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Improve C_3a lower bound to 1.21418 via two-temperature cyclic compression #148

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af612db
Improve C3a candidate lower bound to 1.21418
carson-olaf Aug 19, 2026
62a01c5
Link C3a submission pull request
carson-olaf Aug 19, 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 | 1.19102809 (1.19519192*) | 1.33333 |
| [3a](https://teorth.github.io/optimizationproblems/constants/3a.html) | Gyarmati-Hennecart-Ruzsa sum-difference constant | 1.19102809 (1.21418*) | 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 DownExpand Up@@ -153,6 +153,7 @@ Bounds for which the level of available verification is currently at minimal lev
- [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.
- [15a](https://teorth.github.io/optimizationproblems/constants/15a.html) **improved upper bound:** $C_{15a} \leq 2.371177$ by [E. Dupont, M. Eisenberger, B. Kozlovskii, A. Mehrabian, F. J. R. Ruiz, A. See, R. Zhou, J. Alman, V. Vassilevska Williams, M. Balog](https://arxiv.org/abs/2608.16884), 17 Aug 2026.
- [3a](https://teorth.github.io/optimizationproblems/constants/3a.html) **improved lower bound (limit value):** $C_{3a} \geq 1.21418*$ by [C. Olafson](https://github.com/carcango), 19 Aug 2026.

## Maintainers

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3 changes: 3 additions & 0 deletions constants/3a.md
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Expand Up@@ -32,6 +32,7 @@ $$ |A-B| \gg |A+B|^{C_{3a}}.$$
| $1.187326127925948$* | [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. |
| $1.19102809$ ($1.19519192$*) | [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]. |
| $1.21418$* | [O2026] | Two-temperature cyclic-quotient construction with $M=\langle4,5\rangle\cap[0,80]$, $\lvert M\rvert=75$, $\lvert M+M\rvert=155$, $\lvert M-M\rvert=161$, difference tilt $x=8149/10000$, sum tilt $y=42571/50000$, and quotient rate $R=249293/10000$. Exact rational reconstruction gives $\rho=3.216039541306344\dots$, $\sigma=2.527224394365747\dots$, and the fixed-rate exponent $1+(\rho-\sigma)/\log R=1.214180896097785\dots>1.21418$. The package also verifies a finite-size correction for a canonically specified, unmaterialized finite GHR seed; the bound is recorded conservatively as a limit value pending review of the written theorem. |


## Additional comments and links
Expand All@@ -43,6 +44,7 @@ $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$
- [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.
- [O2026] introduces a two-temperature quotient theorem in which the difference Gibbs tilt and the sum upper-bound tilt are optimized independently. Its gapless specialization gives $2-\log 2/\log(1+\sqrt2)=1.213560298642605\dots$; the numerical-semigroup alphabet $\langle4,5\rangle\cap[0,80]$ improves this to $1.21418$.
- [AlphaEvolve repository page for this problem](https://google-deepmind.github.io/alphaevolve_repository_of_problems/problems/44.html)

## References
Expand All@@ -58,3 +60,4 @@ $C_{3a} \geq 1 + \log( \lvert U-U \rvert /\lvert U+U \rvert )/\log(2 \max(U)+1)$
- [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).
- [O2026] Olafson, Carson. Two-temperature cyclic compression for the Gyarmati–Hennecart–Ruzsa constant, [proof and verification package](https://github.com/carcango/c3a-two-temperature-bound), [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/148) (2026).