Is the number of columns in align environment restricted?

Question

I am using align environment to write First order conditions for Lagrangian.

This is how my code looks like:

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{derivative} 

\begin{document}

\begin{gather}
\begin{align}
&\left[u_t\right]: &\pdv{\mathcal{L}}{u_t} = 0 &\Rightarrow \theta^t \left[ R_t^k \overline{K}_t - \gamma^\prime (u_t)\overline{K}_t P_t^i \right] = 0  \nonumber \\
&&&\Rightarrow R_t^k = \gamma^\prime\left(u_t\right) P_t^i  \\
&\left[I_t\right]: &\pdv{\mathcal{L}}{I_t} = 0 &\Rightarrow \theta^t \left[ -P_t^i + \lambda_t  \left( 1- \tilde{S} \left(\frac{I_t}{I_{t-1}} \right) + I_t \left(- \tilde{S}^\prime \left(\frac{I_t}{I_{t-1}} \right) \frac{1}{I_{t-1}} \right) \right)\right] \nonumber  \\  
&&&+ \theta^{t+1} E_t \left[ \lambda_{t+1} I_{t+1} \left( -\tilde{S}^\prime \left( \frac{I_{t+1}}{I_t} \right) \frac{I_{t+1}}{I_t^2} (-1)\right) \right] = 0 \nonumber \\
&&&\Rightarrow P_t^i = \lambda_t \left( - \tilde{S} \left( \frac{I_t}{I_{t-1}} \right) - \tilde{S}^\prime \left( \frac{I_t}{I_{t-1}} \right) \frac{I_t}{I_{t-1}} \right)
\end{align}
\end{gather}

\end{document}

In the sample above i have 3 columns in align environment. The only thin I would like to do is to align + sign on the forth line with $ \theta^t $ on the third line. For this, I am adding the forth column by adding & on every line. However, what happens is that everything aligns on the left hand side like this:

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{derivative}

\begin{document}

\begin{gather}
\begin{align}
&\left[u_t\right]: &\pdv{\mathcal{L}}{u_t} = 0 &\Rightarrow &\theta^t \left[ R_t^k \overline{K}_t - \gamma^\prime (u_t)\overline{K}_t P_t^i \right] = 0  \nonumber \\
&&&\Rightarrow &R_t^k = \gamma^\prime\left(u_t\right) P_t^i  \\
&\left[I_t\right]: &\pdv{\mathcal{L}}{I_t} = 0 &\Rightarrow &\theta^t \left[ -P_t^i + \lambda_t  \left( 1- \tilde{S} \left(\frac{I_t}{I_{t-1}} \right) + I_t \left(- \tilde{S}^\prime \left(\frac{I_t}{I_{t-1}} \right) \frac{1}{I_{t-1}} \right) \right)\right] \nonumber  \\  
&&&&+ \theta^{t+1} E_t \left[ \lambda_{t+1} I_{t+1} \left( -\tilde{S}^\prime \left( \frac{I_{t+1}}{I_t} \right) \frac{I_{t+1}}{I_t^2} (-1)\right) \right] = 0 \nonumber \\
&&&\Rightarrow &P_t^i = \lambda_t \left( - \tilde{S} \left( \frac{I_t}{I_{t-1}} \right) - \tilde{S}^\prime \left( \frac{I_t}{I_{t-1}} \right) \frac{I_t}{I_{t-1}} \right)
\end{align}
\end{gather}

\end{document}

I am really confused why does align do this. It seems to me I am doing everything well. Is a way to have + on the third align aligned with $ \theta^t $ and have all the equations centered at the same time.

I have tried all possible combinations for the last 2 hours but could not find any solution. So, would be grateful if more advanced and experienced user of align environment suggested his/her solution.

Update: It seems that I solved the problem by removing & from the beginning of each line, but the it looks still not elegant. It seems align environment aligns each part of the line to the left with each column. Is there a way to align in the center of each column?

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{derivative} 



\begin{document}


\begin{gather}
\begin{align}
\left[u_t\right]: &\pdv{\mathcal{L}}{u_t} = 0 &\Rightarrow &\theta^t \left[ R_t^k \overline{K}_t - \gamma^\prime (u_t)\overline{K}_t P_t^i \right] = 0  \nonumber \\
&&\Rightarrow &R_t^k = \gamma^\prime\left(u_t\right) P_t^i  \\
\left[I_t\right]: &\pdv{\mathcal{L}}{I_t} = 0 &\Rightarrow &\theta^t \left[ -P_t^i + \lambda_t  \left( 1- \tilde{S} \left(\frac{I_t}{I_{t-1}} \right) + I_t \left(- \tilde{S}^\prime \left(\frac{I_t}{I_{t-1}} \right) \frac{1}{I_{t-1}} \right) \right)\right] \nonumber  \\  
&&&+ \theta^{t+1} E_t \left[ \lambda_{t+1} I_{t+1} \left( -\tilde{S}^\prime \left( \frac{I_{t+1}}{I_t} \right) \frac{I_{t+1}}{I_t^2} (-1)\right) \right] = 0 \nonumber \\
&&\Rightarrow &P_t^i = \lambda_t \left( - \tilde{S} \left( \frac{I_t}{I_{t-1}} \right) - \tilde{S}^\prime \left( \frac{I_t}{I_{t-1}} \right) \frac{I_t}{I_{t-1}} \right)
\end{align}
\end{gather}

\end{document}

Answer 1

I think you need only one column of & alignment points.

Observe the use of explicit parenthesis-sizing instructions and of \mleft and \mright in lieu of \left and \right; they serve mainly to conserve on horizontal whitespace.

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,mleftright,derivative}

\begin{document}
\begin{align}
[u_t]: \quad \pdv{\mathcal{L}}{u_t} = 0 
&\Rightarrow \theta^t \bigl[ R_t^k \overline{K}_t - 
 \gamma'(u_t)\overline{K}_t P_t^i \, \bigr] = 0  \nonumber \\
&\Rightarrow R_t^k = \gamma'(u_t) P_t^i  \\
[I_t]: \quad \pdv{\mathcal{L}}{I_t} = 0 
&\Rightarrow \theta^t \biggl\{ -P_t^i + \lambda_t  
 \mleft[ 1- \tilde{S} \Bigl(\frac{I_t}{I_{t-1}} \Bigr) + I_t 
 \mleft(- \tilde{S}'\Bigl(\frac{I_t}{I_{t-1}} \Bigr) \frac{1}{I_{t-1}} \mright) 
 \mright]\biggr\} \nonumber  \\  
&\qquad + \theta^{t+1} E_t \biggl[ \lambda_{t+1} I_{t+1} 
 \mleft( -\tilde{S}'\Bigl( \frac{I_{t+1}}{I_t} \Bigr) 
 \frac{I_{t+1}}{I_t^2} (-1)\mright) \biggr] = 0 \nonumber \\
&\Rightarrow P_t^i = \lambda_t \mleft[ - \tilde{S} 
 \Bigl( \frac{I_t}{I_{t-1}} \Bigr) - 
 \tilde{S}'\Bigl( \frac{I_t}{I_{t-1}} \Bigr) \frac{I_t}{I_{t-1}} \mright]
\end{align}
\end{document}