IDOS Energy, Coherence, and Supporting Dynamics — Part 61

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$\phi(r)=r\log r-r+1$; integrals use $d\mu_r$. For $\tau>0$ its unique positive bounded optimizer satisfies $H+C_dyt+\tau b_0\log r+\nu=0$. Put $dw=(r/b_0)d\mu_r$, $\bar f_w=\int f\,dw/\int dw$, $f_0=f-\bar f_w$, $V=\int t_0^2dw>0$, $B_f=\int t_0f_0dw$, $A_f=\int f_0^2dw$. These covariance integrals are unnormalized. For bounded frozen forcing $H\mapsto H+\alpha f$, with $b_0,\kappa,C_d$ fixed,

$$y_\alpha=-\frac{B_f}{\tau+C_dV},\quad r_\alpha=-\frac r{\tau b_0}\left[f_0-\frac{C_dB_ft_0}{\tau+C_dV}\right],\quad \mathcal F_{\alpha\alpha}=-\frac1\tau\left[A_f-\frac{C_dB_f^2}{\tau+C_dV}\right]\le0.$$

With $k=b_0\log r$, the same formulas replace $f$ by $k$ for $\tau$ response. In particular $\mathcal F_\tau=\int b_0\phi(r)\ge0$ and $\mathcal F_{\tau\tau}\le0$. Both inequalities are strict except when $r=1$, equivalently $H+C_d\bar y t$ is constant, $\bar y=\int t\,d\mu_r$. There is no universal sign of $y_\tau$. Also $y_{C_d}=-yV/(\tau+C_dV)$, $\mathcal F_{C_d}=y^2/2$ and $\mathcal F_{C_dC_d}=-y^2V/(\tau+C_dV)$. These are parameter responses, not evolution or an observed pair force. Changes of physical mass that also change $b_0,\kappa$ need their extra chain-rule terms; varying $C_d$ alone does not compare distinct $\eta$ actions.

For $g=H+C_d\bar y t$, $d\varpi=d\mu_r/b_0$ and $\bar g=\int g\,d\varpi/\int d\varpi$,

$$r=1-\frac{g-\bar g}{\tau b_0}+O(\tau^{-2}),\qquad \mathcal F=\mathcal F_\infty-\frac1{2\tau}\int(g-\bar g)^2d\varpi +O(\tau^{-2}),\quad \mathcal F_\infty=\int H\,d\mu_r+C_d\bar y^2/2.$$

The zero-entropy endpoint $G_0$ additionally inherits continuous feature potential and full support of the feature measure from the preceding convex-envelope theorem; bounded measurability alone does not justify a

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