Scretching–Planck Density and Phase in Supersymmetry and String Theory I: Superpartners from Densities, Vacuum Quantum Terms, Critical Dimensions, T-Duality and the Dilaton

This paper applies the Scretching–Planck (SP) density–phase representation, \(\psi=\sqrt P\,e^{iS/\hbar}\), to supersymmetric quantum mechanics and selected string models. For a real, nodeless ground state, the balance \(E_0=V+Q_{\mathrm{SP}}[P_0]\) expresses the superpotential as \(W=-\sqrt{m/2}\,u_0\), where \(u_0\) is the osmotic velocity of the ground-state density. The same density gives the partner potential \(V-(\hbar^2/2m)(\ln P_0)''\). Five numerical benchmarks agree within \(2.3\times10^{-4}\) at the grid resolution used, and the iterated factorization reproduces the first six hydrogen energies within \(6.2\times10^{-6}\) Ry. For the factorization branch considered, a normalizable zero mode signals unbroken supersymmetry. The paper then examines the cancellation of bosonic and fermionic zero-point energies. A cutoff estimate with superpartners near \(2\) TeV gives a residual vacuum energy \(6.4\times10^{59}\) times the observed dark-energy density. In the string examples, regularized oscillator sums contribute to the intercepts in the critical bosonic and superstring theories, with \(D=26\) and \(D=10\), respectively. The bosonic level counts are \(24\) and \(324\), and the oscillator contributions cancel in the Ramond sector. For circle compactification, SP phase circulation quantizes momentum, and T-duality exchanges momentum and winding. Four additional massless vectors appear at the self-dual radius in the bosonic circle model. The paper compares the charge-matched Kaluza–Klein scale of \(5.215\times10^{17}\) GeV with the coupling-dependent heterotic scale \(g(5.265\times10^{17}\,\mathrm{GeV})\). The scales agree within about 1% at \(g=1\), while the separate self-dual matching condition gives \(g^2=8\pi\alpha\). Finally, writing the dilaton as \(P=e^{-2\Phi}\) turns its string-frame kinetic term into the density-gradient expression \((\nabla P)^2/P\) and gives the Einstein-frame conformal factor \(P^{2/(D-2)}\). Ten solved problems are included.