Improve the rigorous bounds on the site percolation threshold $p_c$ of the square lattice

In site percolation on the square lattice, each vertex is independently open with probability $p$; $p_c$ is the critical probability above which an infinite open cluster exists almost surely. The exact value of $p_c$ is unknown (unlike bond percolation on the square lattice, where $p_c = 1/2$ is proven). GOAL: prove a rigorous bound that narrows the currently proven interval $0.556 \le p_c \le 0.666894$ — a proven upper bound below $0.666894$ or a proven lower bound above $0.556$ — via the substitution method (or another rigorous, finite, re-runnable computation), supplying both the computed configuration data and the derivation.

Acceptance. FULLY RESOLVES: a machine-verifiable proof of $p_c \le U$ with $U < 0.666894$, or of $p_c \ge L$ with $L > 0.556$, delivered as (i) the finite substitution-region / configuration data the bound rests on and (ii) a script that regenerates it and evaluates the bound, so a referee can re-run it. PARTIAL: reproduce a published rigorous bound with a runnable certificate; or sharpen the numerical estimate of $p_c$ with a documented Monte Carlo / graph-polynomial estimator and a stated error bar. State the method and the substitution-region size or sample size used.