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 (MZ) ≈ 128. 0 and predicts high-energy saturation: alpha^-1 (inf) ≈ 127. 24 The central falsifiable prediction is that alpha^-1 (E) doesnot run without bound at extreme energies, but insteadapproaches a finite saturation value near 127. This updated version additionally includes a master index ofsupplementary BGRT exploratory notes, tracing the proposedstructural route from the alpha-running formula to candidateStandard-Model-like internal features. These supplementary notes discuss: a candidate geometric origin of N = 111 a 3+1 gauge-like pattern from a signed direction networkwith symmetry Z₂³ semidirect S₃ a su (2) -like algebraic structure a chirality-grading operator candidateGamma⁵BGRT on Hₑff = H_+ oplus H_- an internal doublet space Hᵢnternal a charge-label operator candidate Qcandidate an inverse reconstruction map between charge spectraand the structural coefficients a, b, c, d These structures are presented as candidate precursors toStandard Model gauge, chirality, and charge features, not ascompleted derivations. The connection to the full StandardModel remains open and unproven.
Kenichi Ueki (Sat,) studied this question.