Improve the rigorous bounds on the self-avoiding-walk connective constant $\mu$ of the simple cubic lattice $\mathbb{Z}^3$

For self-avoiding walks (SAWs) on the simple cubic lattice $\mathbb{Z}^3$, let $c_n$ be the number of $n$-step walks from the origin that visit no vertex twice, and let $\mu = \lim_{n\to\infty} c_n^{1/n}$ be the connective constant (limit exists by subadditivity). No lattice in dimension $d\ge 2$ has a proven value of $\mu$ except the honeycomb lattice. GOAL: prove a strictly tighter rigorous upper OR lower bound on $\mu$ for $\mathbb{Z}^3$ than is currently known, via a finite, re-runnable computation (finite-memory / finite-automaton enumeration for the upper bound; bridge concatenation for the lower bound). Report the bound together with a certificate.

Acceptance. FULLY RESOLVES (either direction): a machine-verifiable proof of $\mu \le U$ with $U < 4.7114$, or of $\mu \ge L$ exceeding the current best rigorous lower bound — with (i) the finite automaton / enumeration data and (ii) a script that regenerates it and evaluates the bound. PARTIAL: reproduce a published rigorous bound with a runnable certificate, or improve the numerical estimate of $\mu$ with a documented estimator and error bar (state the walk length / method). All counts must be exact integers a referee can recompute.