IDOS Energy, Coherence, and Supporting Dynamics — Part 68

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growth. For $\beta_l<0$, put $\Gamma_l=\sqrt{-\beta_l}$: the spectral interval $0\le\alpha<4\Gamma_l^2$ grows, with rates $\sqrt{\Gamma_l^2-\alpha/4}$. This interval has infinitely many spectral directions, and the constant attains the maximal weighted rate $\Gamma_l$. The discriminant is still $1+2\epsilon_kQ_l$.

Thus the new covariance coefficient and strengthened selectors retain the preceding growing-origin-direction conclusion at every non-origin singleton. The old constant coefficient retains oscillatory, threshold and growing alternatives. Negative shape curvature alone does not determine which alternative occurs.

For every fixed real $s\ge0$, the scalar empty group also has the ordinary relative Sobolev bound:

$$\|S_l(t)\|_{H^{s+1}\times H^s} \le C_s(1+|t|/(2L))^p e^{\Gamma_l|t|}$$

in the growing case. A backward-cone multiplier $[\sqrt{L^2+R^2}+(T-t)/2]^{2p}$ cancels the bad flux boundary term and gives the weight comparison; commuting powers of $A_w$ and interpolating gives every fixed real Sobolev order $s\ge0$. For a positive block the same argument gives a polynomial bound, and at threshold displacement costs one further power. A uniform $Ce^{\Gamma_l t}$ bound without a polynomial loss is false in ordinary relative $L^2$: positive compact data have a nonzero weighted constant limit after division by $e^{\Gamma_l t}$, whereas that constant is not in ordinary $L^2$. This does not contradict finite physical energy.

A Separate Action With Uniform Vector-Support Speed

The second change replaces only $|\mathscr B|^2/(1+W^2)$ by $|\mathscr B|^2/W^2$ in the support action, with the existing coefficient $\mu_s/8$. To specify the changed action completely, retain the preceding $m,\Xi_q,\epsilon,b,\kappa$ and write

$$\begin{aligned} \mathcal L={}&\rho(U\cdot a-h-\Xi_q) -b[\rho\log(\rho/w)-\rho+w] +\frac{\epsilon w}{2}(|K|^2-|\mathscr R|^2/4) +\mathcal L_s+\mathcal L_E+\mathcal L_U+\mathcal L_{\rm con},\\ h={}&m+\frac{\sqrt{1+4\kappa^2|A|^2}-1}{4\kappa},\qquad a_\alpha=\langle Je,\partial_\alpha e\rangle,\quad P=g+UU,\quad A=Pa,\\ K={}&\partial_Ue-(U\cdot a)Je,\\ \mathscr R_\alpha={}&P_\alpha{}^\beta(\partial_\beta e-a_\beta Je) -\ell^{-1}J\sum_iE_{i\alpha}(T_i-\bar k_i)e. \end{aligned}$$ $$\begin{aligned} \mathcal L_s={}&\frac{\mu_s}{2}[(\partial_UW)^2+|P\nabla_UZ|^2] -\frac{\mu_s}{8}(|\mathscr A|^2+|\mathscr B/W|^2) -\frac{\kappa_G}{2}\mathscr G^2,\\ \mathscr A={}&P\,dW+8Z/L,\\ \mathscr B_{\alpha\beta}={}&WP_\alpha{}^\gamma P_\beta{}^\delta \nabla_\gamma Z_\delta -(W^{9/4}P_{\alpha\beta}-10Z_\alpha Z_\beta)/L,\\ \mathscr G={}&Z^2-W^{9/4}+W^{5/2},\\ \mathcal L_E={}&\frac{\kappa_Ew}{2}\sum_{i<j} [(U\cdot\Omega_{ij})^2-c_E^2P^{\alpha\beta} \Omega_{\alpha,ij}\Omega_{\beta,ij}],\quad \Omega_{\alpha,ij}=E_i\cdot\nabla_\alpha E_j,\\ \mathcal L_U={}&\frac{\kappa_Uw}{2} [|\nabla_UU|^2-c_U^2P^{\alpha\beta} \nabla_\alpha U\cdot\nabla_\beta U]. \end{aligned}$$

Use signature $(-,+,\ldots,+)$ and retain $|e|=1$, $U^2=-1$ with future $U$, $U\cdot Z=U\cdot E_i=0$ and $E_i\cdot E_j=\delta_{ij}$. All ten flow components and 45 spatial frame directions remain. The positive constants $\kappa_G,\kappa_E,\kappa_U$ and speeds $0<c_E,c_U\le1$ are common inputs. Here $m=\mu/\sum p_k^2$ and $\Xi_q=\lambda\mu[V+(1+\eta)V^q+\eta V_\phi]/(\sum p_k^2)^3$ use the preceding literal occupation tensors, $q=2,4$, $\eta\ge0$ and $\lambda>1$. The field-dependent horizontal choice is $\epsilon=\omega_*/[4(\omega_*\mu+\operatorname{tr}C/\ell^2)]$; the alternative constant choice obeys $\epsilon M^2/\ell^2\le\omega_*/4$. These are distinct kinetic laws.

The exact action difference is

$$\Delta\mathcal L=-\frac{\mu_s|\mathscr B|^2}{8W^2(1+W^2)}.$$

Its full value and first variation vanish on every residual-zero branch, including metric, frame, flow and support variations. The count clock,

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