A Graph Path-Sum Framework for the Anderson Localization-Length Exponent from a Critical Green Function

We develop a graph path-sum framework for constructing determining equations for the Anderson localization-length exponent from a critical Green-function sector. The path sum is an all-depth representation; finite depth enters only as an explicit computational truncation. The critical Green function generates contact-irreducible interference shells; forest subtraction removes inherited lower-topology pieces; orbit reduction quotients finite path symmetries; and weighted Krylov--Lanczos compression maps the resulting hierarchy to a Jacobi operator. At finite depth the exponent is then defined by a finite Schur-complement equation, without assuming that the all-depth hierarchy has a simple closed form. For the three-dimensional orthogonal box-disorder problem we evaluate the hierarchy through depth four and obtain a working largest Jacobi eigenvalue $\lambda_4\simeq0.0345442$. Under an explicitly conditional critical-mode detuning-renewal closure and a Gaussian reference conversion $d_w^{(0)}=2$, the reference sequence $1.5000\to1.5227\to1.5398\to1.5528\to1.5537$ moves toward the independent transfer-matrix benchmark $\nu\simeq1.571$ at every computed depth. Because the nested Jacobi edges obey interlacing and the auxiliary renewal map is increasing, the monotonic direction is partly structural; the useful diagnostic is the size and stabilization of the finite-depth corrections. This auxiliary sequence is not a numerical root of the physical Schur equation, not a convergence theorem, and not an identification of the physical critical walk dimension with $2$. A separate transfer-Ward projection remains near $1.0225$, showing that Ward and physical disorder-detuning projections must be distinguished. The physical detuning tangent is generated internally by differentiating the fixed-disorder resolvent and exact box-cumulant functional, so the remaining microscopic task is a singly marked shell calculation rather than an additional empirical input. The graph-path formulation also exposes geometry that a one-parameter Euclidean dimensional expansion does not retain by itself. At minimal single-tail level, $\nu_0\sim\frac{2}{d_w(d_s-2)},$ so a walk/path dimension enters before higher-shell corrections. For the same normalized critical path kernel, spectral/return, walk, and volume-growth exponents obey $d_s d_w^*=2d_V$. Noninteger effective dimensions can therefore be carried by path statistics instead of by a formally noninteger number of Cartesian integration variables. The three-dimensional demonstration uses local vertex data, path scaling, and finite irreducible topology, but not finite-size flow data, a full finite-sample adjacency matrix, or a full lattice spectrum as fitting inputs. The same critical relation can serve as the middle, critical-region anchor in finite weak--critical--strong gluing, while the graph-path architecture provides a concrete extension route to other Anderson symmetry classes and to other kernel/resolvent theories after their theory-specific propagators and local vertices are rederived.