Spectral Geometry on 𝑇5 with SU(3): Algebraic Identities and Their Physical Consequences A geometric framework relating 𝛼, 𝑚𝑝/𝑚𝑒, and Δ𝑚𝑛-𝑝 to the topology of a five-torus

A geometric framework on the five-torus T⁵ with SU(3) gauge field that derives three fundamental constants — the fine-structure constant α, the proton-to-electron mass ratio m_p/m_e, and the neutron-proton mass splitting Δm — from first principles, with zero free parameters. The core result is a transcendental equation for α whose coefficients are not fitted but computed: the instanton action S₀ = π² (a theorem of differential geometry), the spinor trace b₁ = 4 (from five-dimensional Clifford algebra), and the zero-mode measure ν = 2 (from the Atiyah-Singer index theorem). The unique physical solution gives α ≈ 1/137.036 to 37 ppm at one loop and sub-ppm with the derived two-loop coefficient c₂ = (5/2)ln 2 − 3/8. The mass ratio follows from the exact Weyl identity 6π⁵(1 + α²/√8) = 1836.15268, matching experiment to 0.002 ppm. Twenty parameter-free empirical consequences are tested, spanning precision values (V1, V3: sub-ppm), structural integers (N_c = 3, d = 5, |W| = 6), and qualitative features (θ_QCD = 0, proton stability). All claims are classified into proven theorems (Tier 1), conditional propositions (Tier 2), and explicit conjectures (Tier 3). One axiom (Working Hypothesis M) remains unproven; its removal would complete the derivation. Full computational reproduction: 36 Python scripts, 46 numerical results, 15 figures. All code at https://github.com/SFST-2026/Spectral-Geometry