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Improve C_3a lower bound via Lean-formalized controlled-carry semigroup mask-digit construction #146

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0b24d2a
Improve lower bound for C_3a
kleinwaks Aug 14, 2026
265fb64
Improve lower bound for C_3a
kleinwaks Aug 14, 2026
99521b6
Improve lower bound for C_3a
kleinwaks Aug 14, 2026
569192c
Update 3a.md reference [K2026b] with PR URL
kleinwaks Aug 14, 2026
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3 changes: 2 additions & 1 deletion README.md
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Expand Up @@ -17,7 +17,7 @@ Bounds for which the level of available verification is currently at minimal lev
| [1a](https://teorth.github.io/optimizationproblems/constants/1a.html) | Sidon set autocorrelation constant | 1.2802 (1.292*) | 1.502862 |
| [1b](https://teorth.github.io/optimizationproblems/constants/1b.html) | Erdős minimum overlap constant | 0.379005 | 0.380868 |
| [2](https://teorth.github.io/optimizationproblems/constants/2a.html) | Crouzeix constant | 2 | 2 |
| [3a](https://teorth.github.io/optimizationproblems/constants/3a.html) | Gyarmati-Hennecart-Ruzsa sum-difference constant | 1.1835129324 (1.19102809*) | 1.33333 |
| [3a](https://teorth.github.io/optimizationproblems/constants/3a.html) | Gyarmati-Hennecart-Ruzsa sum-difference constant | 1.19102809 (1.19519192*) | 1.33333 |
| [3b](https://teorth.github.io/optimizationproblems/constants/3b.html) | Kakeya sums-differences constant | 1.77898 (1.77898884*) | 1.83333 |
| [3c](https://teorth.github.io/optimizationproblems/constants/3c.html) | 4-slope Kakeya-type sum-difference constant | 1.67473389 | 1.75 |
| [3d](https://teorth.github.io/optimizationproblems/constants/3d.html) | Single-set sum-difference exponent | 2 | 2 |
Expand Down Expand Up @@ -151,6 +151,7 @@ Bounds for which the level of available verification is currently at minimal lev
- [3b](https://teorth.github.io/optimizationproblems/constants/3b.html) **improved lower bound:** $C_{3b} \geq 1.77898884$ by Mosaic Intelligence, [entropy certificate](https://doi.org/10.5281/zenodo.20794135) on a 13-point support.
- [10a](https://teorth.github.io/optimizationproblems/constants/10a.html) **improved upper and lower bounds:** $\frac{6\pi}{11} \leq C_{10a} \leq \frac{\pi}{2\log(1+\sqrt{2})} - 10^{-4}$ by [R. Saha, A. Li, A. Xue, S. Chaudhuri, A. Klivans, P. K. Kothari, R. Meka](https://arxiv.org/abs/2608.11158), 11 Aug 2026 — determines the tenths digit of $C_{10a}$ to be $7$.
- [2](https://teorth.github.io/optimizationproblems/constants/2a.html) **solved:** $C_2 = 2$ — Crouzeix's conjecture, by [Jin](https://www.preprints.org/manuscript/202607.1919) (July 2026); a second proof followed shortly after in [E. Lorist and F. L. Schwenninger](https://arxiv.org/abs/2608.03841), 4 Aug 2026, which acknowledges Jin.
- [3a](https://teorth.github.io/optimizationproblems/constants/3a.html) **improved lower bound (limit value):** $C_{3a} \geq 1.19519192*$ by [L. Kleinwaks](https://github.com/kleinwaks/masked-digit-sum-difference-bound), 14 Aug 2026.

## Maintainers

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3 changes: 3 additions & 0 deletions constants/3a.md
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Expand Up @@ -31,6 +31,7 @@ $$ |A-B| \gg |A+B|^{C_{3a}}.$$
| $1.1835129324$ | [MI2026] | Base-$33$ digit construction with exact counting certificate. |
| $1.187326127925948$* | [Num2026] | Capped base-$89$ digit construction (max digit $44$, sparse 29-letter alphabet); certified as the large-deviation LIMIT of the exact per-depth lemma values $\theta(U_d)$, each valid for every $d$ and increasing to the limit (the same limit-as-lower-bound principle as [Z2025]); interval-arithmetic certificate, replayable checker included. |
| $1.19102809$* | [K2026] | Base-$34065$ masked-digit limit construction with $M=\langle1518,1524,1587,2024,2032,2116\rangle\cap[0,17032]$ and a directed-rounding certificate. |
| $1.19102809$ | [K2026b] | Lean-formalized proof of $1.19519192$* via controlled-carry masked-digit limit construction with $M=\langle1971,2016,2100,2628,2688,2800\rangle\cap[0,26972]$ in base $27022$ and Lean-formalized explicit finite construction of $1.19102809$ with $M=\langle1518,1524,1587,2024,2032,2116\rangle\cap[0,17032]$ in base $34065$ using $10^{13}$ digits. Also includes formalization of necessary results from [GHR2007].|


## Additional comments and links
Expand All @@ -41,6 +42,7 @@ $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$
- **Reproducing the $1.14465$ of [GHR2007].** The final construction of [GHR2007, §2] is recorded there with $\lvert U+U\rvert = 4455634$, $\lvert U-U\rvert = 110205905$ and $q = 2\max U + 1 = 5723906483$, and the exponent is stated as $\theta = 1.144655$ (Theorem 1 asserts $\theta_0 > 1.14465$). Substituting those three values into the lemma gives $1 + \log(110205905/4455634)/\log(5723906483) = 1.142789\dots$, which is smaller. The same substitution reproduces the paper's other stated exponents to six decimal places ($1.1078\dots$, $1.1165757\dots$ and $1.135589\dots$ against a printed $1.135596$), so the method of checking appears sound; the printed multiplier sequence $\Lambda$ for $m = 11$ may not be the one that produced the record. All later entries in the table exceed $1.14465$ regardless, so nothing downstream depends on this.
- [K2026] generalizes the bounded-digit limit construction in [Z2025] by replacing bounded digits with digits restricted to a finite mask.
- The record mask in [K2026] is generated by the product grid $\\{3,4\\}\times\\{506,508,529\\}$; its column semigroup is the simple gluing $\langle506,508,529\rangle=23\langle22,23\rangle+508\mathbb N_0$.
- [K2026b] generalizes [K2026] and [Z2025] by allowing constructions in arbitrary bases greater than the maximum element of the digit mask and accounting for carrying. The record mask is different from that of [K2026], but the semigroup generators again form a product grid.
- [AlphaEvolve repository page for this problem](https://google-deepmind.github.io/alphaevolve_repository_of_problems/problems/44.html)

## References
Expand All @@ -55,3 +57,4 @@ $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$
- [Z2025] Zheng, Fan. Sums and differences of sets: a further improvement over AlphaEvolve, 2025. [arXiv:2506.01896](https://arxiv.org/abs/2506.01896).
- [G2026] Griego, Sebastian. Base-$21$ digit construction certificate for $C_{3a}$, [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/71) (2026).
- [K2026] Kleinwaks, Logan. A masked-digit lower bound for the Gyarmati–Hennecart–Ruzsa sum–difference constant, [proof and verification package](https://github.com/kleinwaks/masked-digit-sum-difference-bound), [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/134) (2026).
- [K2026b] Kleinwaks, Logan. Improved lower bound for the Gyarmati–Hennecart–Ruzsa sum–difference constant using masked digits and controlled carries, [proof, verification package, and Lean formalization](https://github.com/kleinwaks/masked-digit-sum-difference-bound), [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/146) (2026).