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 Nₜˡᵒᶜ(x,y)=(Aᵗ)xy, while the long-step continuation on the cubic reference graph is modeled by Nₜᵃˢʸ(x,y)=χxy(t)Cxy6ᵗ t-dₛ/2 exp\![-aₛT(x,y)/t], T(x,y)=max_T∈ IₚᵣₑPT(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 dₛ, $T(x,y)$, and one matching amplitude. In a fourteen-displacement clean-reference pilot, the median absolute relative error of the truncated normalized generating factor at η=0.05 is 3.17×10⁻⁴. 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.