Parameter-Free Derivation of the Proton Charge Radius from M-Theory Compactification on G₂ Manifolds

The proton charge radius has been derived from first principles within the Quantum Geometric Unification framework, yielding the parameter-free prediction r_p = √2 ℏc/Λ_QCD = 0.8405 fm, in 0.11% agreement with the CODATA 2018 world average 0.8414 ± 0.0019 fm and 0.04% agreement with the CREMA muonic hydrogen value 0.84087 ± 0.00039 fm. The factor √2 has arisen from the canonical normalisation constraint C_QG + C_LIG = 1/√2 of the four-dimensional N = 1 supergravity effective theory, derived from the Kähler potential K = -(7/3)ln[(T+T̄)/2] of the G₂-holonomy compactification. The QCD confinement scale Λ_QCD = 332 MeV has been determined independently from the Yang-Mills mass gap theorem (DOI: 10.5281/zenodo.19464295) and the AMSB-determined strong coupling constant α_s(M_Z) = 0.1170. A lepton-dependent correction from Born-Oppenheimer modulus dynamics has predicted a 2.4% reduction in the radius measured by unstable probes (muonic hydrogen) relative to stable probes (electronic hydrogen), with the direction consistent with the historical proton radius puzzle. Seven appendices provide self-contained derivations of all QGU ingredients for readers unfamiliar with the framework. This constitutes prediction number 31 of the QGU programme across 10 physical sectors with zero adjustable parameters. 1805 lines, 4 figures, 3 tables, 42 equations, 7 appendices, 53 references.