The mass gap problem in quantum Yang–Mills theory remains one of the most profound unresolved problems in modern physics. In this work, using Scale Gauge Theory (SGT) Goto2025a, Goto2025b, Goto2026a, in which the scale of spacetime is gauged, we demonstrate that mass is a ``hierarchical memory of the vacuum. ''Based on the latest Particle Data Group (PDG 2024) analysis reporting ΛQCD~ 330MeV, we compute the packing of the vacuum energy density of the preceding hierarchy (the QCD phase) into the proton volume, and demonstrate that the dominant part of the proton’s inertial mass is naturally explained as a geometric ``hierarchical residue. ''This provides a direct physical answer to the freezing of the Yang–Mills coupling constant and the generation of the mass gap.
Goto Susumu (2026) studied this question.