It's the molecular emission spectrum of PAH (polycyclic aromatic hydrocarbons) plus a blackbody. Both of these are known quite well to atmospheric scientists. The only thing to fit was how much of each to add and what the total flux normalization was.

(The blackbody is a smooth single hump, while the PAH spectrum are the jagged atomic and molecular lines coming from quantum mechanical effects, which have discrete energies.)

- "The only thing to fit was how much of each to add and what the total flux normalization was."

As well as assigning the blackbody temperature (T = 500 K). That's I understand three free parameters for three points?

- "The only thing to fit was how much of each to add and what the total flux normalization was."

And the assignment of blackbody temperature (T = 500 K). This is how many free parameters, fit to three points?

Look at https://www.researchgate.net/publication/1773854_Carriers_of... for a PAH spectrum. In context of what queuebert already wrote, it seems to make sense to me.

I'm asking how you sensibly fit three points to a model with at least three free parameters.

(I'm aware that the emission spectrum for (their specific chosen model of) PAH dust has a fixed shape. That's not my confusion).

Three points is the minimum necessary to fit three parameters, so as long as nothing is degenerate, you should be fine. Now that being said, fits to nonlinear models can be tricky, because you can get stuck in local minima, but there are techniques for avoiding that.

That the model can perfectly fit any data is a *problem* if we're asking, as this paper is, whether the model describes reality or not! It's not falsifiable!

Yes, it's falsifiable. If you cannot fit the model to the data, then the model must be wrong. That's the direction science technically works, but almost never how it is reported in the media. The fit here is considered a good thing, but cannot conclusively prove the model (as I think you are saying).

In principle, you could fit every single atomic/molecular spectrum to the data and falsify everything else based on some goodness of fit test, but no one does this.

Another thing about these molecular spectra is this: if there is an emission line missing, then it cannot be the spectrum you are fitting. These things do not change with environment, and can only be red or blue shifted due to special relativity. Every PAH spectrum must have the same lines. It's like a fingerprint.