= Solution
The purpose of the <RSW lemma> is to turn local crossing information into control at every fixed shape and scale. Work with independent <bond percolation> on the <square lattice>. Let $H(a,b)$ be the event of an open left-to-right <graph path> in $[0,a]\times[0,b]\cap\mathbb Z^2$, and write $h_p(a,b)=\mathbb P_p(H(a,b))$.
A useful precise uniform version of the <RSW lemma> is the following <uniform RSW crossing estimate>. If for a fixed parameter $p$ and some $\delta>0$,
$$
\inf_{n\geq1}h_p(n,n)\geq\delta,
$$
then for every fixed aspect ratio $\rho>1$ there is $c(\delta,\rho)>0$, independent of $n$, such that
$$
\boxed{\inf_{n\geq1}h_p(\lceil\rho n\rceil,n)\geq c(\delta,\rho).}
$$
In particular this bounds length $2n$ in terms of the uniform square bound at length $n$. The constants need not be sharp. The hypotheses used in the proof are planarity, translation and reflection symmetries, the <Harris-FKG inequality> and <independence> of unexplored <edges>. At a self-dual parameter the same argument for dual crossings gives an upper bound strictly below one as well. Neither an exact value of $p_c$ nor the existence of an infinite cluster is an assumption of this estimate.
Here is the gluing picture behind the <RSW lemma>. For increasing crossing or attachment events, the <Harris-FKG inequality> provides a lower bound on their joint <probability>. If an event of <probability> at least $\delta$ is a union of two reflection-related increasing alternatives $A_1,A_2$, the <square-root trick for positively associated events> gives
$$
\mathbb P_p(A_i)\geq1-\sqrt{1-\delta}.
$$
Indeed their decreasing complements are positively associated too, so $\mathbb P_p(A_1^c\cap A_2^c)\geq\mathbb P_p(A_1^c)^2$. This prevents a square crossing from concentrating all its useful attachment locations on just one side.
The nontrivial first gluing step enlarges a square to aspect ratio $3/2$. Explore an extremal square crossing, revealing the <edges> on the explored side but leaving its other side unexamined. In that unexplored region the conditional law is still independent <bond percolation>. Compare possible attachments to the crossing with their reflected alternatives. Reflection symmetry and the preceding square-root estimate give a positive bound, depending only on $\delta$, for the required attachment after averaging over the explored crossing. Carry out the reflected construction at the other end and use the <Harris-FKG inequality> to combine the increasing attachment events with the original crossing. Planarity ensures that the relevant transverse paths actually meet. This <RSW reflection extension lemma> yields a lower bound $u(\delta)>0$ for the $3n/2$-by-$n$ crossing. The exploration is important: a reflection compares the laws of fresh configurations; reflecting the picture of an open path does not make the reflected <edges> open.
From that first extension, longer rectangles are obtained by a genuinely transverse gluing. Two $3n/2$-by-$n$ rectangles shifted by $n/2$ overlap in an $n$-by-$n$ square. Require a horizontal crossing in each long rectangle and a vertical crossing of the overlap square. Each horizontal crossing crosses that overlap from left to right, and therefore meets its vertical crossing. Their union crosses the $2n$-by-$n$ rectangle. The <Harris-FKG inequality> gives a lower bound $u(\delta)^2\delta$. Repeat a bounded number of times for any fixed $\rho$. Integer rounding uses neighboring lattice rectangles and bounded additional gluing steps; finitely many smallest scales can be absorbed into the constant. The constants deteriorate with $\rho$ but not with $n$. Merely requiring horizontal crossings in adjacent squares would not suffice, since their endpoints need not coincide; the overlap crossing solves that problem.
I apply the <RSW lemma> to the exact threshold for <bond percolation> on the <square lattice>. The <planar duality for rectangle crossings> says that an open horizontal crossing and a closed dual vertical crossing are complementary. At $p=1/2$, the dual <edge> states have the same law as the primal ones. For the balanced lattice rectangle with side lengths $n+1,n$, the rotated dual crossing rectangle has those same side lengths: its transverse lengths before rotation are $n,n+1$. Thus the <exact self-dual rectangle crossing probability> is $1/2$. Restricting a crossing of this rectangle to its first visit to the shorter vertical side gives
$$
h_{1/2}(n,n)\geq\frac12.
$$
The one-unit balance avoids assuming an exact one-half <probability> for every finite vertex-square convention. The <RSW lemma> now gives scale-independent positive bounds for all fixed-aspect-ratio primal and closed dual rectangle crossings.
