsilon_k.$$

The derivative includes the actual $\mu'(P)$ wherever nonzero. The general singleton calculation makes $Q_l$ affine in the empty label except at a bounded exceptional set; its finite minimum procedure does not enumerate the entire large carrier. It need not select the origin. For example, $k=2e_1$, constant $\mu=1$, $\lambda=2$, $q=2$, $\eta=1$ gives $Q_0=-20$ but $Q_{e_1}=-22$, so the covariance law gives respectively $\Gamma_0^2=144$ and $\Gamma_*^2=160$.

At a singleton the covariance law has $\varepsilon_k=1/(4\mu_k)$. The origin-direction bounds give

$$\Gamma_*^2\ge32\mu_k^2(\lambda-1)>0\quad(q=2),$$ $$\Gamma_*^2\ge8\mu_k^2[\lambda(156+159\eta)-4]>0\quad(q=4).$$

Thus every non-origin singleton has a growing direction for this stated law. The constant-horizontal alternative instead retains growing, threshold and oscillatory cases according to the actual minimum of $\beta_l=\varepsilon_c^{-2}+2Q_l/\varepsilon_c$. Only $\min\beta_l<0$ supports the departure theorem below. The other two linear alternatives are not nonlinear stability conclusions.

Compact Preparations At Exactly The Conserved Levels

Choose a minimizing empty label $l_*$, put $d_*=l_*-k$, and write $A=a f_R$, $P=A^2$, where $f_R=1$ on $B_R$ and vanishes outside $B_{2R}$. All coefficients below are their actual finite-$P$ values. Set

$$\mathcal D(P)=\frac{\sqrt{1+4\kappa(P)^2P^2|d_*|^2/\ell^2}-1} {4\kappa(P)},\qquad \bar\omega=\mathcal G_{l_*}(P)+\mathcal D(P),$$ $$\omega_{\rm rel}=-\frac1{\varepsilon_h(P)(1-P)},\qquad g(P)=-\frac{2(1-P)[\mathcal G_{l_*}(P)+\mathcal D(P)-\Omega]} {\varepsilon_h(P)P}-\varepsilon_h(P)^{-2}.$$

The apparent quotient is removable, with $g(0)=\Gamma_*^2>0$. The pairs are $(\sqrt{1-P},A\exp[Jd_*\cdot x/\ell])$. Their canonical

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