Electromagnetic Induction Back-Action as a Passive Mechanical Memory Kernel: From Lumped RL Maxwell Elements to Conducting Half-Space Diffusion (and Flux-Quantized Superconducting Rings)

Electromagnetic induction forces produced by moving magnets near conductors are frequently approximated as either conservative stiffness (magnetostatics) or viscous damping (eddy-current loss). Both are controlled limits of a stricter statement: Maxwell--Faraday induction plus finite magnetic energy storage generates a \emph{causal, passive mechanical memory kernel}. This paper develops that kernel viewpoint in a hierarchy of models of increasing physical fidelity. We begin with a dipole--lumped-loop system, where the exact small-signal dynamic stiffness is $K_{\mathrm{em}}(s)=G^2 s/(R+sL)$, mechanically identical to a Maxwell element with stiffness $G^2/L$ and dashpot coefficient $G^2/R$. We then move beyond single-pole phenomenology by treating real conductors as distributed eddy-current continua. For a magnetic dipole oscillating normal to a conducting half-space, we derive an exact quasi-static frequency-domain kernel using Hankel (Sommerfeld) spectral methods. The resulting stiffness is an explicit passive branch-cut (diffusion) function of $s$ governed by the dimensionless parameter $\Omega=s\,\mu\sigma h^2$, where $h$ is the dipole height and $\mu\sigma$ sets magnetic diffusion. Low- and high-frequency asymptotes recover viscous and image-spring limits, while the intermediate regime reflects the continuous relaxation spectrum of diffusion. Finally, for superconducting rings we incorporate fluxoid quantization $Li+\Lambda(x)=n\Phi_0$ and show that flux jumps (phase slips) create discrete-state hysteretic magnetomechanical memory beyond any linear kernel.