teorth · JD-Jones-ASES · Sep 13, 2026
diff --git a/README.md b/README.md
@@ -23,7 +23,7 @@ Bounds for which the level of available verification is currently at minimal lev
 | [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 |
-| [4b](https://teorth.github.io/optimizationproblems/constants/4b.html) | Furstenberg–Sárközy square-difference constant | 0.733412 | 1 |
+| [4b](https://teorth.github.io/optimizationproblems/constants/4b.html) | Furstenberg–Sárközy square-difference constant | 0.753742 | 1 |
 | [5a](https://teorth.github.io/optimizationproblems/constants/5a.html) | Sidon set size constant | 0 | 0.97633 |
 | [5b](https://teorth.github.io/optimizationproblems/constants/5b.html) | Sidon set density inside (4,5) sets | 0.5294 | 0.5714 |
 | [6](https://teorth.github.io/optimizationproblems/constants/6a.html) | Union-closed sets conjecture constant | 0.38271 | 0.5 |
@@ -158,6 +158,8 @@ 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.
+- [4b](https://teorth.github.io/optimizationproblems/constants/4b.html) **improved lower bound:** $C_{4b} \geq 0.752796$ by [D. Krachun](https://arxiv.org/abs/2608.01325), 2 Aug 2026 — the first bound past $3/4$.
+- [4b](https://teorth.github.io/optimizationproblems/constants/4b.html) **improved lower bound:** $C_{4b} \geq 0.753742$ by [JD Jones](https://github.com/JD-Jones-ASES/fs-lower-bound), 23 Aug 2026, with a Lean 4 formalization registered as Palomar entry [PALOMAR-2026-08-26-000004](https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-26-000004&version=1) (standard axioms only).
 
 ## Maintainers
 

diff --git a/constants/4b.md b/constants/4b.md
@@ -17,6 +17,8 @@ Let $r(N)$ be the maximum size of a subset $A\subset\\{1,\dots,N\\}$ with no non
 | $\tfrac12$ | Trivial / folklore (see [BG2008]) | Can use an arithmetic progression of spacing $p \asymp \sqrt{N}$ |
 | $\frac12\!\left(1+\frac{\log 7}{\log 65}\right)\approx 0.733077$ | [Ruz1984] | Base-expansion construction |
 | $\frac12\!\left(1+\frac{\log 12}{\log 205}\right)\approx 0.733412$ | [Lew2015] | Improves modulus and residue set in base expansion |
+| $\approx 0.752796$ | [Kra2026] | First bound past $3/4$: ten Paley chains (tuples in $\mathbb{F}\_p$, $p \equiv 3 \pmod 4$, every forward difference a nonzero square) at $p = 3, 7, 11, 19, 23, 31, 43, 59, 71, 103$ of lengths $t = 2, 3, 4, 5, 5, 7, 7, 9, 9, 11$, glued by CRT; $\alpha^\star = \big(\sum\_i \log(p\_i t\_i)/\log t\_i\big) \big/ \big(1 + 2\sum\_i \log p\_i/\log t\_i\big)$. |
+| $\approx 0.753742$ | [Jon2026] | Krachun's gluing over eleven blocks: his nine chains with $p \neq 23$, plus two acyclic square-difference digraphs on square-free composite moduli (any nonzero $z^2 \bmod m$ counts as a square) — on $235$ the residues $\\{0, 22, 50, 64, 67, 73, 92, 110, 112, 126, 136, 148, 155, 189, 193, 196, 224\\}$ (size $17$, longest path $H = 11$) and on $299$ the residues $\\{7, 21, 25, 40, 46, 78, 83, 116, 153, 161, 165, 206, 207, 210, 212, 244, 264, 289, 292\\}$ (size $19$, $H = 12$); a chain of length $t$ has $H = t$. Lifted to even-digit blocks by one new lemma; $\alpha\_\infty = \big(\sum\_i \log(m\_i t\_i)/\log H\_i\big) \big/ \big(1 + 2\sum\_i \log m\_i/\log H\_i\big) = 0.753741541837329405\ldots$. Lean 4 formalization (fs-formal, commit `4e79656`; Palomar PALOMAR-2026-08-26-000004) certifies the liminf statement and $0.7537 < \alpha\_\infty$. |
 
 ## Additional comments and links
 
@@ -37,8 +39,10 @@ Let $r(N)$ be the maximum size of a subset $A\subset\\{1,\dots,N\\}$ with no non
 - [Lew2015] Lewko, M. *An improved lower bound related to the Furstenberg–Sárközy theorem.* Electron. J. Combin. **22** (1) (2015), Paper P1.32. DOI: 10.37236/4656
 - [Rice2019] Rice, A. *A maximal extension of the best-known bounds for the Furstenberg–Sárközy theorem.* Acta Arith. **187** (2019), 1–41. DOI: 10.4064/aa170828-26-8
 - [BG2008] Beigel, R.; Gasarch, W. *Square-Difference-Free Sets of Size \(\Omega(n^{0.7334\ldots})\).* 2008. [arXiv:0804.4892](https://arxiv.org/abs/0804.4892)
+- [Kra2026] Krachun, D. *Square-difference-free sets beyond the three-quarter barrier.* 2026. [arXiv:2608.01325](https://arxiv.org/abs/2608.01325)
+- [Jon2026] Jones, JD. *A lower bound for the Furstenberg–Sárközy problem.* 2026. Note, data and verification script: [fs-lower-bound](https://github.com/JD-Jones-ASES/fs-lower-bound) (commit `c2c0687`). Lean 4 / Mathlib formalization: [fs-formal](https://github.com/JD-Jones-ASES/fs-formal) (commit `4e79656`), Palomar registry entry [PALOMAR-2026-08-26-000004](https://palomar-registry.org/entry.html?id=PALOMAR-2026-08-26-000004&version=1). AI-generated with a human managing the workflow; see the repository's `DISCLOSURE.md`.
 
 
 ## Contribution notes
 
-ChatGPT 5.2 Pro was used to prepare an initial version of this page.
+ChatGPT 5.2 Pro was used to prepare an initial version of this page. The [Kra2026] and [Jon2026] rows were drafted with Claude (Anthropic) for the contributor, who verified the references; both constants in those rows were recomputed from the data as stated.