Improve the rigorous bounds on Klarner's constant $\lambda$ (the polyomino / lattice-animal growth constant)

A polyomino (fixed lattice animal) of size $n$ is an edge-connected set of $n$ cells on the square lattice, counted up to translation only. Let $A(n)$ be the number of such polyominoes (OEIS A001168). Klarner's constant is $\lambda = \lim_{n\to\infty} A(n)^{1/n}$; the limit's existence was proven by Madras (1999). Its exact value is unknown. GOAL: prove a strictly tighter rigorous bound than the current interval $4 < \lambda \le 4.5252$ — i.e. a proven upper bound below $4.5252$ or a proven lower bound above $4.00253$ — via a finite, re-runnable computation (e.g. a concatenation/recurrence argument or a transfer-matrix bound over bounded-width animals), with the enumeration data and the derivation both provided.

Acceptance. FULLY RESOLVES: a machine-verifiable proof of $\lambda \le U$ with $U < 4.5252$, or of $\lambda \ge L$ with $L > 4.00253$, supplied as (i) the finite enumeration / transfer-matrix data used and (ii) a script that regenerates it and evaluates the bound formula. PARTIAL: reproduce a published rigorous bound with a runnable certificate, or extend the exact series $A(n)$ by one or more new terms (each an exact integer, cross-checked by two independent enumeration codes and matching OEIS A001168 on the overlap). State the polyomino size $n$ reached and the method.