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First obtain a polynomial critical one-arm upper bound from part (a). Fix a sufficiently large integer radius $r$. Use rotated-coordinate boxes $B_r=[-r,r{]}^2$ for this geometric construction, in the axes of part (a). Surround $B_r$ by a square ring inside $B_{3r}$. It can be assembled from the four stripsThe first two require horizontal dual crossings; the last two require vertical dual crossings. Their corner overlaps are squares, so the adjacent long crossings intersect there and their union contains a dual graph cycle surrounding the inner box. Translate the strips by the half-lattice displacement of the planar dual graph and adjust the endpoints by one lattice unit if needed. All aspect ratios remain bounded. Part (a) and the same overlap gluing give a uniform lower bound $c_1>0$ for each dual crossing at $1/2$. Applying the Harris-FKG inequality to these decreasing events in the primal configuration gives a closed dual circuit with probability at least $\delta =c_1^4>0$. We may replace $\delta$ by a smaller number in $(0,1)$.

Use the independent annular barriers for percolation at radii $r_j=r_{0}10^j$. Their supporting edge sets are disjoint, so the circuit events are independent. In the original unrotated coordinates, $B_{3r}$ lies inside ${\Lambda }_{5r}$ and the ring still separates the origin from infinity. These fixed geometric comparisons let us bound the original $g_N$. An open graph path from $0$ to distance $N$ must avoid every such closed dual circuit inside ${\Lambda }_N$. Thus, for some constants $C<\infty$ and $\alpha >0$,where the first bound is used only when at least one of the indicated rings fits; increasing $C$ handles smaller $N$.

Now let $\epsilon =p-1/2>0$ and use the monotone coupling of Bernoulli percolation on edges. The event $\{0\leftrightarrow \partial {\Lambda }_N\}$ only needs edges whose endpoints lie in ${\Lambda }_N$, because a first boundary hit gives an internal graph path. There are exactly $4N(2N+1)$ such edges. Each changes state between parameters $1/2$ and $p$ with probability $\epsilon$. The union bound therefore gives the finite-box comparison for percolation parametersFor sufficiently small $\epsilon$, choose $N=\lfloor {\epsilon }^{-1/(\alpha +2)}\rfloor$, so that $N$ is at least half the unrounded value. Both terms then have order ${\epsilon }^{\alpha /(\alpha +2)}$. In particular,for a finite $A$. Increase $A$ to cover the remaining compact range of parameters by $\theta (p)\le 1$. This near-critical percolation power upper bound requires only a positive crossing constant, not the exact value of a critical exponent. It also implies $\theta (1/2)=0$ through the critical one-arm probability bound.