What Price Settlement Speed? A Gabrielli–von Kármán Frontier for Financial Settlement Systems, from Babylonian Storehouses to Proof-of-Stake

In 1950 Gabrielli and von Kármán asked "what price speed?" and answered with a single log-log chart: specific power plotted against velocity for every class of vehicle, bounded below by one limit line. This paper transposes that device to financial settlement. We define settlement velocity v = 1/τ as the inverse of the time to final settlement and specific exchange drag ε as the resource cost of settling one unit of value, in basis points, decomposed into four additive frictions: direct fees, risk and adverse-selection reserves, energy and hardware, and the opportunity cost of locked collateral. Placing six pre-industrial exemplars (c. 1750 BCE–1880) and fifteen modern systems on one diagram yields a nonparametric GvK Limit Line, the lower convex hull in log-log space, and a set of institutional regime envelopes. Two analytical results organise the diagram. First, the elasticity of drag with respect to the policy rate equals the collateral share of drag, which explains why netting shields systems from rate shocks. Second, when faster settlement destroys netting, transaction-window and balance-sheet accounting of collateral costs can disagree in sign; we call this the settlement liquidity paradox and apply it to the 2024 U.S. move to T+1. Third, a hedonic decomposition of drag into core settlement cost and bundled services (credit float, dispute rights, novation, censorship resistance) shows that the Limit Line is an upper bound on the frontier of pure settlement cost, and splits each system's distance from the line exactly into a within-class distance and a cross-class wrapper premium, following the class-by-class logic of Ashby's material charts. We illustrate the results on a calibrated structural simulation panel of 1,466 system-months (2018-2026). The panel is synthetic and benchmarked to public magnitudes; we therefore report no statistical inference on it, and confine a parametric response surface of the calibration to an appendix. We set out what an empirical test would require, and draw implications for regulatory sandboxes evaluating instant, tokenised and T+0 settlement.