Fractional Quantum Hall States as Spatial Phase Closure of the Existence Equation

Fractional Quantum Hall States as Spatial Phase Closure of the Existence Equation Evidence Paper II of the Existence Equation series In the main paper 1, the Aharonov-Bohm effect is demonstrated directly from the Existence Equation: a 3D simulation of two parallel vortex lines shows that the closed-loop integral ∮∇Φ·dl = 2πn around any path enclosing the vortex cores, independent of loop size, position, or z-slice. The vector potential is not a separate entity acting on a particle. It is the phase gradient of the deviation field itself. The Aharonov-Bohm phase is Stokes' theorem applied to phase circulation. In the same paper, the phase stiffness energy ½A²|∇Φ|² is shown to diverge when two identical half-winding structures approach each other — a 134-fold energy barrier that constitutes the Pauli exclusion principle from topology. In Evidence Paper I 2, this exclusion, projected onto a one-dimensional lattice in the strong-coupling limit of −α|Ψ|²Ψ, becomes nearest-neighbor exclusion ni·ni+1 = 0, which mechanically produces the PXP Hamiltonian and temporal phase closure (quantum many-body scars). This paper asks: what happens when the same nearest-neighbor exclusion operates in two dimensions, and the Aharonov-Bohm phase is added? On a two-dimensional torus with NN exclusion and Peierls magnetic phase — the lattice encoding of the AB phase circulation demonstrated in 1 — the constrained Hamiltonian is constructed mechanically. No Coulomb interaction is used. No Laughlin wavefunction is assumed. No composite fermions are invoked. No Chern-Simons gauge field is postulated. The magnetic field curves all propagation into cyclotron orbits. The constraint forbids overlap of neighboring orbits. With propagation curved and overlap forbidden, the deviation field cannot spread in any direction. It must close spatially. A vortex is what deviation becomes when it closes on itself in space — the spatial counterpart of the temporal closure that produced scars in EP I. EP I (PXP)EP II (FQHE)EP III (Helix) Dimension1D chain2D torusQuasi-1D ladder ObstructionSpatial spreading blockedLinear propagation blockedBoth axes blocked ResponseCloses in timeCloses in spaceSpirals between both SignatureFidelity revivalTopological degeneracySelective scarring + chirality StructureScar towerVortex latticeHelical orbit Emergent symmetryApproximate SU(2)Topological orderChiral degeneracy (±q) The results span four tests: TestWhat is measuredResult 1. DegeneracyGround-state multiplicity at ν = 1/33-fold exact (< 10−8 splitting), gap increases with system size 2. Filling scanα = p/q → q-fold?4/4 commensurate MATCH; 1/1 incommensurate MISMATCH (null control) 3. Twist boundaryGap survives adiabatic deformation?Δmin = 0.135 Ω (topologically protected) 4. Wilson loopAnyon braiding phase2π/q with 0.0% error across all configurations The Wilson loop computation — the non-Abelian Berry phase of the ground-state manifold under flux insertion θx: 0 → 2π — yields the anyon braiding phase 120.0° at ν = 1/3 (three system sizes, D = 117 to 26,937), 120.0° at ν = 2/3 (particle-hole conjugate), and 72.0° at ν = 2/5 (Jain sequence). The error is 0.0% in every case. The structural interpretation connects back to the Aharonov-Bohm effect demonstrated in 1. The AB phase is the integer topological response of the ED field to enclosed winding: ∮∇Φ·dl = 2πn. When the same field is subjected to NN exclusion on a 2D torus, the constraint forces the phase circulation to close in a way that accommodates the restricted Hilbert space. The full 2π AB phase becomes the fractional 2π/q. An anyon is not an exotic quasiparticle. It is the fractional Aharonov-Bohm phase left as a geometric footprint by a constrained spatial domain. The causal chain within the ED framework is: ED phase circulation (AB) → Peierls phase (lattice AB) → NN exclusion (phase-stiffness divergence) → constrained Hilbert space → spatial closure → vortex lattice → q-fold degeneracy → Wilson loop → anyon braiding phase 2π/q Every link in this chain is verified numerically. No link is postulated. References 1 J.-A. Shin, "The Existence Equation: The Grammar of Persistence," Zenodo (2026). doi: 10.5281/zenodo.18639316 2 J.-A. Shin, "Quantum Many-Body Scars as Temporal Phase Closure of the Existence Equation," Evidence Paper I, Zenodo (2026). doi: 10.5281/zenodo.19327776 All simulation code and raw data are publicly available at https://github.com/Galileo-leo/existence-equation.