Finite Gluing of Anderson Green Functions from Clean Crystals to Strong Disorder

We present a finite-gluing framework for Anderson Green functions that retains the clean-crystal and strong-disorder limits as distinct analytic anchors. The weak anchor is a resolvent expansion about a freely chosen clean reference Hamiltonian. The strong anchor is a random atomic/corridor resolvent that exactly resums walks internal to a finite endpoint-conditioned corridor. These two anchors define the minimal construction; no middle sector is required. As an optional refinement, we construct an intermediate target one level earlier in step space. For fixed endpoints $x,y$, short reference walks are kept exactly through $N_t^{\rm loc}(x,y)=(A^t)_{xy},$ while the long-step continuation on the cubic reference graph is modeled by $N_t^{\rm asy}(x,y)=\chi_{xy}(t)C_{xy}6^t t^{-d_s/2}$ $\exp\!\left[-a_{\d_s}\frac{T(x,y)}{t}\right],$ $T(x,y)=\arg\max_{T\in\mathcal I_{\rm pre}}P^T(x,y).$ The two descriptions are glued over a finite step window, so $t=0$, minimum-step constraints, and parity remain in the exact sector, whereas the long-walk tail is compressed into $d_s$, $T(x,y)$, and one matching amplitude. In a fourteen-displacement clean-reference pilot, the median absolute relative error of the truncated normalized generating factor at $\eta=0.05$ is $3.17\times10^{-4}$. The principal Anderson result is stability, not a precision transition estimate. In the fixed-center $L=7,9$ pilot, the ratio of middle-refined to two-anchor global endpoint-block error lies in $0.9909$--$1.0016$ with median $0.99998$, while the mean absolute change in the size-normalized decay diagnostic is $0.0084$. Thus the path-count refinement can be inserted without destabilizing the recorded endpoint gluing. Self-consistent recentering is retained only as a stress test because it may converge when the finite-size crossing is itself unreliable. No critical exponent is estimated from the present small, low-statistics data. The architecture is intended to support future calculations that start from material- or symmetry-specific clean tight-binding models, retain a strong-disorder description in the same representation, and optionally use independently established universal critical data as gluing constraints.