The Geometry of Quantum Mechanics: Hilbert Structure, Unitary Dynamics, Spin, Measurement, and the One-Particle Quantum Theory in Six-Throat Directed Spacetime

Paper XXII of the Six-Throat Directed Geometry program derives the mathematical structure of one-particle quantum mechanics from the directed spinor dynamics established on the regular closed six-throat spacetime. Starting from the conserved curved-spacetime Dirac current, the construction produces a positive and slice-independent inner product and its Hilbert-space completion. The corresponding geometric propagator is unitary and satisfies composition, inverse, and adjoint closure. Real physical observables are represented by self-adjoint operators with real spectra and orthogonal spectral resolutions. The compact global geometry yields an exact periodic commutator algebra, while the standard canonical position-momentum relation is recovered on regular local charts. The Robertson and Schrödinger-Robertson uncertainty relations then follow from the same Hilbert-space structure. The tetrad Clifford algebra generates the intrinsic spin-one-half representation, including the expected spin magnitude, the spinor sign reversal under a full two-pi rotation, and the rotational multiplets obtained by coupling orbital and intrinsic angular momentum. Detector-surface current is promoted from the direct Born-flux construction of Paper XXI to a general positive measurement structure described by a non-projective POVM and the trace probability rule. Density operators, purity, von Neumann entropy, conditional access states, and Kraus dephasing channels arise within the same measurement framework. The positive-energy Dirac branch reduces at low energy to Pauli dynamics and, when the spin-covariant terms commute appropriately, to the Schrödinger Hamiltonian. The same unitary framework yields the Liouville-von Neumann equation, the Heisenberg operator equation, and the Ehrenfest relations in the local directed potential. A forty-three-condition numerical closure audit satisfies all 43 declared conditions across the derived one-particle structure. Representative numerical results include Hilbert-product and unitary residuals at approximately machine precision, exact spin-one-half algebra to numerical precision, POVM completeness at machine precision, low-energy wave-packet infidelity of 3.16 × 10⁻¹⁰ for the stated audit configuration, and Schrödinger-Heisenberg expectation agreement at 5.66 × 10⁻¹⁷. The result is a single geometric chain from conserved spacetime flux to one-particle state space, unitary dynamics, observables, commutators, uncertainty, spin, measurement operators, statistical states, and the low-energy quantum equations. Composite systems, entanglement, exchange symmetry, quantum statistics, and variable-particle-number field theory remain subsequent structural layers and are not claimed as part of the present one-particle closure.