Square-lattice self-avoiding walk connective constant $\mu_{\mathbb{Z}^2}$

Upper bound, last-listed row

| $2.679193$ | [FV2017] | Reported as a rigorous upper bound in [FV2017] (attributed there to [PT2000]). [FV2017-bounds-square] [FV2017-ref-277] |

exact pinned row

| $2.679193$ | <a href="#FV2017">[FV2017]</a> | Reported as a rigorous upper bound in <a href="#FV2017">[FV2017]</a> (attributed there to <a href="#PT2000">[PT2000]</a>). <a href="#FV2017-bounds-square">[FV2017-bounds-square]</a> <a href="#FV2017-ref-277">[FV2017-ref-277]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $2.6273856$ | [Num2026] | Kesten irreducible-bridge renewal exactly as in [Jen2004-lb], extended two span levels past the 2004 computation (span $L=17$, series length $N=260$) by an exact cell-by-cell finite-lattice transfer matrix (CRT big-integer arithmetic); the certificate is a single exact big-integer inequality (self-contained checker included), and secondary certificates at Jensen's own span ($L=15$, $N=260$ gives $2.6256270$) and at $L=16$ ($2.6265705$) independently also exceed $2.625622$. [Num2026-bound] |

exact pinned row

| $2.6273856$ | [Num2026] | Kesten irreducible-bridge renewal exactly as in [Jen2004-lb], extended two span levels past the 2004 computation (span $L=17$, series length $N=260$) by an exact cell-by-cell finite-lattice transfer matrix (CRT big-integer arithmetic); the certificate is a single exact big-integer inequality (self-contained checker included), and secondary certificates at Jensen's own span ($L=15$, $N=260$ gives $2.6256270$) and at $L=16$ ($2.6265705$) independently also exceed $2.625622$. <a href="#Num2026-bound">[Num2026-bound]</a> |

value changed 2026-08-02