A First-Principles Field Theory of Magnetic Flux Noise in Superconducting Quantum Interference Devices (SQUIDs) and Flux-Tunable Qubits: Inter-Site Surface-Spin Hamiltonians, Vector Central Limit Theorems, and the Millikelvin Quantum Floor

Low-frequency magnetic flux noise remains a persistent performance bottleneck in dc Superconducting Quantum Interference Devices (SQUIDs) and flux-tunable superconducting qubits across dilution refrigerator base temperatures. Traditional models treat surface paramagnetic spins as independent, non-interacting fluctuators, failing to explain strong exchange interactions, sub-Kelvin temperature insensitivity, and the universal Clarke R/W perimeter scaling law. Here, we resolve these empirical anomalies from first principles by embedding surface spins into a continuous inter-site electrostatic and magnetic potential manifold. Applying the Lindeberg-Feller Vector Central Limit Theorem to random surface dipolar fields yields an emergent Gaussian Density of States (GDOS) for spin inversion barriers. Integrating this disordered manifold across the SQUID washer geometry via magnetostatic reciprocity reproduces the universal R/W scaling law and predicts an absolute spectral amplitude of approximately 1.4 to 1.7 micro-Phi-0 per square root of Hz at 100 mK. Furthermore, accounting for non-cancellation phase agitation within the open spin saddles proves that soft-magnon and phonon bremsstrahlung sustain an irreducible millikelvin quantum flux noise floor down to absolute zero. In order to bridge microscopic theory with practical engineering, the manuscript includes a complete, modular MATLAB simulation toolbox (compatible with MATLAB R2016a/b and later) that computes closed-form classical flux noise spectral densities, frequency sweeps, and absolute spectral amplitudes for device optimization.