The entropy-rate dimension formula for self-similar measures

What OpenAI claims

For every self-similar measure on the line generated by finitely many contracting similarities, proves dim_Hμ=min {1,h_RW/χ}, where h_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)

Paper

1. The entropy-rate dimension formula for self-similar measures on the line

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.