IDOS Energy, Coherence, and Supporting Dynamics — Part 43

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In the same two-region control, allow the actual low-region charge fraction $y$ to differ from its background weight $t$. Fixed total $Q$ gives $n_H/\rho=(1-y)/(1-t)$ and $n_L/\rho=y/t$. Constant relative density in each region minimizes its strictly convex entropy. The horizontal Gram remains weighted by $t$, while the charge deficit is weighted by $y$. Define

$$C_{\rm affine}(t)=\frac{B}{2\epsilon[(1-t)A_H+tA_L]},\qquad C_{\rm pair}(t)=\frac{D_H/D_L-1}{2\epsilon(1-t)}.$$

The complete bulk energy difference per $Q$ is

$$F_t(y)=-\Delta y+C(t)y^2 +b_H\left[(1-y)\log\frac{1-y}{1-t}+y-t\right] +b_L\left[y\log\frac yt-y+t\right].$$

Keep the linear entropy terms when $b_H\ne b_L$. Both common positive $b$ and the smooth law $b_A=m_A/128$ are covered. Direct differentiation gives

$$F_t'(y)=-\Delta+2Cy+b_L\log(y/t) -b_H\log[(1-y)/(1-t)],\qquad F_t''(y)=2C+\frac{b_L}y+\frac{b_H}{1-y}>0.$$

The derivative has opposite infinite limits at the two endpoints, so there is one interior minimizer. It obeys $y_*>t$ exactly when $\Delta>2Ct$; at equality it equals $t$, and under the reversed inequality it is smaller. In a centered inversion-symmetric bulk control, scalar recoil adds $C_Py^2$, where $C_P=159B/(2\omega_{\min}\ell^2)$, without changing this convex structure. A nonsymmetric wall needs its actual angular and momentum terms.

For the separate check $C=0$, $b_H=b_L=b$,

$$y_* =\frac{t e^{\Delta/b}}{1-t+t e^{\Delta/b}},\qquad \min F_t=-b\log(1-t+t e^{\Delta/b}).$$

For either actual phase-edge cost, at fixed finite parameters as $t\to0$, $y_*/t\to e^{\Delta/b_L}$ and $\min F_t/t\to-b_L(e^{\Delta/b_L}-1)$. The compensation first contributes at order $t^2$. A huge exponential requires a correspondingly smaller $t$ before this asymptotic is accurate; there is no uniform count limit or runaway total charge. The principal two-region-density-relaxation.md and its independent 199-check review retain 16 finite controls and the analytic checks. No interface, evolution, decay rate or uniform participating-energy clock follows.

Local Nonlinear Evolution Of Small Compact Relative Data

The linear energy theorem has a separate nonlinear continuation counterpart for small compact data. It applies to the constant-positive-support-inertia SF-DW action on its positive chart, with the entire finite carrier, support, frame and independent flow retained. Let $w_0=(1+|x|^2/L^2)^{-4}$, $b_0=x/[L(1+|x|^2/L^2)]$, and use a real relative chart

$$e=R_0(t,x)\frac{e_*+u}{\sqrt{1+|u|^2}},\quad u\perp e_*; \quad W=w_0(1+s),\quad Z_{\rm spatial}=w_0(b_0+z).$$

Use $U=(\sqrt{1+|v|^2},v)$, its boosted orthonormal frame followed by $\exp f$ with $f$ skew, and the temporal support component imposed by $U\cdot Z=0$. These coordinates solve the algebraic constraints. Empty carrier labels remain regular real tangent coordinates. Eliminate density exactly as $n=\rho\exp(F/b)$, with $F=U\cdot a-h_{\rm clock}-\gamma V$ and $\rho=n_{\rm peak}W^2$. The reduced Lagrangian is $b\rho[\exp(F/b)-1]+\rho\mathcal Q+\mathcal L_s$.

For its derivative Hessian set $P=\mathcal L_{q_tq_t}$, $B^i=\mathcal L_{q_tq_i}+\mathcal L_{q_iq_t}$ and

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