Finite Hierarchical Running of the Fine-Structure Constant in the BGRT Framework

This preprint proposes a finite hierarchical running modelfor the fine-structure constant within the BGRT(Boundary Geometry Response Theory) framework. Instead of treating the inverse fine-structure constantalpha^-1(E) as an unbounded logarithmic flow, the modelinterprets running as a finite reduction of boundary hierarchygoverned by a structural number N = 111. The proposed formula reproduces: - the low-energy value alpha^-1 ≈ 137.036 - the Z-pole value alpha^-1(M_Z) ≈ 128.0 and predicts high-energy saturation: - alpha^-1(inf) ≈ 127.24 The central falsifiable prediction is that alpha^-1(E) doesnot run toward zero at extreme energies, but insteadapproaches a finite saturation value near 127. The same boundary hierarchy motivates: - a 3+1 gauge-like pattern from a signed direction network with symmetry Z_2^3 semidirect S_3 - a chirality-grading operator candidate Gamma^5_BGRT on H_eff = H_+ oplus H_- These are presented as candidate precursors to Standard Modelgauge and chirality features, not as completed derivations.The connection to the full Standard Model remains unproven.