Artian Completed-Event Hamiltonian Theorem: Primitive Direction, Magnitude No-Go, and the Record-Spend Closure Gate

When one completed quantum event fixes a Hamiltonian direction but not its strength Version: 1.0 Concept DOI: 10.5281/zenodo.21428033 Author: Ali Attar Website: quantumtraction.org Main Book anchor: Quantum Traction Theory: Main Book v10.01 Take one homogeneous edge, one quantum of support, equal endpoint labels, and a self-adjoint real-\(J\) generator. Endpoint symmetry leaves exactly one physical reversible direction after the common identity phase is removed: \[ \boxed{ \mathrm{Comm}(S_e)_{\rm sa}/\mathbb{R}I \cong \mathbb{R}S_e } \] The direction is unique. Its magnitude is not. The complete local premise set admits the continuum \[ \boxed{ H_{C_H}^{\rm tot} = E_* \left[ \left(1-\frac{C_H}{2}\right)I + \frac{C_H}{2}S_e \right], \qquad \mathrm{spec}\,H_{C_H}^{\rm tot} = \{(1-C_H)E_*,E_*\}, \quad 0\lt C_H\le1 } \] Every member carries the same completed upper state and obeys the same finite spectral ceiling. The printed local premises therefore do not select \(C_H=1\). The exact coherent readout makes the event meaning testable: \[ \boxed{ H_e=\frac{\hbar\Omega_e}{2}S_e, \qquad p_{u\to v}(T) = \sin^2\!\left(\frac{\Omega_eT}{2}\right) } \] At \(C_H=\Omega_e\widetilde t_A=1\), one tick gives \(p_{u\to v}=0.229848847065930\ldots\), not an exact swap. An orthogonal unitary transfer requires at least \[ \boxed{ \int \mathrm{osc}(H(T))\,\mathrm dT\ge\pi\hbar } \] The missing magnitude theorem is therefore an allocation theorem. Event records must first be grouped by which funded transaction they represent: \[ \boxed{ \mathfrak P_{\mathcal E} = \mathfrak R_{\mathcal E}/\!\sim_{\rm fund} } \] If that quotient has one non-null funded class, if the class is primitive reversible transport, and if its action is faithfully represented by shift-invariant spectral contrast, then the closure gate is immediate: \[ \boxed{ \widetilde t_A\, \mathrm{osc} \!\left( P_{\rm prim}H_{\mathcal E}P_{\rm prim} \right) = \hbar \quad\Longrightarrow\quad C_H=1 } \] The paper also proves coherent/dissipative type separation, distinguishes conductance from Hamiltonian frequency, gives the exact three-address counterexample, defines the finite no-double-spend network polytope, and proves that continuous-time nearest-neighbor hopping does not have exact compact support at positive time. Machine audit: 500 endpoint-commutant trials, 500 phase-gauge trials, 1,000 legal coefficient countermodels, 300 three-site coherent trials, 500 spectral non-additivity trials, 400 network packets, graph distances 1 through 7, and 14/14 detected mutations. Headline status: SIGMA-CEH-PRIMITIVE-HAMILTONIAN-DIRECTION-UNIQUE-ON-DECLARED-CLASS - GREEN SIGMA-CEH-LOCAL-COEFFICIENT-NONIDENTIFIABILITY-NOGO-CLOSED - GREEN SIGMA-CEH-ONE-TICK-EXACT-UNITARY-RELOCATION-NOGO-CLOSED - GREEN SIGMA-CEH-RECORD-SPEND-QUOTIENT-PENDING - AMBER SIGMA-CEH-C-H-EQUALS-ONE-CONDITIONAL-ON-CEAC-SISAE - AMBER Related QTT anchors: Artian Gravity Reference Framework Artian Hamiltonian Framework Artian Lagrangian Framework QTT Derivation of the Principle of Least Action Gravity Authority Derivation Atlas Included public files: the PDF paper and one full reconstruction package containing LaTeX sources, verifier, machine receipts, scientific audit, render audit, metadata, and SHA-256 manifest.