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 \(R\mapsto R^{1/2}(I-C^{*}_{j}C_{j})R^{1/2}\) 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 \(x\mapsto \left\langle x,D_{w,j}x\right\rangle \) (defining the measures \(\nu _{x}\) ) or, in the trace-class setting, by \(\textrm{tr}\left( D_{w,j}\right) \) (yielding a reference measure \(\nu _{\textrm{tr}}\) ). When \(R_{0}\in S_{1}\left( H\right) _{+}\) , we show that \(\nu _{\textrm{tr}}\) dominates the family \(\left\{ \nu _{x}\right\} \) and identify \(d\nu _{x}/d\nu _{\textrm{tr}}\) as a canonical martingale limit of cylinder likelihood ratios. Along \(\nu _{\textrm{tr}}\) -almost every branch the residuals decrease to a terminal trace-class random variable \(R_{\infty }\) , which we interpret as the WR boundary variable. We then disintegrate \(\nu _{\textrm{tr}}\) over \(\sigma \left( R_{\infty }\right) \) , obtaining a boundary law \(\mu _{\textrm{tr}}=\left( R_{\infty }\right) _{\#}\nu _{\textrm{tr}}\) and conditional path measures \(\left\{ \nu ^{T}_{\textrm{tr}}\right\} \) . Finally, we show that each \(\nu _{x}\) admits a boundary representation as a mixture of \(\left\{ \nu ^{T}_{\textrm{tr}}\right\} \) with an explicit boundary density \(h_{x}=d\mu _{x}/d\mu _{\textrm{tr}}\) , thereby organizing the family of intrinsic WR path measures by a single trace-biased boundary disintegration.
