IDOS Energy, Coherence, and Supporting Dynamics — Part 66

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$\sigma\varepsilon_p^pL^{2p}R^{-2p}$. The elementary bounds $(1+s)^{-p}\le F_p(s)\le\max(1,s)^{-p}$ for $s>0$ imply

$$\sigma a_p(1+t_R)^{-p}\le w_0\le \sigma b_p(1+t_R)^{-p},\qquad a_p=\min(1,\varepsilon_p)^p,\quad b_p=(2+\varepsilon_p)^p.$$

Every spatial integral is ten-dimensional, including for active depths $2,3,4,5,6,10$. The complete branch has finite action and energy precisely when $p>5$. More generally, $\int R^a w_0\,d^{10}x<\infty$ for $a\ge0$ precisely when $a<2p-10$. Equality gives logarithmic divergence. The common choice $p=8$ makes $\mu_s(W_0)$ bounded above and away from zero, with limit $C_s\sigma\varepsilon_p^8$ at infinity. It also gives finite moments of every order below six.

The logarithmic derivative has a uniform bound:

$$\Lambda(W)=W\partial_W\log w(W) =\frac p4(s+2/\varepsilon_p)\frac{\psi'(s)}{1+\psi(s)}, \qquad 0\le\Lambda\le B_p=\frac p4(1+2/\varepsilon_p),$$

with $\Lambda=0$ in the core and $\Lambda\to p/4$ at vacuum. Every fixed positive-order logarithmic derivative is bounded. Thus for $u=\log(W/W_0)$,

$$\left|\log\frac{w(W)}{w_0}\right|\le B_p|u|, \qquad \rho=w(W)e^{F/b},\quad F=U\cdot a-h-\Xi_q.$$

A bounded relative support and bounded $F/b$ control the positive density ratio uniformly in spatial location. Density reconstruction retains one derivative because $F$ contains derivatives; it is not a derivative-free map on the carrier coordinates.

The common total action is explicit. Define

$$M_{j,p}=\int_0^\infty s^jF_p(s)\,ds,\qquad J_p=\varepsilon_p\sum_{j=0}^4\binom4j\varepsilon_p^jM_{j,p}.$$ $$\frac{\Gamma(j+1)\Gamma(p-j-1)}{\Gamma(p)} \le M_{j,p}\le\frac1{j+1}+\frac1{p-j-1}, \qquad \mathcal A=\int w_0\,d^{10}x =\frac{\sigma\pi^5L^{10}}{120}(1+5J_p).$$

The exterior-to-core action ratio is $5J_p$. One prescribed common $\mathcal A$ fixes $L=[120\mathcal A/(\sigma\pi^5(1+5J_p))]^{1/10}$ if the other support parameters are specified. Identifying $\mathcal A$ with $\hbar$ is still a normalization premise; no particle-specific field amplitude is chosen.

This improvement is on the positive domain $W>0$. At $p=8$ the extension of $w$ to $W=0$ is $C^2$ but not $C^3$, and that of $\mu_s$ is continuous but not $C^1$. Uniform exterior estimates do not imply a smooth vacuum evolution theorem.

Exact Branches And The Original Polynomial Action

The three support identities

$$\nabla W_0=-8Z_0/L,\qquad W_0\partial_iZ_{0j} =(W_0^{9/4}\delta_{ij}-10Z_{0i}Z_{0j})/L,\qquad Z_0^2-W_0^{9/4}+W_0^{5/2}=0$$

do not depend on the density function. On the centered uniform prefix, all support and horizontal residuals vanish, $\rho=w_0$ and $F=0$. Coefficient variations multiply these zero residuals; the density term's support variation is $b(\rho/w-1)w'(W)=0$. Varying the full action before taking branch values therefore retains

$$D=\binom{N+d}{d},\qquad \Omega=\mu_dD,\qquad j^\alpha=w_0U^\alpha,\qquad T^{\alpha\beta}=w_0\Omega U^\alpha U^\beta,\qquad E_{\rm rest}=M_{\rm boost}=\mathcal A\Omega$$

in characteristic-speed-one units. The equal participating allocation and flat occupied spectrum are those of the preceding participation Hamiltonian.

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