The entropy-rate dimension formula for self-similar measures
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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. (OpenAI’s Lean note)
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For every finite nonempty indexed family \varphi_i(x)=r_i x+t_i on \R with 0<|r_i|<1, and every strictly positive probability vector p, its self-similar measure satisfies
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.
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For every finite nonempty indexed family \varphi_i(x)=r_i x+t_i on \R with 0<|r_i|<1, and every strictly positive probability vector p, its self-similar measure satisfies
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
For every finite nonempty indexed family \varphi_i(x)=r_i x+t_i on \R with 0<|r_i|<1, and every strictly positive probability vector p, its self-similar measure satisfies
SelfSimilarCorollaries.lean, the release lists it as a supporting statement.
theoremhomogeneous_dimension_direct {ι : Type uι} [Fintype ι] [Nonempty ι] (S : System ι) {lam : ℝ} (hlam : 0 < lam) (hr : ∀ i, S.ratio i = lam) (μ : Measure ℝ) [IsProbabilityMeasure μ] (hμ : S.SelfSimilar μ) (h : NoExactOverlaps S) : lowerHausdorffDimension μ = ENNReal.ofReal (min 1 (symbolEntropy S / Real.log (1/lam))) := bysorry
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Independent comparisons and proof-verification reports
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MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:entropy_rate_dimension gives lowerHausdorffDimension μ = ofReal (min 1 (entropyRate / lyapunov)) for every System (signed ratios 0<|r_i|<1, positive weights, arbitrary offsets) and every probability measure with μ = Σ w_i f_i μ, with no separation hypothesis, so exact overlaps are allowed. The two corollaries (homogeneous ratio, attractor formula under no exact overlaps) are stated in SelfSimilarCorollaries.
Definitions the group asks readers to compare with the paper:lowerHausdorffDimension μ := ⨅ E measurable with 0 < μ E, dimH E (the inf-dimension, as the docs say; equals dim_H for the exact-dimensional self-similar measure); entropyRate S := sInf (range fun n => walkEntropy (n+1)/(n+1)) with walkEntropy the base-2 entropy of the law of the composed affine map (words with equal composition merged), and lyapunov also in base 2: custom but matches the standard h_RW/chi
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The formalization proves $\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$ 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 $h_{\mathrm{RW}}$ is the entropy rate of random affine compositions and $\chi$ 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 $\dim_H K=\min\{1,s\}$, where $s\ge0$ is the unique solution of $\sum_i|r_i|^s=1$. Absolute continuity is outside these statements.
AI-drafted comparisons, with no recorded human review
Entropy-rate dimension formulaSelfSimilar
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the lower Hausdorff dimension of every self-similar measure on the line is min{1, h_RW/χ}, with signed unequal ratios and exact overlaps allowed.
Homogeneous-measure and self-similar-attractor dimension formulasSelfSimilarCorollaries
Unreviewed AI draft: states a consequence of a paper’s theorem. States two consequences of the main theorem without exact overlaps: the entropy-over-Lyapunov formula for a common contraction ratio and the attractor formula dim K = min{1, s}.
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Entropy-rate dimension formulaSelfSimilar
Two reports, all at the release’s recorded version.
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Other people’s accounts of running proof-verification software, with links to their reports.
Constantin Kogler, Samuel Kittle, ckkogler/exact-overlaps-one-dim-lean: Lean formalisation of the exact overlaps conjecture in dimension one, 4 October 2026. Source
Strict Comparator over eight Challenge declarations accepted by Lean, NanoDa and con-ron; CI green on 7 and 10 Oct.
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
Follow-up work by others
Samuel Kittle, Constantin Kogler, The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line, 7 October 2026. Source
Human-written, AI-assisted proof posted on arXiv on 7 Oct 2026 of the same conjecture as OpenAI’s paper 148.
Samuel Kittle, Constantin Kogler, The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line, 7 October 2026. Source
The paper cites a separate Lean formalisation of the exact overlaps conjecture in dimension one by Kogler and Kittle at github.com/ckkogler/exact-overlaps-one-dim-lean (repository URL returned HTTP 200).
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
From terrytao.wordpress.com, 8 October 2026. Source
Joerg Neunhaeuserer says they spent much of their career on a problem now claimed by OpenAI (they cite the number 148; read here as the family on dimension of self-similar measures), is relieved not to have to do that work, and is proud their intuition about the answer was right.
Samuel Kittle, Constantin Kogler, The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line, 7 October 2026. Source
Kittle and Kogler independently prove the exact overlaps conjecture in dimension one by a different method, compare their dates with OpenAI’s, and call their entropy inequality a quantitative form of a lemma in OpenAI’s paper 148.
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The challenge statements imply the headline claim after unpacking definitions or one routine step.
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An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
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Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.