For every self-similar measure on the line generated by finitely many contracting similarities, proves \dim_{\mathrm H}\mu=\min\{1,h_{\mathrm{RW}}/\chi\}, where h_{\mathrm{RW}} is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios. (Lean)
Manuscript
The entropy-rate dimension formula for self-similar measures on the line
September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026}
We prove that the Hausdorff dimension of every finite real self-similar measure equals the minimum of one and its random-walk entropy rate divided by its Lyapunov exponent. Exact overlaps are allowed, and the contraction ratios may be unequal and negative. This resolves the entropy-rate dimension conjecture.
Result 148 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md