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

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 $H_{\epsilon }$ whose images are $\epsilon$-dense in intrinsic distance. A standard covering/tube estimate gives $L_g[H_{\epsilon }]\ge C_g{\epsilon }^{1-d}-O(\epsilon )$ 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 $\zeta$ coincides with, or uniformly dominates, the coverage metric $g$, Cauchy–Schwarz for the quadratic slow-driving action gives $W_{\mathrm{e}\mathrm{x}}^{(2)}\ge L_{\zeta }^2\mathrm{/}\tau =\mathrm{\Omega }({\epsilon }^{2(1-d)}\mathrm{/}\tau )$. Equivalently, at fixed quadratic excess-work budget, maintaining slow driving requires $\tau =\mathrm{\Omega }({\epsilon }^{2(1-d)})$. We derive microscopic friction metrics for a detailed-balance three-state Markov jump process, ${\zeta }_{ij}=(\beta \mathrm{/}\gamma )({\pi }_i{\delta }_{ij}-{\pi }_i{\pi }_j)$, and for an overdamped harmonic trap, $\mathrm{d}{\mathrm{\ell }}_{\zeta }^2={\mu }^{-1}\mathrm{d}{a}^2+(4\beta \mu k^3{)}^{-1}\mathrm{d}{k}^2$. In the trap, a raster scan gives $L_{\zeta }\sim {\mathrm{\Delta }}_g^{-1}$ and fixed-time $W_{\mathrm{e}\mathrm{x}}^{(2)}\sim {\mathrm{\Delta }}_g^{-2}$, while fixed dwell time shifts the cost to acquisition time. A laboratory or simulation floor cuts off the continuum divergence as $L_{\mathrm{o}\mathrm{p}}=\mathrm{\Theta }(\max \{\epsilon ,{\mathrm{\Delta }}_g{\}}^{1-d})$. 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.