Anderson Beta Flows on Hierarchical Weighted Tori

We construct a deterministic family of finite hierarchical weighted tori whose finite-scale effective spectral dimension can be tuned continuously over approximately dₛᵉᶠᶠ=1.65--$1.90$, and study a symplectic Anderson model on this family. The hierarchy is carried by deterministic bond weights on a torus with periodic boundary conditions, while quenched disorder is introduced independently as a scalar onsite potential. The use of a finite-scale effective spectral dimension is deliberate: asymptotic spectral dimension is a coarse-geometric invariant under broad classes of finite-scale graph deformations, whereas the present construction is designed to control path statistics over the same finite size window probed electronically. A small boundary twist defines a level-curvature Thouless number, from which we estimate the finite-size beta flow βₓ(L\!→\!2L)=<ln gₓ(2L)>-<ln gₓ(L)>/ln 2. At dₛeff1.75, the L=64→128 data exhibit multiple zero/fixed-point-like candidates and a finite disorder interval with positive or near-zero beta. Two-zero and critical-phase flows are known in symplectic fractals; the present result is therefore interpreted as their possible realization in a qualitatively different, periodically closed weighted-hierarchy architecture rather than as the first observation of such a flow. The data do not establish an asymptotic fourth phase. A matched-dₛᵉᶠᶠ comparison between layered and de-layered hierarchies shows that the finite-size Anderson response is not fixed by dₛᵉᶠᶠ alone. This is consistent with earlier topology dependence found on bifractals, while the present controls expose specific accompanying quantities---gap scaling, transverse conductivity, and weakest-cut bypassability. A third, intermediate hierarchy can be recalibrated to the same dₛᵉᶠᶠ while changing the weakest-cut ratio by orders of magnitude, establishing path geometry as an independently designable coordinate. A fixed-Lanczos, same-seed 16-realization curvature audit on these three geometries does not yet resolve a monotonic dependence of β or conductance variance on bypassability, so transport-design claims are kept exploratory. Replacing deterministic spin-orbit hopping directions by bond-random Haar-SU(2) matrices strongly alters the flow on the same weighted graph. Finally, a dimension sweep identifies a low-disorder near-critical candidate at dₛᵉᶠᶠ1.85 and W/Bclean0.109. The individual ingredients of this program have important precedents; the targeted literature audit performed for this draft does not, however, identify a single prior study combining the same weighted-torus construction, disorder separation, matched-dₛᵉᶠᶠ geometry controls, bond-randomness control, and dimension-resolved closed-system beta-flow analysis. The resulting platform therefore complements self-similar fractal and long-range hierarchical Anderson models at the level of the integrated numerical design rather than by claiming novelty for each ingredient separately.