Why the Squared Loss

The noise assumption, stated precisely

From likelihood to a sum

What survives

The negative log-likelihood under Gaussian noise contains the term $\frac{1}{2}\log(2\pi\sigma^2)$ summed over examples, along with the squared-error term. The first term is dropped when minimizing over $\mathbf{w}$ and $b$. Which reason is correct?

Under Gaussian noise with mean $\mathbf{w}^\top\mathbf{x} + b$ and variance $\sigma^2$, the density is proportional to $\exp\!\left(-\frac{1}{2\sigma^2}(y - \mathbf{w}^\top\mathbf{x} - b)^2\right)$. With $\sigma = 2$, the exponent for an observation with residual $y - \mathbf{w}^\top\mathbf{x} - b = 4$ equals ____.