Scretching–Planck Density and Phase in Quantum Theory III

The Scretching–Planck (SP) density–phase action is locally equivalent to the Schrödinger equation wherever the density is positive. Reconstructing a global wavefunction requires single-valuedness and compatible nodal and boundary conditions. Relativistic and gauge extensions require independently specified fields and dynamics. Varying the gauged matter action together with a separately specified parity-even gauge kinetic term yields the classical Yang–Mills equations. On a finite spatial lattice, the physical spectral gap is determined by the vacuum-weighted Poincaré constant and the Hamiltonian’s kinetic prefactor. The four-dimensional continuum Yang–Mills construction and a uniform positive gap in physical units remain open. In the separate two-dimensional \(O(N)\) model, the strict large-\(N\) Gaussian saddle has a gap bounded below by \(0.9624236501\) times the fixed saddle mass over the cutoff range studied. The reduced-mass Coulomb calculation for hydrogen gives \(E_1=-13.5982873\) eV and a \(1S\)–\(2P\) interval of \(10.1987154\) eV before additional recoil and level corrections. The muon lifetime serves as a closure calculation, and the true-muonium results are potential-model estimates.