Narrow the rigorous bounds on the self-avoiding-walk connective constant $\mu$ of the square lattice

A self-avoiding walk (SAW) of length $n$ on the square lattice $\mathbb{Z}^2$ is a nearest-neighbour lattice path from the origin that visits no vertex twice. Let $c_n$ be the number of such $n$-step walks. The connective constant is $\mu = \lim_{n\to\infty} c_n^{1/n}$ (the limit exists by subadditivity, since $c_{m+n}\le c_m c_n$). Its exact value is unknown for $\mathbb{Z}^2$ — in contrast to the honeycomb lattice, where $\mu=\sqrt{2+\sqrt{2}}$ is proven (Duminil-Copin & Smirnov). GOAL: produce a rigorous, certificate-backed bound that narrows the currently proven interval for $\mu$ — either a proven upper bound strictly below $2.662343$, or a proven lower bound above the current best — via a finite computation (finite-lattice / transfer-matrix enumeration, an irreducible-bridge count a la Kesten, or a finite-automaton method) whose validity does not rest on unproven series/scaling assumptions.

Acceptance. FULLY RESOLVES (either direction): a machine-verifiable proof of $\mu \le U$ with $U < 2.662343$, or of $\mu \ge L$ with $L$ exceeding the current best rigorous lower bound — delivered as (i) the finite enumeration / transfer-matrix / automaton data the bound is derived from (SAW counts $c_n$, bridge counts by span, or the automaton), and (ii) a script that regenerates that data and evaluates the bound, so a referee can re-run it end to end. PARTIAL: independently reproduce a published rigorous bound (e.g. recompute $\mu \le 2.662343$) with a runnable certificate; or tighten the best numerical estimate of $\mu$ with a documented estimator and a stated error bar. State precisely the method and the walk length / bridge span $n$ reached.