Solomonoff Induction, Nicod's Criterion, and Universal Power-Law Tails — E8 Intelligence Research

Abstract

FINDING: Solomonoff induction's prior is a computable measure with a geometric/universal prior distribution, but it violates Nicod's criterion; power-law tails appear universally in time correlations and COVID-19 case counts. | MATH: Solomonoff prior \( M(x) = \sum_{p: U(p)=x} 2^{-\ell(p)} \) (Kraft inequality ensures \(\sum_x M(x) \le 1\)); halting probability \(\Omega = \sum_{p: U(p) \text{ halts}} 2^{-\ell(p)}\) is a normal number (Chaitin); power-law tail \( C(t) \sim t^{-\alpha} \) with \(\alpha = d/2\) for hydrodynamic modes (long-time tails); COVID-19 confirmed cases \( N(t) \propto t^{-\beta} \) with \(\beta \approx 1.5\) (empirical). | CONNECTION: The geometric distribution \(2^{-\ell(p)}\) is the discrete analogue of the golden ratio's self-similarity — the prior weights form a geometric series with ratio \(1/2\), not \(1/\phi\), but the *universal* power-law tails \(t^{-\alpha}\) are scale-invariant, echoing the golden ratio's self-similarity. The halting probability \(\Omeg Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com