IDOS Energy, Coherence, and Supporting Dynamics — Part 79

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$$Z_R=-c_R(10+x\cdot\nabla\log w)-2c_R'(s-s_0).$$

Here $c_R'\le0$, so $Z_R\ge5c_R$ beyond one fixed radius. Its compact negative part is controlled by $N_h$. The integration by parts is legitimate before physical phase integrability is known: a further cutoff at radius $T$ has error bounded by $C_RT^{-2}\int_{T<|x|<2T}w\chi^2/(1+|x|^2)dx\to0$. There is no boundary at the smooth plateau join.

Uniformly in $R\ge1$, direct differentiation gives

$$\frac{|\nabla F_R|^2}{F_R}\le C,\quad \frac{|\nabla^2F_R|^2}{F_R}\le C/s,\quad \frac{|\nabla\cdot(w\nabla F_R)|}{w}\le C,\quad \frac{|\nabla F_R|^2}{c_R}\le Cs,\quad \|V\||\nabla F_R|\le Cc_R/\sqrt s.$$

Integrating $E_1$ by parts bounds it by $CN_h$. Young's inequality and (N11) give $|E_2|\le\delta\int wF_R|\nabla X|^2dx+C_\delta N_h$ and $|E_3|\le CN_h$. The tested source is at most

$$\delta\int wF_R|X|^2dx+\delta\int wc_R\chi^2dx+ C_\delta\int ws|f|^2dx.$$

Choose $\delta$ smaller than the positive leading energy coefficients and the exterior phase coefficient. Absorption yields, independently of $R$,

$$\int wF_R(|X|^2+|\nabla X|^2+|\nabla\chi|^2)dx+ \int wc_R\chi^2dx\le C\left(N_h+\int ws|f|^2dx\right).$$

As $R\to\infty$, $F_R\uparrow s$ and $c_R\uparrow1$, proving (N10). The additional source moment and absence of a phase source are essential. This is not a bounded inverse for arbitrary physical $L^2$ forcing, and it is consistent with the unchanged essential zero threshold.

Physical Domain, Chain Counts And Inertia

Estimate (N10), multiplication by $\sqrt w$, and the bounded logarithmic derivative of $w$ place the solution in the physical $H^1$ form domain. The normalized affine forcing lies in $L^2$. The weak equation therefore places the solution in $\mathcal D(\mathscr A)=H^2$. Equivalently, the normalized principal Laplacians have positive constant coefficients and all remaining first-order and zeroth-order coefficients are bounded; the equation puts their Laplacians in $L^2$, giving $H^2$ by the full-space Fourier characterization. Thus the result solves the physical operator equation, rather than only a weighted formal divergence equation. The reduced solution is unique modulo constant common phase; testing the difference with itself sets $X=0$ and $\nabla\chi=0$. Complete correctors are unique modulo the full static kernel.

Every kernel direction with zero common-phase coefficient now has a partner, after subtracting (N6). The independent support obstruction excludes every direction with nonzero common-phase coefficient. The exact classification is

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