Human contribution summary on Math-2688: Samuel Kittle, Constantin Kogler

Human Contribution: Samuel Kittle and Constantin Kogler

Related independent contribution: Kittle--Kogler $2026$

Paper and relation to this problem.

Samuel Kittle and Constantin Kogler, The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line, arXiv:2610.10511v1 (posted October 7, 2026), give an independent proof of the one-dimensional entropy-rate dimension formula. Their Corollary 1.2 settles exactly the attractor question recorded here: for a finite family $g_i(x)=r_i x+b_i$ with $0<|r_i|<1$ and no exact overlaps, if $K$ is the attractor and $s\geq 0$ satisfies $\sum_i|r_i|^s=1$, then \[ \boxed{\dim_H K=\min\{1,s\}.} \] This imposes no exponential separation or algebraicity hypothesis and allows unequal and orientation-reversing contraction ratios.

Stronger measure theorem (Theorem 1.1).

For a finitely supported probability law $\theta=\sum_i p_i\delta_{g_i}$ on contracting real similarities, let $\nu$ denote its stationary self-similar measure. Write \[ h_\theta=\lim_{n\to\infty}\frac{H(\theta^{*n})}{n}, \qquad \chi_\theta=-\sum_i p_i\log|r_i|>0, \] where $H(\theta^{*n})$ counts the entropy of the distinct composed affine maps, not merely symbolic words. Kittle--Kogler prove \[ \boxed{\dim_H\nu=\min\left\{1,\frac{h_\theta}{\chi_\theta}\right\}} \] with exact overlaps permitted. This has the same conclusion and scope as the main theorem of OpenAI's Result 148. If there are no exact overlaps, $h_\theta=H(p)$; choosing $p_i=|r_i|^s$ gives $H(p)=s\chi_\theta$, yielding the lower bound for $\dim_H K$, while the standard cylinder cover gives the matching upper bound. Thus neither contribution leaves a case of the recorded one-dimensional attractor problem unresolved by the other.

Additional result: entropy loss under addition (Theorem 1.3).

A distinct contribution of Kittle--Kogler is a quantitative inequality for arbitrary independent, finitely supported real random variables. For a compactly supported probability law $\rho$, put \[ V_\rho(r)=\frac{4}{r^3}\int_{\mathbb R} \inf_{c\in\mathbb R}\int_{[a,a+r)}(x-c)^2\,d\rho(x)\,da, \qquad \mathcal E(\rho)=\int_0^\infty V_\rho(r)\,\frac{dr}{r}. \] Here $\mathcal E$ is the variance energy, which accumulates normalized local variance across logarithmic scales. For independent finitely supported $X_1,\ldots,X_m$, their Theorem 1.3 gives \[ \boxed{ \sum_{j=1}^m H(X_j) -H\!\left(\sum_{j=1}^m X_j\right) \leq 30m\sum_{j=1}^m\mathcal E(\operatorname{Law}(X_j)). } \] The authors identify this as an explicit quantitative strengthening of the finite-law estimate underlying OpenAI's Lemma 3.2. In their comparison, it implies a strengthened version of that lemma with constant $C_m=120m^2$, whereas the original proof method yields a constant of order $O(m^3)$. These constants refer to the corresponding finite-law bounds and should not be confused with the $30m$ coefficient in Theorem 1.3 itself.

Proof strategies and attribution.

The arguments are different. OpenAI's Result 148 uses randomized nested partitions, fair two-point submeasures, conditioning on block symbol counts, and telescoping entropy gains at disjoint fine-scale windows. Kittle--Kogler prove their entropy-loss inequality by Poisson partitions and a Margulis--Russo differentiation argument, and deduce the dimension formula using Hochman's entropy theorem and a variance-summation method developed in earlier work of the authors. Kittle--Kogler explicitly acknowledge OpenAI's independently announced Result 148 and describe the finite-law estimates as the central new ingredients of both proofs.

Chronology, machine assistance, and formalization.

OpenAI dates Result 148 to September 24, 2026 and released it publicly on October 6, 2026. Kittle--Kogler report discovering their initial proof on September 27, 2026 and posted their paper on October 7, 2026. Their machine-contribution statement credits GPT-6 Astra with the initial proof and entropy inequality, and describes subsequent human rewriting and an alternative proof of the dimension deduction. They report an approximately 47,000-line Lean formalization, available at https://github.com/ckkogler/exact-overlaps-one-dim-lean. The OpenAI contribution separately links its own Lean scope and comparator files. Neither dimension theorem, by itself, asserts absolute continuity, and the existence of proof code should not be mistaken for an independent review by this site.

Sources for this comparison.

• Kittle--Kogler, manuscript (Theorems 1.1 and 1.3; Corollary 1.2; machine-contribution statement): https://arxiv.org/pdf/2610.10511.
• OpenAI, The entropy-rate dimension formula for self-similar measures on the line, Result 148 (September 24, 2026): 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.
• Kittle--Kogler, Lean formalization: https://github.com/ckkogler/exact-overlaps-one-dim-lean.