What PCA Throws Away

A dimensionality-reduction pipeline is fitted on this training set of $5$ samples with $2$ features:

The pipeline runs the four standard steps:

1. Centre the data: $c_i = x_i - \bar{x}$.
2. Build the scatter matrix $S = \sum_{i=1}^{5} c_i\, c_i^{\top}$ (a plain sum of outer products, with no division).
3. Take $u$, the unit eigenvector of $S$ with the largest eigenvalue, as PC1.
4. Compress each sample to the single number $t_i = u^{\top} c_i$.

Each sample is then reconstructed back into $2$-D as $\hat{x}_i = \bar{x} + t_i\, u$.

What is the total squared reconstruction error over the whole training set?

$E = \sum_{i=1}^{5} \left\| x_i - \hat{x}_i \right\|^{2}$