Remark: Family 148. The dimension formula for self-similar measures on the line

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 $\chi$ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.