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C_3c: improved lower bound 1.6747338950414058 on a 147-point support #185

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C_3c: improved lower bound 1.6747338950414058 on a 147-point support
CoolRmal Sep 9, 2026
2812c47
C_3c: cite the submitting pull request in the [L2026] reference
CoolRmal Sep 9, 2026
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3 changes: 2 additions & 1 deletion README.md
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Expand Up @@ -19,7 +19,7 @@ Bounds for which the level of available verification is currently at minimal lev
| [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 |
| [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 |
| [3c](https://teorth.github.io/optimizationproblems/constants/3c.html) | 4-slope Kakeya-type sum-difference constant | 1.67473389 (1.6747338950414058*) | 1.75 |
| [3d](https://teorth.github.io/optimizationproblems/constants/3d.html) | Single-set sum-difference exponent | 2 | 2 |
| [3e](https://teorth.github.io/optimizationproblems/constants/3e.html) | Unnormalized single-set sum-difference exponent | 1.27155 | 1.33333 |
| [4a](https://teorth.github.io/optimizationproblems/constants/4a.html) | Cap set constant | 2.2203 | 2.756 |
Expand Down Expand Up @@ -158,6 +158,7 @@ Bounds for which the level of available verification is currently at minimal lev
- [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.
- [43](https://teorth.github.io/optimizationproblems/constants/43a.html) **improved lower bound (unverified):** $C_{43} \geq 0.860*$ (exact $43/50$; certificate-layer result conditional on the lemma set of [KHSHGW2026](https://arxiv.org/abs/2601.22365)) by [J. Savva](https://doi.org/10.5281/zenodo.22223485), 1 Sep 2026.
- [88a](https://teorth.github.io/optimizationproblems/constants/88a.html) **improved upper bound:** $C_{88a} \leq 186$ via $\mathrm{DHL}[40,2]$, by [OpenAI](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf), 30 Aug 2026, with a Lean 4 formalization conditional on three declared axioms.
- [3c](https://teorth.github.io/optimizationproblems/constants/3c.html) **improved lower bound:** $C_{3c} \geq 1.6747338950414058$ by Y. Lin, [entropy certificate](https://gist.github.com/CoolRmal/5368357cd781d7e5c676c9d68ad24d22) on a 147-point support, 9 Sep 2026.

## Maintainers

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3 changes: 3 additions & 0 deletions constants/3c.md
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Expand Up @@ -31,6 +31,7 @@ $$ A \stackrel{G}{\pm} rB := \{ a \pm rb: a \in A, b \in B\}.$$
| $1.67471$ | [A2026] | |
| $1.67473389$ | [G2026] | Entropy construction on a 26-point support. |
| $1.6747338950208249$ | [MI2026] | Entropy construction on a 95-point support. |
| $1.6747338950414058$ | [L2026] | Entropy construction on a 147-point support, exact rational weights at denominator $10^{320}$. Certified $C_{3c} \geq 1.674733895041405870063135756722213999136383713818148696811828$, with $H(X-Y) = 1.90118266043426947934\ldots$ and $H(X) = H(Y) = H(X+Y) = H(X+2Y) = 1.13521477415805507651\ldots$ (natural logarithm), so no constraint is slack. Improves [MI2026] by $2.06 \times 10^{-11}$. |



Expand All @@ -51,6 +52,7 @@ $$ H(X-Y) \leq C_{3c} \max( H(X), H(Y), H(X+Y), H(X+2Y)).$$ This entropy formula
- [GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. [arXiv:2511.02864](https://arxiv.org/abs/2511.02864)
- [GR2019] Green, B.; Ruzsa, I. Z. On the arithmetic Kakeya conjecture of Katz and Tao. Periodica Mathematica Hungarica, Volume 78, Issue 1, pp 135–151 (2019). DOI: 10.1007/s10958-018-2003-3.
- [L2015] Lemm, Marius. New counterexamples for sums-differences. Proceedings of the American Mathematical Society, Vol. 143, No. 9 (SEPTEMBER 2015), pp. 3863-3868 (6 pages). DOI: 10.1090/proc/12731.
- [L2026] Lin, Yongxi. 147-point entropy certificate for $C_{3c}$, [certificate and verifiers](https://gist.github.com/CoolRmal/5368357cd781d7e5c676c9d68ad24d22) (pinned revision [`62621d2`](https://gist.github.com/CoolRmal/5368357cd781d7e5c676c9d68ad24d22/62621d2d30df0a4405dce760cde8a5b59af51281)), [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/185) (2026).
- [MI2026] Mosaic Intelligence ([@111111](https://x.com/111111)). 95-point entropy certificate for $C_{3c}$, [certificate archive](https://doi.org/10.5281/zenodo.20794135), [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/93) (2026).
- [KT1999] Katz, Nets Hawk; Tao, Terence. Bounds on arithmetic projections, and applications to the Kakeya conjecture. Math. Res. Lett. 6 (1999), no. 5-6, 625-630. DOI: 10.4310/MRL.1999.v6.n6.a3.
- [KT2002] Katz, N. H.; Tao, T. New bounds for Kakeya problems. J. Anal. Math. 87 (2002), 231–263. DOI: 10.1007/BF02792310.
Expand All @@ -59,3 +61,4 @@ $$ H(X-Y) \leq C_{3c} \max( H(X), H(Y), H(X+Y), H(X+2Y)).$$ This entropy formula
## Contribution notes

- Used formatting of 3b.md
- The [L2026] row was prepared with AI assistance; the certificate is machine-checkable by the two independent verifier scripts linked above.