Kardashev's Conundrum: Statistical Falsification of Exponential Energy Growth, the Non-Observation of Type II Technosignatures, and the Kardashev-Sagan-Nakamoto Resolution

We test the standard Kardashev conjecture – that a technologically advanced civilisation’s energy production grows at a fixed one percent per year – against six decades of global energy-production data (1965-2024), with consequences for the search for extraterrestrial intelligence. Markov Chain Monte Carlo inference falsifies the one-percent rate, returning a posterior $r=2.01\mathrm{\%}\pm 0.03\mathrm{\%}\mathrm{y}{\mathrm{r}}^{-1}$; a linear model and a free-rate exponential fit the data equally well and are statistically indistinguishable. Extrapolated to the Type~II threshold (the solar luminosity), the linear form implies a transition time of order ${10}^5$ Hubble times, while the exponential branch is excluded as a sustained trajectory by waste-heat and carrying-capacity limits. Neither adequate description therefore reaches energy-scaling Type~II coherently – Kardashev’s Conundrum. The observational consequence is that Type~II technosignatures, Dyson spheres in particular, are unlikely on physically realistic trajectories, offering a candidate explanation for the null results of Dyson-sphere surveys and a data-driven entry in Cirkovic’s ``logistic’’ category of Fermi-paradox solutions. The root cause is that the Kardashev state variable, power, is dimensionally incomplete: it captures energy throughput but not the information richness of its use. We therefore propose the Kardashev-Sagan-Nakamoto renormalisation, dividing energy production by the global proof-of-work hashrate to define $B=P\mathrm{/}H$ in $\mathrm{J}\mathrm{H}\mathrm{a}\mathrm{s}{\mathrm{h}}^{-1}$ – the KarNak unit – a dimensionally complete information–energy variable motivated by the Landauer limit, on which the Type~II concept is preserved.