Result 148, Dynamical systems and ergodic theory

The entropy-rate dimension formula for self-similar measures

For every self-similar measure on the line generated by finitely many contracting similarities, proves dim⁡Hμ=min⁡{1,hRW/χ}\dim_{\mathrm H}\mu=\min\{1,h_{\mathrm{RW}}/\chi\}, where hRWh_{\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 formalization Proof

The bigger picture

Why it matters

Repeated random shrinking can produce probability distributions concentrated on fractals. The central claim is that their dimension, which measures concentration across scales, is determined by the balance between uncertainty in the resulting maps and geometric shrinkage.

What changes?

The manuscript reports that every self-similar measure on the real line, generated by finitely many contracting similarities chosen with fixed probabilities, has Hausdorff dimension equal to the smaller of one and its random-walk entropy rate divided by its average logarithmic contraction. Entropy rate measures uncertainty per step in the composed map; logarithmic contraction measures average shrinkage. No separation is required: different sequences can produce exactly the same map, and contraction ratios can be unequal or negative.

What does that help mathematicians do?

The formula gives a precise criterion for full dimension: the entropy rate must be at least the average logarithmic contraction. Below that threshold, their ratio gives the dimension exactly. Thus, even with exact overlaps, researchers could determine dimension from these two quantities without separately controlling how the pieces intersect. Computing the entropy rate remains a separate task; the formula does not supply an algorithm for it.

Are there practical applications?

The immediate value is foundational, connecting the geometry of fractal probability measures to the entropy of random map compositions. For researchers studying repeated random contractions on the line, the claimed identity would turn estimates of entropy rate and average shrinkage into dimension estimates, including when overlapping pieces prevent separation-based arguments.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

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

September 24, 2026 22 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026,
  author = {{OpenAI}},
  title = {{The entropy-rate dimension formula for self-similar measures on the line}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026/main.pdf}{OAI:The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/148.md.

The entropy-rate dimension formula for self-similar measures

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves dim⁡Hμ=min⁡{1,hRW/χ}\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\} for the lower Hausdorff dimension of every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one. Here hRWh_{\mathrm{RW}} is the entropy rate of random affine compositions and χ\chi is the corresponding Lyapunov exponent in the same logarithmic base. Ratios may be signed and unequal, and exact overlaps are allowed.

It also gives two consequences without exact overlaps: the usual entropy-over-Lyapunov formula for a common positive contraction ratio, and the attractor formula dim⁡HK=min⁡{1,s}\dim_H K=\min\{1,s\}, where s≥0s\ge0 is the unique solution of ∑i∣ri∣s=1\sum_i|r_i|^s=1. Absolute continuity is outside these statements.

Comparator links

Result Comparator statement
Entropy-rate dimension formula SelfSimilar.lean
Homogeneous-measure and self-similar-attractor dimension formulas SelfSimilarCorollaries.lean

Posts about this result

Constantin Kogler @ConstantinKogl, Mathematician, Institute for Advanced Study

I have now published on my website a paper, jointly written with Samuel Kittle, on the same main result as Paper 148 of the @OpenAI announcement. The first proof I found was discovered by GPT-6 Astra on September 27, 2026. So I don't feel too bad that OpenAI published the result before us, yet I would have preferred it the other way around. My collaborator Samuel Kittle and I spent most of our time over the past 10 days to understand the proof and rewrite part of the argument using our own language. The paper will go on the arXiv tomorrow and I will post more on the mathematics of that paper soon. The paper solves one of the best-known questions on self-similar measures, but only in dimension one. The method extends to dimension two, as we discuss in our version of the paper, but not to dimension three or higher. Paper: constantinkogler.com/Files/ExactOverlapsDim1.pdf Supplementary Material: github.com/ckkogler/kk26-supplementary-material Lean: github.com/ckkogler/exact-overlaps-one-dim-lean

Oct 7, 2026, 12:44 AM ET

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.