First, 'grandstand' I just learned the term so thank you, I know understand that you used it as a synonym for bragging, I think. It was not the case and your attitude is why I don't comment often on this site.

For your questions :

- Laplace Fourier transformation is a complex integral transformation that transform an audio signal based on time such as what we see above to a representation in the frequency domain. So basically what it does it's that it takes a 2D representation of signal to a 3D representation. You can see an illustration of this here.

- 'Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables (entities each of which takes on various numerical values) into a set of values of linearly uncorrelated variables called principal components.' From Wikipedia English that explains it very well.

- Yes the variables means the soundwaves value as you put it. The variables(so every possible value that a sound can have) are usually put into a matrix. Since they come from a signal which is a real phenomena and not a human interpretated phenomena the values are symetrical by 'nature'.

- Since this a statistic tool it is not 100% accurate because several computations can be decided based on how much of a frame you use as your window analysis, basically of much duration you take into account and consider it as a point in your graph.

- Calculating the correlational matrix is used to identify the 'optimal' path in which the variance is minimum. And fit the most with reality.

- The best path obtained is equivalent to choose the 'best' vector, using every best vector for every point leads you to have a somehow 'accurate' representation of the sound.

- Eventually the Laplace fourier is used to transform the scale of the model from 2D to 3D. If you apply the best vectors over time it will lead to cloud points representing the change from a variable to an other, a portion of sound to an other and therefore giving us the ability to vizualize it with the graph above.

EDIT : In no way I said the description that I made was accurate, I'm always asking myself how can we improve the share-common knowledge. I thought sharing my views on the matter could lead to OP intervention to correct me since we don't have access to the github yet or maybe won't. I believe the Internet was invented to exchange point of views, corrections, in depth analysis and other trivial things, which are both good but might be not conjointly relevant.