The Scretching–Planck Equations: All of Quantum Theory

Description We derive all of quantum theory from a single Scretching–Planck (SP) density–phase action: The SP pair is exactly Schrödinger and Heisenberg quantum mechanics. Its relativistic polar forms are Klein–Gordon and Dirac. Making its phase local yields Maxwell's field and QED. Making its weak-isospin and colour orientations local yields the weak, electroweak and strong interactions. The non-abelian field equations, known since 1954 as the Yang–Mills equations, are derived from SP + non-abelian Maxwell in seven steps rather than assumed. All results are recomputed from CODATA and PDG values. Quantum mechanics. SP is exactly Schrödinger's equation; quantising (P, S) gives Heisenberg's [x, p] = iħ. The SP Fisher term gives Δx·Δp ≥ ħ/2, and hydrogen 1s satisfies V + Q_SP = −½ a.u. at every point. Relativistic quantum mechanics. Hydrogen 1s is −13.598287 eV (Schrödinger), −13.599193 eV (Klein–Gordon) and −13.598468 eV (Dirac). The 2p fine structure is 10.94 GHz (measured 10.969). QED. 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. It gives v = 246.22 GeV and a tree-level M_W of 79.83 GeV (measured 80.369). Yang–Mills equations derived. Seven steps lead from a local SP orientation to D_νF^{νμ} = J^μ and the Bianchi identity, with F = ∂A − ∂A − ig[A, A]. On their own, those equations do not derive quantum mechanics, relativistic quantum mechanics or the matter side of QED; the SP route derives all three. QCD. The predictions match experiment: α_s(M_Z) = 0.1180 (measured 0.1180 ± 0.0009), a gluon colour factor C_A = 3 (measured ≈ 3), σ(e⁺e⁻ → W⁺W⁻) = 17.77 pb at 200 GeV (measured ≈ 17), and a proton mass of 938 MeV generated by the field. Deuteron (both SP branches). From scattering data alone, the binding energy is B = 2.2266 MeV (measured 2.2246, 0.09%) and the deuteron mass is correct to 2 keV. The D/S ratio is η = 0.0257 (measured 0.0256). Mass gap. The gap is the Poincaré constant of the SP vacuum density, Δ = λ(P₀), and is proved positive on every finite lattice. Its four-dimensional continuum limit remains open. The work includes twenty tabulated successes, 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 derivation; QCD; deuteron; mass gap