Finite-resolution exhaustive traversal of thermodynamic state spaces has divergent thermodynamic length

Abstract: Continuous space-filling maps can be surjective onto higher-dimensional regions, but thermodynamic protocols are rectifiable finite-resolution paths. We study exhaustive traversal of a compact $d$-dimensional thermodynamic state-space window $(\mathcal{M},g)$ by curves $H_\varepsilon$ whose images are $\varepsilon$-dense in intrinsic distance. A standard covering/tube estimate gives $L_g[H_\varepsilon]\ge C_g\varepsilon^{1-d}-O(\varepsilon)$ for every regular $d>1$ window. The geometry is classical; the contribution is to turn it into an operational resource law for thermodynamic coverage. When the physical friction tensor $ζ$ coincides with, or uniformly dominates, the coverage metric $g$, Cauchy--Schwarz for the quadratic slow-driving action gives $W_{\rm ex}^{(2)}\ge L_ζ^2/τ=Ω(\varepsilon^{2(1-d)}/τ)$. Equivalently, at fixed quadratic excess-work budget, maintaining slow driving requires $τ=Ω(\varepsilon^{2(1-d)})$. We derive microscopic friction metrics for a detailed-balance three-state Markov jump process, $ζ{ij}=(β/γ)(π_iδ{ij}-πiπ_j)$, and for an overdamped harmonic trap, $\mathrm d\ellζ^2=μ^{-1}\mathrm da^2+(4βμk^3)^{-1}\mathrm dk^2$. In the trap, a raster scan gives $L_ζ\simΔg^{-1}$ and fixed-time $W{\rm ex}^{(2)}\simΔ_g^{-2}$, while fixed dwell time shifts the cost to acquisition time. A laboratory or simulation floor cuts off the continuum divergence as $L_{\rm op}=Θ(\max{\varepsilon,Δ_g}^{1-d})$. Controlled singular response-proxy metrics diagnose critical prefactors and directional integrability, but are not physical friction tensors unless derived from microscopic dynamics. Morton/Z-order preserves the exponent while increasing locality-dependent amplitudes.