IDOS Energy, Coherence, and Supporting Dynamics — Part 59

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only $b=\tau b_0$, retaining the associated $\kappa$ and all other coefficients. Use $\rho=(1+R^2)^{-8}$, $Q=\pi^5/2520$, $d\mu_r=210R^9(1+R^2)^{-8}dR$, and $F_B(z)=21z^5-35z^6+15z^7$, $z=R^2/(1+R^2)$. Let $S$ be the preceding smooth step, $\delta=10^{-12}$, $h_s=10^{-4}$, $\beta=\delta+(1-\delta)S(R-1000)$, $\chi=e^{-1/(1-R^2)}$ for $R<1$ and zero otherwise. For each pinned rational radius $R_0\in(2.47414,2.47883)$ and weight $w$, set

$$B=S(2(R-R_0+h_s)/h_s)S(2(R_0+h_s-R)/h_s),\quad I=\int B\,d\mu_r,\quad j_1=\int\chi\,d\mu_r,\quad r=\beta+\frac{1-w-\int\beta\,d\mu_r}{j_1}\chi+\frac wI B.$$

Exact integrals define this density: its core coefficient is positive, $\int r\,d\mu_r=1$, $r\ge\delta$, and $r=1$ for $R\ge1001$. The complete energy, including wall and the appropriate pair-charge or affine-charge core compensation, is

$$E_\tau/Q-D=E_{\rm wall}+\int(H-D)r\,d\mu_r+ \tfrac12 C_d(\int tr\,d\mu_r)^2+\tau\int b_0(r\log r-r+1)\,d\mu_r.$$

Here $D=\binom{d+2}d$, $H$ includes the actual flattened mass, saturated drift and occupation/phase-edge potential, and $t$ is the carrier feature, not squared radius. The previous core construction gives

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