Artian Action-Rotor Reference Framework: Hamilton's Stationary-Action Recovery, the Real-J Path Sum, and the Capacity/Canonical Action Ledger

Abstract

One Phase Turn, Three Different Ledgers Version: 3.0 Concept DOI: 10.5281/zenodo.20098143 Author: Ali Attar Website: quantumtraction.org Main book: Quantum Traction Theory: Main Book v10.01 A unit quantum phase can be written entirely with a real two-component rotor, \[ R_J(\theta)=\cos\theta\,I+\sin\theta\,J, \qquad J^2=-I, \qquad R_J(\theta_2)R_J(\theta_1)=R_J(\theta_1+\theta_2). \] Version 3.0 asks what this exact representation does and does not imply for action. Its central result is a typed separation of winding degree, canonical phase action, and positive capacity spend: \[ \boxed{ \nu_A=\frac{1}{2\pi}\oint d\theta\in\mathbb Z, \qquad S_{\rm can}=2\pi\hbar\nu_A, \qquad C_A=\hbar\sum_j|\nu_{A,j}| } \] For one funded primitive completion, \[ \boxed{ |\nu_A|=1, \qquad C_A=\hbar, \qquad |S_{\rm can}|=2\pi\hbar=h, \qquad \frac{C_A}{|S_{\rm can}|}=\frac{1}{2\pi} } \] The factor \(1/(2\pi)\) is the normalized circle generator that converts angular holonomy into integer winding. The dimensionful A6 capacity token is a separate physical input. An explicit degree-two countermodel, \[ U_2(s)=R_J(4\pi s), \qquad U_2(0)=U_2(1)=I, \qquad \nu_A[U_2]=2, \] shows that compactness and endpoint closure do not select one primitive turn. Primitive funding is A6 content, not a hidden consequence of (SO(2)). With the action-phase bridge, short-time kernel, measure, endpoints, and convergence hypotheses printed explicitly, the time-sliced laboratory shadow is \[ K_J(x_f,T_f\mid x_i,T_i) = \int \mathcal D_{\rm slice}x\; R_J\!\left(\frac{S_{\rm can}[x]}{\hbar}\right). \] The stationary-action limit is controlled by variation of phase across a history family, not by a large constant offset: \[ \frac{S_{\rm can}[\gamma_\varepsilon]}{\hbar} =\Theta_0+\lambda f(\varepsilon), \qquad \lambda\gg1 \quad\Longrightarrow\quad \delta S_{\rm can}=0. \] This yields the standard Euler-Lagrange equations under the usual regularity and nondegeneracy conditions. The same framework records the exact real-(J) form of the Feynman pole prescription, the clock-reparametrized action, gauge holonomy without double counting, and a bounded A6 spectral-window class whose detailed profile remains a separate source theorem. Scientific status CLOSED: real-(J) phase representation and integer-winding classification. CLOSED: three-ledger type separation and the degree-two countermodel. CONDITIONAL STANDARD RECOVERY: time-sliced path integral, stationary phase, and Euler-Lagrange mechanics. QTT SOURCE LAW: positive A6 funding and no stacking, anchored to the independent A6 theorem. PENDING: unique Feynman measure from source law, unique A6 spectral profile, populated A7U residual, and independent empirical selection of the ontology. Related anchors A6 Quantum-Capacity Necessity and Independence Theorem Completed-Event Hamiltonian Theorem Artian Hamiltonian Framework Artian Lagrangian Framework QTT Derivation Atlas QTT Lexicon The release includes the PDF, LaTeX source, scientific audit, deterministic verifier, machine receipt, render audit, checksums, and reconstruction package.