Algebraic Grand Unification via the Noncompact Unitary Group U (1, 3) × U (1, 3): A Complete Topos-Theoretic Framework Unifying Baryon Land´e Factors, Electroweak Mass Spectrum, Lepton Masses, QCD Asymptotic Freedom, Quantum Gravity, and Intrinsic String Theory

We establish a deep algebraic isomorphism among seven domains of physics: baryon Land´e factors, the electroweak gauge boson mass spectrum, the logarithmic spectrum of charged lepton masses, QCD asymptotic freedom, quantum gravity via mass-spin coupling, the intrinsic form of string theory, and the topos-theoretic structure of physical reality. we formulate and rigorously prove a complete system of theorems within the noncompact unitary group G = U (1, 3) × U (1, 3). All theorems are derived from the representation theory of G without any fitted parameters. All numerical relations are consistent with the CODATA 2022 and PDG 2024 data within the stated uncertainties. The framework demonstrates that the numbers {3, 4, 5, 7, 8, 9, 11, 17, 26, 33} constitute a closed algebraic network generated from the single integer Nc = 3. In particular, the three spectra — electroweak (7, 9, 17), QCD β-function (11), and lepton logarithms (26) — share the common denominator 9 = N 2 c , satisfying the homology relations 7 + 9 + 17 = 33 = 11 × 3, 9 + 17 = 26. This “three-spectra homology” is shown to be a projection of the A3 root system of u(1, 3). Furthermore, we establish a complete extrinsic-intrinsic correspondence between external wave mechanics and the intrinsic representation space of U (1, 3). This mapping repositions string theory as the intrinsic form of wave mechanics with string tension Tint = Λ2 int/(2π) where Λint = p11/6 MZ ≈ 150 GeV, completely decoupled from the Planck scale.