Dual-Anchor Finite Gluing of Anderson Green Functions on General Finite Graphs

We formulate and numerically test a dual-anchor finite-gluing construction for disorder-dependent Anderson Green functions on general finite graphs. For a common realization-dependent Hamiltonian $H_\omega(W)=H_{\rm crys}+WU_\omega,$ the weak-disorder anchor is the ordinary resolvent perturbation about the clean/reference-graph Green function, while the strong-disorder anchor is a hopping (locator) expansion about the random atomic one-site limit. The disorder realization is held fixed while $W$ is varied, so both anchors describe the same random landscape from opposite limits. The exact Hamiltonian, the two endpoint truncations, and the finite-gluing prescription are kept logically distinct. For finite graphs we give explicit Neumann-series error bounds in the endpoint domains, a sufficient endpoint-inheritance proposition for relation-level gluing, a shortest-path proposition for the strong locator term, and a sufficient causality theorem for positive relation-level gluing. The gluing kernel is a prescribed methodological choice rather than a claimed optimal or variational principle; a common weight-kernel sensitivity audit tests the dependence of the numerical conclusions on that choice. The construction is tested on four representative graph classes: a periodic dimerized one-dimensional chain, a periodic square lattice, a periodic cubic lattice, and a nonperiodic 4-regular random graph. The weak anchor exhibits the expected $W^{K+1}$ truncation law and accurately retains distance-resolved propagation beyond a single observation cell. In the strong-disorder regime, eigenstate envelopes are exponentially localized and the localization length grows away from the one-site limit as $W$ decreases; the graph-distance-matched locator term reproduces the inverse localization length increasingly well toward the atomic limit. The approximately linear strong-disorder behavior of $\xi^{-1}$ in $\log W$ is used only as a consistency check of the locator picture. The unoptimized two-anchor construction is not a precision description of the intermediate regime, but it remains finite and quantitatively nontrivial. For the distance-resolved local-scale metric, the recorded $W=4$--$8$ median errors range from about $1.3\%$ to $19.4\%$ across the four graph classes; a small common-kernel audit gives aggregate intermediate medians of about $12.5\%$--$16.9\%$ while retaining sub-percent endpoint medians. Causality, high-frequency normalization, and disorder-averaging order are separately audited. The result is a methodological proof of principle, formulated on general finite graphs and tested on representative classes, that clean-graph perturbative and random-atomic localized descriptions of the same disorder realization can be retained in their natural domains and assembled in disorder space by finite relation-level gluing. We do not claim graph universality, graph-specific optimality, or a precision determination of an Anderson transition.