Source authenticated

The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture)

Theorem 1.1: for every finite system of contracting similarities of $\mathbb R$ and every positive probability vector, $\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$ , with no separation or arithmetic assumption; exact overlaps, repeated maps, negative and unequal ratios are allowed. Without exact overlaps this gives $\min\{1,H(p)/\chi\}$ (the exact overlaps conjecture in dimension one) and the attractor formula $\dim_HK=\min\{1,s\}$ with $\sum|r_i|^s=1$ . It says nothing about absolute continuity (for instance of Bernoulli convolutions) and does not treat higher-dimensional or non-conformal systems.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture) · Entropy-rate dimension / exact overlaps conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against both formulations. It states $\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$ for every finite indexed family with $0<|r_i|<1$ and every strictly positive weight vector, signed and unequal ratios and exact overlaps allowed; with no exact overlaps $H(G_n)=nH(p)$ , giving the exact overlaps conjecture on the line. Lean: lean/formalization.yaml lists comparator SelfSimilar with declaration OAI.EntropyRateDimension.entropy_rate_dimension. The statement ComparatorChallenges/SelfSimilar.lean was read here: for a finite system with nonzero ratios of modulus below one and positive weights summing to one, and any probability measure satisfying the self-similarity equation, the lower Hausdorff dimension (infimum of dimH over positive-measure Borel sets) equals min(1, entropy rate / Lyapunov exponent), both in base 2, the entropy rate defined as the infimum of $H(G_{n})/n$ over complete affine maps. This states the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Source ↗

The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture)

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean proof (OAI.EntropyRateDimension.entropy_rate_dimension)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Comparator statement: SelfSimilar.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Comparator statement: SelfSimilarCorollaries.lean (no-overlap and attractor forms)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Varju, Entropy rates in the dimension theory of self-similar measures (Conjecture 3)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof (OAI.EntropyRateDimension.entropy_rate_dimension)

lean artifact · pending

Artifact ↗

Comparator statement: SelfSimilar.lean

lean artifact · pending

Artifact ↗

Comparator statement: SelfSimilarCorollaries.lean (no-overlap and attractor forms)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

Comparator statement: SelfSimilarCorollaries.lean (no-overlap and attractor forms) evidence for this event

Comparator statement: SelfSimilar.lean evidence for this event

Varju, Entropy rates in the dimension theory of self-similar measures (Conjecture 3) evidence for this event

The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture) evidence for this event

The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture) parent of this event

VibeMathed record: The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture) evidence for this event

Lean proof (OAI.EntropyRateDimension.entropy_rate_dimension) evidence for this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph