IDOS Energy, Coherence, and Supporting Dynamics — Part 52

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$U_t=E_t=0$. Let $p$ be the complete carrier translation covector, whose physical momentum is $-p$. Allow every finite-kinetic-energy scalar velocity $W_t$ and laboratory vector velocity $V=Z_t$. The two isotropic translation Grams are

$$G_W=g_WI,\quad G_Z=g_ZI,\qquad g_W=\frac{Q\omega_{\rm floor}}{159},\quad \frac{g_Z}{g_W}=\frac{15}{44},\quad g_{\rm tot}=\frac{59Q\omega_{\rm floor}}{6996}.$$

These follow from the exact ordinary-$\mathbb R^{10}$ integrals $\int(\partial_1W)^2=\pi^5L^8/1890$ and $\int\sum_j(\partial_1Z_j)^2=\pi^5L^8/5544$, multiplied by $M_W=M_Z=n_{\rm peak}L^2\omega_{\rm floor}/212$. Orthogonal projection in the supporting kinetic norm gives

$$W_t=-\frac p{g_{\rm tot}}\cdot\nabla W,\qquad V_j=-\frac p{g_{\rm tot}}\cdot\nabla Z_j,\qquad E_{\rm rec,min}=\frac{|p|^2}{2g_{\rm tot}}.$$

Every other admissible velocity adds half its squared orthogonal kinetic norm. This is the exact minimum over these $W/Z$ velocities, with a cost $44/59$ of scalar recoil. It is not a minimum over carrier/frame/flow velocities, positions or all fields, and is not a boost-derived inertia.

For a physical antisymmetric rotation $A$, radial $W$ and $Z=xh(R)$ obey $(Ax)\cdot\nabla W=0$ and $(Ax)\cdot\nabla Z=AZ$. Thus $\int M_ZV\cdot[AZ-(Ax)\cdot\nabla Z]=0$ pointwise. Intrinsic and orbital vector terms may each be nonzero and cancel. In material variables $z=E^TZ$, the induced frame covector $M_ZV\cdot(\delta E)z$ is essential, including at $E_t=0$. These velocities preserve zero carrier angular momentum but cannot repair a nonzero one at fixed radial positions.

For a compact radial cutoff, define $g_{Aj}=M_A\int\chi^j|\nabla A|^2/10$ for $A=W,Z$ and $j=1,2$, with the vector derivative norm summed over its components. In the stated family $W_t=\chi v_W\cdot\nabla W$, $V=\chi(v_Z\cdot\nabla)Z$, the exact result is

$$g_\chi=\frac{g_{W1}^2}{g_{W2}}+\frac{g_{Z1}^2}{g_{Z2}},\qquad v_A=-\frac{g_{A1}}{g_{A2}}\frac p{g_\chi},\qquad E_{\rm rec,\chi}=\frac{|p|^2}{2g_\chi},\quad g_\chi\le g_{\rm tot}.$$

Separate coefficients are needed; a common velocity coefficient is generally suboptimal. For nonzero $p$ this improves the scalar-core family. Existing phase restoration remains unchanged. A genuine frame rotation $F=\chi F_0$ can separately cancel carrier angular covector $B_0$, with $(F_0)_{ab}=-(B_0)_{ab}/(\kappa_FQ_1)$ and $z_t=E^TV-Fz$. Its extra energy is $Q_2\sum_{a<b}(B_0)_{ab}^2/(2\kappa_FQ_1^2)$; no additional vector/frame cross energy remains in laboratory velocities.

For the preceding whole-space common-phase endpoint, replace only $C=159/\omega_{\rm floor}$ by $C_{\rm tot}=6996/(59\omega_{\rm floor})$. Its convex minimization, bounds and conditional threshold retain their form. The optimized cost strictly decreases when the nonzero charge offset is fixed. Positive phase edges and $N_L>0$ are still required for that endpoint. Compact high-core compensation retains both symmetry branches. The amended complete-support-recoil/report.md and independent 340-check review establish these initial-jet results; they do not establish evolution or new physical particle properties.

Density Concentration And A Convex-Envelope Bound

In the fixed radial family, vary only the entropy coefficient to $b=\tau b_0$, holding $H,C$ and the associated $\kappa$ branch fixed.

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