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 $H_\varepsilon$ 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 $\zeta$ coincides with, or uniformly dominates, the coverage metric $g$, Cauchy--Schwarz for the quadratic slow-driving action gives $W_{\rm ex}^{(2)}\ge L_\zeta^2/\tau=\Omega(\varepsilon^{2(1-d)}/\tau)$.

Equivalently, at fixed quadratic excess-work budget, maintaining slow driving requires $\tau=\Omega(\varepsilon^{2(1-d)})$.

We derive microscopic friction metrics for a detailed-balance three-state Markov jump process, $\zeta_{ij}=(\beta/\gamma)(\pi_i\delta_{ij}-\pi_i\pi_j)$, and for an overdamped harmonic trap, $\mathrm d\ell_\zeta^2=\mu^{-1}\mathrm da^2+(4\beta\mu k^3)^{-1}\mathrm dk^2$.

In the trap, a raster scan gives $L_\zeta\sim\Delta_g^{-1}$ and fixed-time $W_{\rm ex}^{(2)}\sim\Delta_g^{-2}$, while fixed dwell time shifts the cost to acquisition time.

A laboratory or simulation floor cuts off the continuum divergence as $L_{\rm op}=\Theta(\max\{\varepsilon,\Delta_g\}^{1-d})$.

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.