SP + Non-Abelian Maxwell: Quantum Mechanics to QCD from the Scretching–Planck Equations

Description We derive quantum mechanics, relativistic quantum mechanics, quantum electrodynamics, the weak and electroweak interactions, and quantum chromodynamics from a single Scretching–Planck (SP) density–phase action. Making the SP phase local yields Maxwell's field. Making the SP weak-isospin and colour orientations local yields the non-abelian Maxwell field with its self-interaction. The resulting field equations are derived step by step and coincide with the Yang–Mills equations of 1954, obtained here from SP rather than assumed. All results are recomputed from CODATA and PDG values. One chain. The SP action ψ = √P e^{iS/ħ}, E = V + Q_SP, is exactly equivalent to Schrödinger and Heisenberg quantum mechanics. Its Fisher term gives Δx·Δp ≥ ħ/2. Its relativistic polar forms are the Klein–Gordon and Dirac (Takabayasi) equations; the Dirac hydrogen 1s energy is −13.598468 eV and the 2p fine structure is 10.94 GHz (measured 10.969). SP + Maxwell = QED. A local phase gives minimal coupling. The Lamb shift is 1050.5 MHz at leading order (measured 1057.8), and a_e = α/2π = 0.0011614 (measured 0.0011597). Weak interaction. W exchange gives G_μ/√2 = g²/8M_W². The tree-level muon lifetime is 2.1873 μs (measured 2.19698). Electroweak. The SP polar form of the Higgs doublet is the unitary gauge. This gives v = 246.22 GeV and a tree-level M_W = 79.83 GeV (measured 80.369). SP + non-abelian Maxwell. The colour and weak-isospin fields, and their self-interaction G = ∂A − ∂A + g f^{abc}AA, are derived from the local SP orientation. Without the self-interaction the theory fails four measurements: α_s(M_Z) = 0.904 (measured 0.1180), C_A = 0 (measured ≈ 3), σ_WW(200 GeV) = 41.2 pb (measured ≈ 17), and a proton mass of 9 MeV (measured 938). With it, all four are reproduced. What the gauge equations alone do not derive. They take quantum mechanics and the Dirac matter field as given. The SP route derives quantum mechanics, relativistic quantum mechanics and the matter side of QED, and then generates the gauge fields. Mass gap. On every finite lattice the mass gap is proved positive and equals the Poincaré constant of the SP vacuum density, Δ = λ(P₀). It is computed in SU(2), SU(3), Migdal–Kadanoff and O(N) models. The four-dimensional continuum statement (Conjecture C) remains the open Clay problem. Both SP branches. The Wronskian of the two SP solutions gives the deuteron. The binding energy B = 2.2266 MeV follows from scattering data alone (measured 2.2246), the deuteron mass to within 2 keV, and η = 0.0257 (measured 0.0256). It also gives a no-threshold lemma for confinement. Includes 60 solved problems and all code and data. Keywords: Scretching–Planck; density–phase action; Schrödinger; Heisenberg; Klein–Gordon; Dirac; QED; weak interaction; electroweak; non-abelian Maxwell; Yang–Mills equations; QCD; deuteron; mass gap