Mathematical justification for zero padding?
Resolution (in the context of spectral analysis) is the ability to distinguish two or more closely spaced sinusoidal components in a spectrum. If you can't resolve them, this means that you only see one maximum instead of two or more. Resolution is determined by the window length, i.e., by the number of time domain samples (given a certain sampling rate). Windowing will smear out or broaden any narrow band components in the spectrum. Remember that windowing, i.e., multiplication in the time domain, corresponds to convolution in the frequency domain, and this convolution with the spectrum of a standard window is a sort of averaging operation in the frequency domain, hence broadening any (frequency domain) impulses.

Resolution can only be improved by adding more data points, i.e., more information, to your analysis window. Zero-padding obviously doesn't add any information. What zero-padding does is increase the sampling density of the discretet-time Fourier transform (DTFT) of the windowed signal. For a length $N$ signal, the DTFT is given by

$$\tilde{X}(\omega)=\sum_{n=0}^{N-1}x[n]e^{-jn\omega}\tag{1}$$

The discrete Fourier transform is

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{2}$$

Comparing $(1)$ and $(2)$, we see that the DFT is just a sampled version of the DTFT:

$$X[k]=\tilde{X}\left(\frac{2\pi k}{N}\right)\tag{3}$$

Zero-padding increases the number of equidistant samples taken from the DTFT $\tilde{X}(\omega)$.

A detailed analysis of the fact that zero-padding corresponds to interpolation in the frequency domain is given in this answer.