IDOS Energy, Coherence, and Supporting Dynamics — Part 71

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the positive relative chart of the preceding uniform-vector construction. In particular retain the changed spatial support square $|B_{\rm sup}/W|^2$, rather than the older degenerate denominator. Its exact support and physical density are

$$W_0=(1+|x|^2/L^2)^{-4},\qquad Z_0=\frac{x}{L}(1+|x|^2/L^2)^{-5},$$ $$w_0(x)=\sigma\left[1+\psi\left( \frac{|x|^2/L^2-1}{\epsilon_p}\right)\right]^{-8}, \qquad \psi(s)=\begin{cases}s e^{-1/s},&s>0,\\0,&s\le0.\end{cases}$$

All integrals are over ordinary physical $\mathbb R^{10}$. The positive density has a constant core and $w_0\asymp\langle x\rangle^{-16}$ outside; $\mathcal A=\int w_0\,dx$ is finite. The action, its support, ten flow variables, 45 physical frame variables, complete carrier and reconstructed density are all varied. Changing the support square is an explicit change of constitutive law; results for this version do not automatically apply to a Gaussian exterior or the original vector square.

For $p_k=|c_k|^2$, put $r=\sum_kp_k^2$, $\bar k=\sum_kp_k k$ and $C=\operatorname{Cov}_p(k)$. The same literal finite-carrier selectors $V,V_\phi$ defined above give the complete shape energy

$$H(p,c)=\frac{\mu(p)}r+ \frac{\lambda\mu(p)}{r^3} \left[V+(1+\eta)V^q+\eta V_\phi\right], \qquad \lambda>1,\quad q\in\{2,4\},\quad\eta\ge0.$$

Keep either positive constant $\mu$ or the stated common smooth unit law at depths $2,3,4,5,6,10$. Its supplied values are not fitted separately for the disturbances. The two entropy alternatives and horizontal alternatives are

$$(b,\kappa)=\left(\frac{\omega_*}{128},\frac8{\omega_*}\right) \quad\hbox{or}\quad \left(\frac m{128},\frac8m\right),\qquad m=\frac\mu r,$$ $$\varepsilon_h(p)= \frac{\omega_*}{4[\omega_*\mu(p)+\operatorname{tr}C(p)/\ell^2]} \quad\hbox{or}\quad\varepsilon_h=\varepsilon_c>0, \qquad \varepsilon_c M^2/\ell^2\le\omega_*/4.$$

Here $0<\omega_*\le\inf\mu$. The constant-horizontal bound is on the whole carrier. Every coefficient variation remains in the equations. Choosing $q$, $\eta$ and an entropy/horizontal alternative fixes one action for both families; no branchwise change is made in the comparison.

For a prefix supported on $d$ axes through degree $N$, $N\le M$,

$$D=\binom{N+d}{d},\quad c_k=D^{-1/2}\exp J\left[\Omega t+ \frac{(k-\bar k)\cdot x}{\ell}+\phi_k\right],\quad \Omega=\mu_dD.$$

The phases are arbitrary constants for $\eta=0$ and affine in $k$ for $\eta>0$. At a maintained positive prefix the first two jets of the actual unit vanish, but its higher off-branch variations are retained. The common $N=0$ origin has $D=1$, unit $\mu_3$ and no distinguishable active depth. The exact branch has

$$\rho=w_0,\qquad j^\alpha=w_0U^\alpha,\qquad T^{\alpha\beta}=w_0\Omega U^\alpha U^\beta,\qquad E_{\rm rest}=M_{\rm boost}=\mathcal A\Omega$$

in the action's unit-speed convention. This complete branch identity does not make the stress dust-like off the branch.

The Complete Prefix Operator Has No Gap

Write $x$ for occupied radial sphere tangents and $h_E$ for all empty pairs. With $\sum x_k=0$, the full shape second jet satisfies

$$Q_H\ge\mu_dD\left[4(\lambda-1)|x|^2+ (\lambda+2)|h_E|^2\right]+\lambda\mu_dD^3\eta Q_\phi.$$

The anchor term alone contributes $D^{-2}[4|x|^2+4D x_0^2+|h_E|^2]$ to $Q_V$; the other residual squares are nonnegative. Empty labels beyond the prefix and on unused axes are included. The full support and horizontal gradient costs obey

$$\frac{21}{212}+\frac14=\frac{37}{106},\qquad \frac78-\frac{37}{106}=\frac{223}{424}>0,$$

which leaves a positive flow remainder. This is a relative weighted inequality, not an absolute positive lower bound at infinity.

After the real reference rotations and complete kinetic normalization, $v=\sqrt{w_0}R_0(x)X$ satisfies

$$v_{tt}+G(x)v_t+Av=0,\qquad A=\mathcal T^*\mathcal T\ge0,\qquad G^*=-G.$$

The bounded invertible $R_0$ includes the support/frame/flow coframe. The full gyro is retained, including empty-pair blocks. The form domain is

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