Hamilton's Principle, Feynman's Path Integral, and the Action Quantum h = 2πℏ as Capacity-Rotor Theorems: A Parameter-Free Derivation from QTT Axioms A1--A7U

Why the path-integral phase is a real action-rotor ledger Version: 2.02 First public release: 9 May 2026 Current release: 26 June 2026 Concept DOI: 10.5281/zenodo.20098143 Author: Ali Attar Website: quantumtraction.org Main book anchor: Quantum Traction Theory: Main Book v10.01 The standard path integral weights histories by a phase, and the classical path is recovered by stationary action. This paper asks whether those two familiar facts are separate postulates or one source-side rotor rule. The QTT action dial begins with \[ d\theta={dS\over\hbar}, \] so the laboratory complex phase is the readout of a real source rotor, \[ R_J(\theta)=\cos\theta\,I+\sin\theta\,J, \qquad J^2=-I, \qquad J^T=-J. \] The path-integral object is then written as a real- \(J\) capacity sum over histories, \[ K_J(b,a) = \int_{\mathfrak H_{a\to b}}\mathcal D\gamma\, R_J\!\left({S[\gamma]\over\hbar}\right), \] with the laboratory exponential recovered as the usual shadow \[ R_J(S/\hbar)\quad\longmapsto\quad e^{iS/\hbar}. \] The action quantum follows from one complete rotor turn: \[ \Delta\theta=2\pi \quad\Longleftrightarrow\quad \Delta S=2\pi\hbar=h. \] The least-action rule is the coherent-sector limit of the same object: \[ \delta S=0. \] Read the paper as a single bridge: completed capacity rotations give the action quantum, the path-integral phase, and Hamilton's stationary-action condition without assigning them to unrelated axioms.