Critical Mode Compression, Self-Kerr Duality, and Single-Mode-to-Droplet Instanton Crossover for Sub-Gap Scalar Fields in Josephson Metamaterials

We develop an exact theoretical and computational framework for trapping, controlling, and probing spatially localized sub-gap resonant modes of relativistic scalar fields in superconducting Circuit-QED metamaterials. Starting from a $(d+1)$-dimensional Klein-Gordon Lagrangian with a spatially graded effective mass profile meff²(x) = m² + V(x), we solve for the discrete bound-state spectrum (ωₙ < m) in 1D and 2D Gaussian potential wells, demonstrating agreement to $<0.06%$ between exact sparse-matrix diagonalization and long-time symplectic lattice integration across both fundamental ($1s$) and dipole-excited ($1p$) sub-gap manifolds. We then construct an exact microscopic mapping between the relativistic scalar field and a transmission line of flux-tunable SQUID resonators, proving that the bound mode's Inverse Participation Ratio, IPR(V₀,σ) = ∫ |f₀(x)|⁴ dᵈ x, governs both the single-photon self-Kerr shift χ₀₀ = -(EC/)(ωp,0/ω₀)²IPRₗₐₜ (recovering the exact Koch transmon limit -EC/ at single-site localization) and the effective anharmonicity λeff = λ\,IPR. Combining narrow-trap Dirac-delta asymptotics (IPR ∝ σ) with wide-trap anharmonic WKB expansions (IPR ∝ σ-1/2), we prove the existence of a non-monotonic Critical Mode-Compression Resonance at σ^* = 0.965\,m⁻¹ (V₀ = 0.80\,m²). In the non-perturbative tunneling regime—realized without Taylor-truncation artifacts in frustrated two-harmonic SQUID arrays or parametrically driven Kerr resonators—projecting the Euclidean action onto f₀(x) yields an analytical collective-coordinate bounce BSMCC(V₀,σ) = 4ω₀³/3λ\,IPR whose Pöschl-Teller (=2) fluctuation determinant gives the exact Callan-Coleman decay rate ΓSMCC = ω₀/π√6 BSMCC\,e^-BSMCC and satisfies the exact one-loop self-Kerr duality δω₀⁽¹⁾/ω₀ = -1/BSMCC. Finally, using the Nehari mountain-pass variational principle and spectral Petviashvili iteration, we prove a universal instanton hierarchy B∞ < Bexact(V₀,σ) ≤ Bbound(V₀,σ) ≤ BSMCC(V₀,σ), showing that Z₂ parity symmetry keeps the single-mode bounce exact across the entire sub-gap manifold below the second even-mode threshold (σ < σ₂ ≈ 2.05\,m⁻¹) before undergoing a dimensional crossover to a localized 2D Weinstein-Coleman droplet (B_∞ = C2D(m²-V₀)/λ). We validate single-excitation sub-band trapping in a 5-qubit XY quantum circuit via Qiskit and provide concrete hardware parameters.