IDOS Energy, Coherence, and Supporting Dynamics — Part 48

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relative density $c_\infty>0$ on both regions. It need not equal one. This exterior mismatch prevents direct use of the compact-data evolution theorem, despite finite physical energy.

Choose a nonnegative smooth core bump $\psi$ with $\int\rho\psi=1$, and a smooth radial cutoff $\eta_R=\eta(|x|/R)$ with $0\le\eta\le1$, equal to one inside $R$ and zero outside $2R$. Take $R$ beyond every fixed deformation and velocity support. Define

$$T_R=\int\rho(1-\eta_R),\qquad A_R=(c_\infty-1)T_R, \qquad r_R=1+\eta_R(r_*-1)+A_R\psi.$$

For sufficiently large $R$ this is smooth and positive, both when the core correction adds density and when it subtracts it. It equals one outside $2R$. The correction has exactly zero total integral and occurs only where $z=z_H$, so

$$\int\rho r_R=Q,\qquad \delta(r_R)=\delta(r_*),\qquad v_R=v_*.$$

Every retained phase, translation and rotation charge is therefore unchanged. No new recoil solve or kinetic energy is needed. With $k(u,v)=u\log(u/v)-u+v$, the complete energy excess is exactly

$$E_R-E_* =\int\rho b\,k(r_R,r_*).$$

For the rational support in ordinary $\mathbb R^{10}$, write $t=(1+R^2/L^2)^{-1}$. Its exact exterior mass is

$$M(R)=\int_{|x|>R}\rho =Q(35t^3-105t^4+126t^5-70t^6+15t^7) \sim\frac{n_{\rm peak}|S^9|L^{16}}{6R^6}, \qquad |S^9|=\frac{\pi^5}{12}.$$

Since $M(2R)\le T_R\le M(R)$, the core entropy correction is $O(R^{-12})$. The full joining annulus and exterior cost instead satisfy, when $c_\infty\ne1$,

$$E_R-E_*=n_{\rm peak}|S^9|L^{16}b_H R^{-6}I_\eta(c_\infty) +O(R^{-8})=\Theta(R^{-6}),$$ $$I_\eta(c)=\int_1^\infty s^{-7} k(1+\eta(s)(c-1),c)\,ds>0.$$

The unchanged positional wall and all support/frame energies remain included. There is no density-gradient term in this particular action. Moreover $n_Rh_A=n_*h_A$ exactly, since changed regions have $a=0$. Thus continuity gives the same $n_t$ there, and physical reconstruction retains $\Psi_t=\sqrt{2n}\,e_t+n_t e/\sqrt{2n}$. Weighted density differences are $O(R^{-6})$ in $L^1_\rho$ and $O(R^{-3})$ in $L^2_\rho$; every fixed physical Sobolev norm of the position and velocity differences is also $O(R^{-3})$. There is no uniform relative convergence at infinity when $c_\infty\ne1$.

The union of smooth compact-relative density classes over all finite supports consequently has the same infimum as the unrestricted density class. It does not attain that value when $c_\infty\ne1$; strict entropy equality would require $r=r_*$. When $c_\infty=1$, the optimum itself already belongs to the compact-relative class. Each fixed admissible $R$ has its own local evolution under the regular positive-chart theorem. Neither a uniform lifespan nor evolution toward the optimum follows.

compact-density-recovery/report.md and its independent 34-check review establish this extension of the accepted amended dual theorem, including both signs of the density correction and both diagonal symmetry branches. The exponent comes from the ambient support tail, not active depth; no depth-five exception or occupied-clock selection is implied.

Common Spatial Phase Improves A Full Low-Prefix Endpoint

A different smooth whole-space family uses positive phase edges and a positive lower prefix $N_L>0$. Keep $n=\rho$, fixed rational support,

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