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 (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.