Helicity structure of massive three-dimensional electric–magnetic duality: preferred-foliation closure, nonlinear no-go theorem, and anyonic charge–flux phase

We revisit the parity-invariant massive Abelian gauge theory in 2+1 dimensions studied by Fayyazuddin [arXiv:1608.00940], whose source-free equations admit an exchange between the gauge-invariant transverse vector ℓ_μ and the dual field strength L_μ. Writing Φ_± = ℓ ± L gives two first-order self-dual equations, curl Φ_± = ± m Φ_±, and identifies the electric–magnetic exchange with the sign reversal of one helicity sector. Among constant real internal maps preserving both the equations and the positive quadratic energy, the symmetry group is exactly ℤ₂ × ℤ₂. We then close two extensions that are usually left open. First, we replace the covariant Stueckelberg projector by the purely spatial transverse projector Π_ij = δ_ij − ∂_i ∂_j / ∇² on a preferred foliation with universal time T. The resulting theory is strictly local in T: A₀ remains an elliptic Gauss multiplier while the single spatially transverse Maxwell mode obeys a massive Klein–Gordon equation. The algebraic first-order split still holds on source-free solutions after setting ℓ₀ = 0, but the physical phase space is not closed under ℓ ↔ L: for a moving mode the transformed configuration violates ℓ₀ = 0 because L₀ = B ≠ 0. Thus the full ℤ₂ × ℤ₂ survives on physical modes if and only if the foliation constraint is duality-invariant, n · ℓ = 0 ⇒ n · L = 0; for the minimal spatial-projector theory this reduces to B = 0 and excludes every nontrivial moving transverse plane wave. Restoring the duality therefore requires an enlarged foliation theory with a second physical mode. Second, for a nonlinear scalar density ℒ(x, y, z) with x = ℓ², y = L², and z = ℓ · L, action antisymmetry alone gives the general local form ℒ = (y − x) F(x + y, (y − x)², z). We define the natural nonlinear constitutive duality map by 𝒟_ℒ(ℓ, L) = (e² ∂ℒ/∂L, −e² ∂ℒ/∂ℓ), which reduces to the linear exchange. Requiring the same discrete duality at equation level, 𝒟_ℒ² = 1, yields a local analytic no-go theorem: the canonical quadratic density is the unique analytic solution near the weak-field point. In particular, the antisymmetric Born–Infeld difference violates constitutive self-duality already at order β⁻², independently of its parameter α. We also give its Hamiltonian and stability inequalities on the physical foliation branch. Finally, the static solution is developed in detail. The identity d[s K₁(s)]/ds = −s K₀(s) gives the finite-radius dual holonomy 1 − m r K₁(m r) and the asymptotic screening flux e² q_e / m. Since no Dirac string is present, the resulting Aharonov–Bohm phase is physical. Without imposing trivial mutual statistics, charge–flux composites have the Wilczek anyonic exchange angle θ = e² q_e q_m / m.