Arrange four appropriately overlapping long rectangles around a square ring. Closed dual crossings along the four sides, with transverse overlap crossings if needed, join to a closed dual <graph cycle> surrounding the inner square. The <Harris-FKG inequality>, applied to the closed dual states, and the <RSW lemma> give a constant $b>0$ for this circuit event, uniformly over ring size. Choose disjoint rings with radii increasing, for example, by a factor of four. Their circuit events depend on disjoint <edge> sets, so they are independent. An open <graph path> from the origin to infinity would have to avoid every one of these dual barriers. Its <probability> is at most $(1-b)^j$ after $j$ rings and therefore zero. Hence
$$
\theta_2(1/2)=0,\qquad p_c(2)\geq\frac12.
$$
This is the <independent annular barriers for percolation> argument. It is worth separating it from the other inequality: the absence of an infinite cluster at one parameter alone does not prove that every larger parameter percolates.
For the reverse inequality use <sharpness of the percolation transition>: below $p_c$, independent <bond percolation> on the <cubic lattice> has exponentially decaying connection <probabilities>. One can see why this is the relevant general ingredient through the <finite-set criterion for percolation sharpness>. For a finite set $S$ containing the origin put
$$
\varphi_p(S)=p\sum_{\substack{x\in S,\ y\notin S\\x\sim y}}
\mathbb P_p(0\leftrightarrow x\text{ using only edges in }S),
\qquad
\widetilde p_c=\sup\{p:\varphi_p(S)<1\text{ for some finite }S\ni0\}.
$$
If $\varphi_p(S)<1$, split a long open <self-avoiding walk> at its first exit from $S$. Its internal connection, exit <edge> and subsequent connection have disjoint witnesses. The <BK inequality> gives a contraction by $\varphi_p(S)$ each time distance decreases by the diameter of $S$ plus a fixed step. Iteration proves <exponential decay of subcritical percolation> at such $p$.
To identify this finite-set threshold with $p_c$, the <Margulis–Russo formula> expresses the derivative of $g_n(p)$ as the sum of pivotal-edge <probabilities>. Explore the cluster attached to the box boundary, and let $S$ be its complement. On failure of the origin-to-boundary event, $0\in S$; all <edges> from $S$ to the boundary cluster are closed, while the internal <edges> of $S$ remain fresh. A boundary <edge> is pivotal precisely when its endpoint in $S$ is connected to the origin inside $S$. Removing the factor $1-p$ for a closed pivotal <edge> yields the general differential inequality
$$
\frac{d}{dp}g_n(p)\geq
\frac{1-g_n(p)}{p(1-p)}
\inf_{\substack{0\in S\subseteq\Lambda_n\setminus\partial\Lambda_n}}
\varphi_p(S).
$$
For $p>\widetilde p_c$ every finite-set quantity on the right is at least one. Integrating from any $p_0\in(\widetilde p_c,p)$ gives a positive lower bound for $g_n(p)$ independent of $n$, and taking $n\to\infty$ gives $\theta(p)>0$. Combined with the contraction below $\widetilde p_c$, this proves $p_c=\widetilde p_c$ and the stated sharpness conclusion. This outline supplies the extra threshold argument rather than assuming the desired critical value.
If $p_c(2)>1/2$, sharpness would give $g_n(1/2)\leq C e^{-cn}$. But a square crossing starting somewhere on the left side entails an open connection from that starting <graph vertex> to distance $n$. A <union bound> over the $n+1$ possible starting <graph vertices> gives
$$
\frac12\leq h_{1/2}(n,n)\leq(n+1)g_n(1/2),
$$
contradicting exponential decay. Therefore $p_c(2)\leq1/2$, and the two directions establish
$$
\boxed{p_c(\mathbb Z^2)=\frac12,\qquad\theta_2(1/2)=0.}
$$
This is the <Harris-Kesten theorem>. The overall mechanism is that self-duality supplies a square crossing, the <RSW lemma> transports it between shapes and creates barriers at every scale, and sharpness converts the finite-scale crossing information into the exact threshold.
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