Quantum Mechanics from Five-Dimensional Spacetime: A Geometric Origin of Probability

We show that the five-dimensional hydrodynamic spacetime framework developed in the companion paper naturally reproduces quantum mechanics as the effective description seen by observers confined to three spatial dimensions. The central insight is that quantum probability arises not from fundamental indeterminacy but from the observer's inability to access the w-coordinate and the w-t orbital phases of particles. Electromagnetic coupling is a binary contact condition at w = w_max; measurement outcomes are selected by the orbital-phase alignment at the instant of contact, with unresolved phases producing the observed statistics. Starting from the classical geodesic equations in five-dimensional spacetime, we derive the Schrödinger equation through dimensional projection and the characteristic orbital radius Δw ~ ℏ/(mc) of the w-t motion. The Planck constant ℏ emerges as a geometric quantity relating mass to orbital radius in the w-direction. Heisenberg's uncertainty principle, wave-particle duality, quantum tunneling, and entanglement all follow from the five-dimensional geometric structure without additional postulates. The measurement problem is resolved: wavefunction collapse is the phase-selective w_max-contact that couples one branch while leaving the others phase-mismatched. Spin-1/2 arises from the five-dimensional Dirac equation, with chirality identified as w-orbital direction and 4π periodicity established through holonomy. Bell inequality violations (S = 2√(2)) are reproduced through non-separable shared w-t orbital states, with no-signaling preserved. The theory predicts that every particle possesses excited states in the w-direction with masses m_n = n m_0 (n = 1, 2, 3, …). For electrons, this predicts states at 1.02 MeV/c², 1.53 MeV/c², and higher energies. Laboratory production of these states is strongly suppressed by viscous flattening of the w-field and dilution of the w-potential by cosmic expansion; the most promising observational signatures may be cosmological imprints of early-universe particle creation. The five-dimensional structure yields quantitative connections to the Standard Model with minimal free parameters. From the universal Reynolds number Re = 2 (a geometric identity: diameter/radius) and the base vortex coupling g_0 = 1/(8π), we derive: the strong coupling constant α_s = 3/(8π) ≈ 0.119 as a parameter-free prediction (1.2% agreement with experiment); the fine structure constant in closed form to the full precision of its experimental determination, α⁻¹ = 128π/3 + 3 - 32πα²(1-27α/16) = 137.035999067 versus the measured 137.035999084(21) (0.8σ; relative precision 1.3×10^(-10)), from the channel decomposition D_eff = 4×(4/3) + 3/(8π) (four transport channels x,y,z,w at radiation-enthalpy weight, plus three gated w-planes), an exact gate self-consistency, and two orders of transport back-reaction whose coefficient is fixed by the doubly measured ledger exchange rate t = 3/4; the Weinberg angle sin²θ_W = 3/8 from counting eight force output vectors; the Higgs field identified with the w-field (v/m_H = Re = 2, 1.7% agreement; λ_H = 1/8, 3.4% agreement); and a particle-internal partition 4/Re = 2 from the drawing cost of four w-faces, numerically close to the present-epoch Ω_DE/Ω_m ≈ 2.17 (the cosmological ratio is time-dependent and the numerical proximity at the present epoch is a coincidence). Parity violation follows from boundary conditions at w=0, and color confinement arises as the local version of the same drawing cost. Two further closed-form results complete the set: the gravitational hierarchy from budget equipartition, m_e/M_P = y_e⁴/√(π) (0.03%) and α_G = y_e⁸/π (0.06%), finalizing the prediction that Newton's constant lies within the CODATA cluster; and the vacuum quantumness derived from the vacuum's two-fluid metabolism, ε = (2m_e/m_H)^(1/3) = 0.0202. These results are now anchored in a single five-dimensional action: the complex w-field with the parameter-free Mexican-hat potential λ(|Φ|²-v²/2)², λ = 1/8 = 1/(2 Re²), whose valley expansion reproduces the metabolism sector and whose topology and boundary structure convert the framework's former structural premises into theorems: the non-transfer rule, the Laue-type protection of the vacuum's circulation, the equipartition machinery, the quantization of circulation with the face inventory C(4,2)=6 (three w-containing color planes), the structural vanishing of θ_QCD, and the identification of the kinematic viscosity with the wave equation's own dispersive coefficient, ν = ℏ/2m_e.