Boundary Disintegration for Weighted Residual Energy Trees

We study iterated weighted residual (WR) splittings generated by a positive operator $R_0\in B{\left(H\right)}_{+}$ and a finite family of contractions $C_1,\dots ,C_m$ in $B\left(H\right)$. The associated residual update $R\mapsto R^{1\mathrm{/}2}(I-C_j^{\ast }{C}_j)R^{1\mathrm{/}2}$ produces an $m$-ary energy tree of residuals $\left\{R_w\right\}$ and dissipated pieces $\left\{D_{w,j}\right\}$ indexed by finite words. From this tree we construct intrinsic path measures on the path space by biasing transitions either by a fixed quadratic form $x\mapsto ⟨x,D_{w,j}x⟩$ (defining the measures ${\nu }_x$) or, in the trace-class setting, by $\mathrm{t}\mathrm{r}\left(D_{w,j}\right)$ (yielding a reference measure ${\nu }_{\mathrm{t}\mathrm{r}}$). When $R_0\in S_1{\left(H\right)}_{+}$, we show that ${\nu }_{\mathrm{t}\mathrm{r}}$ dominates the family $\left\{{\nu }_x\right\}$ and identify $d{\nu }_x\mathrm{/}d{\nu }_{\mathrm{t}\mathrm{r}}$ as a canonical martingale limit of cylinder likelihood ratios. Along ${\nu }_{\mathrm{t}\mathrm{r}}$-almost every branch the residuals decrease to a terminal trace-class random variable $R_{\mathrm{\infty }}$, which we interpret as the WR boundary variable. We then disintegrate ${\nu }_{\mathrm{t}\mathrm{r}}$ over $\sigma \left(R_{\mathrm{\infty }}\right)$, obtaining a boundary law ${\mu }_{\mathrm{t}\mathrm{r}}={\left(R_{\mathrm{\infty }}\right)}_{\mathrm{\#}}{\nu }_{\mathrm{t}\mathrm{r}}$ and conditional path measures $\left\{{\nu }_{\mathrm{t}\mathrm{r}}^T\right\}$. Finally, we show that each ${\nu }_x$ admits a boundary representation as a mixture of $\left\{{\nu }_{\mathrm{t}\mathrm{r}}^T\right\}$ with an explicit boundary density $h_x=d{\mu }_x\mathrm{/}d{\mu }_{\mathrm{t}\mathrm{r}}$, thereby organizing the family of intrinsic WR path measures by a single trace-biased boundary disintegration.