IDOS Energy, Coherence, and Supporting Dynamics — Part 47

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flow, constant frame, and a smooth unit carrier equal to the centered high prefix on a central ball and outside a compact deformation. Work within that finite occupied face. Write $\rho=n_{\rm peak}W^2$, $Q=\int\rho$, $H=h_{\rm clock}+\gamma V$, and $r=n/\rho$. Both smooth positive coefficient laws are allowed, with bounded $H,b$ and $0<b_{\min}\le b\le b_{\max}$. The spatial carrier residual $R_i$ and all occupation terms include the joining layer.

Choose a fixed finite family of smooth core velocities: horizontal carrier phase tangents, scalar-support translations and frame rotations. A nonnegative, nonzero smooth radial cutoff gives full charge rank after common-phase redundancy and inactive generators are removed. Let $v$ be its coefficients and $G>0$ its kinetic Gram. Its charge matrix $B$ retains the phase-to-momentum block, the first cutoff moments in charges and the second moments in energy. Material support velocity includes the frame connection that keeps laboratory $Z_t=0$. Other support positions are unchanged and $U_t=0$.

Use the density features and target

$$z=(q_T,a_i,[x\wedge a]_{ab}),\qquad c=(Qq_H,0,0),\qquad \delta(r)=c-\int\rho r z.$$

The components correspond to the independent diagonal phase charges, minus physical momentum, and minus physical angular momentum. With $e_t=(H+b\log r)Je+K$, the auxiliary equation holds exactly. Continuity determines $n_t$; it is not an independent datum and need not equal $\rho_t$. The exact finite-family velocity minimum is

$$W_c=BG^{-1}B^T>0,\qquad v_*=G^{-1}B^TW_c^{-1}\delta, \qquad E_{\rm vel,min}=\frac12\delta^TW_c^{-1}\delta.$$

$B,G,W_c$ are independent of $r$, since SF-DW derivative weights remain $\rho$. Separate charge and recoil minima would generally miss their cross terms. The complete initial energy is $E_{\rm pos}+F(r)$, where

$$E_{\rm pos}=\frac\epsilon8\int\rho\sum_i|R_i|^2,\qquad F(r)=\int\rho[Hr+b(r\log r-r+1)] +\frac12\delta(r)^TW_c^{-1}\delta(r),\qquad \int\rho r=Q.$$

Its finite-dimensional concave dual is

$$D(\lambda,\nu)=\lambda\cdot c-\frac12\lambda^TW_c\lambda-\nu Q +\int\rho b\left[1-e^{(\lambda\cdot z-H-\nu)/b}\right].$$

Writing $r_d=e^{(\lambda\cdot z-H-\nu)/b}$, its critical equations are $W_c\lambda=\delta(r_d)$ and $\int\rho r_d=Q$. Its Hessian is

$$-\operatorname{diag}(W_c,0) -\int\frac{\rho r_d}{b}(z,-1)(z,-1)^T<0.$$

There is a unique finite maximum. Testing the pointwise entropy infimum with $r=1$ bounds $\lambda$ on every nonempty superlevel set; uniformly for bounded $\lambda$, the linear or exponential term sends $D$ to $-\infty$ at the two ends of the $\nu$ axis. Thus superlevel sets are compact. The resulting $r_*$ is smooth, strictly positive and bounded above and below by finite constants. For every normalized positive trial $r$ and finite dual pair, the exact identity is

$$F(r)-D(\lambda,\nu) =\int\rho b\left[r\log\frac r{r_d}-r+r_d\right] +\frac12[\delta(r)-W_c\lambda]^TW_c^{-1} [\delta(r)-W_c\lambda].$$

It proves strong duality and uniqueness in this fixed-position, fixed-velocity-family class. Numerical evaluation supplies rigorous bounds only if spatial quadrature errors are also bounded. The earlier two-region minimum is recovered only with its matching admitted velocity Gram and recoil convention; a core-only family generally has a different cost.

The accepted amended principal/smooth-profile-density-dual.md and its independent review retain this proof, exact coupled-Gram controls and finite quadrature examples. No optimum over carrier/support positions or all velocity fields has been established.

Compact Relative Densities Recover The Same Energy Infimum

On the unchanged high core and exterior, $H=m_H$, $b=b_H$ and $z=z_H=(q_H,0,0)$. Hence the preceding optimum has the same exact constant

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