IDOS Energy, Coherence, and Supporting Dynamics — Part 49

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rest flow and constant frame. Multiply the centered low carrier by $e^{Jp\cdot x}$. Its occupation potential and horizontal spatial residuals remain zero, while $a=p$. Write

$$\delta=q_H-q_L,\quad B=|\delta|^2,\quad A_L=\frac{N_L(N_L+d+1)}{(d+1)^2(d+2)},\quad f(r)=\frac{v_*^2}{\kappa}\left(\sqrt{1+\kappa^2r^2/v_*^2}-1\right).$$

Minimum affine horizontal compensation restores the high charges at energy $QB/(2\epsilon A_L)$. Its translation covector combines with the common phase to give $Q(p+\delta/\ell)$. Allowing scalar-support recoil, while keeping laboratory vector-support and frame velocities zero, yields the exact restricted-family energy

$$\frac{E(p)}Q=m_L+\frac{B}{2\epsilon A_L} +f(|p|)+\frac C2|p+\delta/\ell|^2, \qquad C=\frac{159}{\omega_{\rm floor}}.$$

All physical angular charges are zero by radial symmetry. This is the scalar-support family; allowing additional velocity channels requires a separate minimization and can lower its cost. For $s=\sqrt B/\ell>0$ the unique minimum has $p_*=-r_*\delta/\sqrt B$, where

$$\frac{\kappa r_*}{\sqrt{1+\kappa^2r_*^2/v_*^2}}=C(s-r_*), \qquad 0<r_*<s.$$

Let $\Phi=f(r_*)+C(s-r_*)^2/2$. Strict convexity gives $0<\Phi<\min(Cs^2/2,f(s))$: the optimum improves both zero common phase and zero recoil. Defining $\kappa_{\rm eff}=2\kappa/[\sqrt{1+\kappa^2s^2/v_*^2}+1]$ gives

$$\frac{C\kappa_{\rm eff}s^2}{2(C+\kappa_{\rm eff})} \le\Phi\le\frac{C\kappa s^2}{2(C+\kappa)}.$$

For small offsets, $\Phi=C\kappa s^2/[2(C+\kappa)] -[\kappa^3/(8v_*^2)][C/(C+\kappa)]^4s^4+O(s^6)$. For large offsets it is $v_*s-v_*^2/\kappa-v_*^2/(2C)+O(s^{-1})$. The separate horizontal compensation remains and is not removed by this linear large-offset behavior. The endpoint has lower energy exactly when

$$m_H-m_L>\frac{B}{2\epsilon A_L}+\Phi.$$

Both $\kappa=8/\omega_{\rm floor}$ and $\kappa=8/m_L$ are covered, with their associated positive $b$ laws. Entropy vanishes at $n=\rho$; its dynamics do not disappear. Continuity generally gives $n_t=h_A\cdot\nabla\rho\ne\rho_t$, accommodated by phase acceleration. The data are not an invariant stationary ansatz. At zero phase edges, a proper lower prefix cannot carry the deleted individual pair charges. At positive edges, $N_L=0$ cannot carry the required nonzero affine charges. These obstructions remain exact.

For a fixed finite comparison set with positive gaps and $\epsilon=\omega_{\rm floor}\ell^2/(4M^2)$, the complete penalty is continuous and strictly decreases from infinity to zero as common $\ell$ increases. A finite common threshold therefore exists within this family; no physical value of $\ell$ has been calibrated. The 24 exact pair/control rows and independent 1,373-check review in phase-recoil-endpoint and phase-recoil-review retain the maintained counts, both coefficient laws and the origin obstruction. These whole-space endpoints change the normalized carrier at infinity; the compact-data evolution theorem does not apply to them directly. No dynamical accessibility, decay, observed-particle identification or physical mass-operator derivation follows from their energy comparison.

Smooth Radial Comparisons At Actual Prefix Counts

The smooth density dual can retain the full joining layer and actual sparse occupation polynomial. Use rational support with $n_{\rm peak}=L=1$, $\rho=(1+R^2)^{-8}$, where $R=|x|$ is ordinary ten-dimensional radius. These common-unit controls set all $\mu_d=1$, $\gamma=\omega_{\rm floor}=1$, $v_*=1/2$ and $\epsilon=\ell^2/(4\,94^2)$. The shared choices $\ell=1,10,100$ are different action controls, not separately fitted particle sizes. Both $(\kappa,b)=(8,1/128)$ and $(8/m,m/128)$ are retained.

For $N_H>N_L\ge1$, put

$$D=\binom{N_H+d}{d},\quad L_c=\binom{N_L+d}{d},\quad E_c=D-L_c, \quad c=\frac{(N_H-N_L)D}{(d+1)E_c}.$$

Let $u_L,u_E$ be real paired unit carriers uniform on the low labels and the remaining high labels, respectively. Every pair retains the same

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