Kardashev's Conundrum: Statistical Falsification of the Standard Kardashev Model and the Kardashev--Sagan--Nakamoto Resolution

Abstract

We test the standard Kardashev one-percent exponential conjecture against six decades of global primary-energy production data (1965-2024; Our World in Data). Markov Chain Monte Carlo inference yields a posterior growth rate of r = 2.01 +/- 0.03% per year (95% credible interval [1.94%, 2.08%]), placing the Kardashev 1% value well outside the credible interval. A linear OLS model fits the data with remarkably low dispersion (R^2 = 0.987) and is preferred over the free-rate exponential by the Widely Applicable Information Criterion (ΔWAIC = 5.5). Year-over-year increments are non-Gaussian (Shapiro-Wilk W = 0.925, p = 0.0014; skewness = -0.664) with identifiable crisis outliers (2008, 2020), rejecting the independent-increment multiplicative structure with positive drift required by Kardashev's (1+x)^t geometric series. Extrapolation of the linear model to the solar luminosity yields a Type II civilisational timescale of approximately 1.6E15 years -- approximately 1E5 times both the age of the Universe and the main-sequence lifetime of the Sun -- a physical reductio we term Kardashev's Conundrum. No functional form fitted to P(t) alone can simultaneously satisfy statistical adequacy and physical coherence: the Kardashev variable is dimensionally incomplete. We propose the Kardashev-Sagan-Nakamoto (KSN) renormalisation B(t) = P(t)/H(t) [J/Hash, the KarNak unit], where H(t) is the annual Bitcoin hashrate. The renormalisation adds no free parameters, is motivated by the Landauer limit, and fulfils Sagan's information-richness requirement. Over 2009-2024, B(t) spans 14 orders of magnitude.