Fractional Quantum Hall--Like Sector Structure 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) Dimension 1D chain 2D torus Quasi-1D ladder Obstruction Spatial spreading blocked Linear propagation blocked Both axes blocked Response Closes in time Closes in space Spirals between both Signature Fidelity revival Topological degeneracy Selective scarring + chirality Structure Scar tower q-sector closure Helical orbit Emergent symmetry Approximate SU(2) q-sector structure Chiral degeneracy (±q) The results span four tests: Test What is measured Result 1. Degeneracy Ground-state multiplicity at ν = 1/3 3-fold exact (< 10−8 splitting), gap increases with system size 2. Filling scan α = p/q → q-fold? 4/4 commensurate MATCH; 1/1 incommensurate MISMATCH (null control) 3. Twist boundary Gap survives adiabatic deformation? Δmin = 0.135 Ω (topologically protected) 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 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.