You didn't really say anything the article didn't say.
"This could mean nothing less than the fall of the Standard Model of particle physics (SM)"
No, it means that the SM is an approximation to a deeper theory, which everybody already believes anyway and has for several decades. Models don't "fall" just because they turn out to be approximations.
"The SM describes what the building blocks of the universe are and how they work, and from there, at least in principle, explains every other phenomenon in nature."
No, it doesn't. The SM doesn't include gravity. It also has twenty-six free parameters that have to be determined from experiments; it doesn't explain why those parameters take the values they do. So it's obviously incomplete, and everybody has known that for several decades.
(Yes, the article does go on to talk about the SM being incomplete. But that just means it contradicts itself.)
In short: the article talks like this discovery is somehow a huge change in how everybody views the SM. In fact it's business as usual: we built experiments to try to discover new physics to help us extend a model we knew was incomplete, and now those experiments are bearing fruit. This is normal science. It is not some huge rift in particle physics.
and
>The SM doesn't include gravity. It also has twenty-six free parameters that have to be determined from experiments; it doesn't explain why those parameters take the values they do.
26 free parameters. Yikes.
At what point does the addition of parameters and particles break through from "yet another particle we expected" to "this is completely unexpected and we need to seriously modify our understanding". Obviously with geocentric vs heliocentric it took a major shift in understanding before epicycles crumbled. Same with newton to einstein -- a major shift in understanding before minor errors were explained.
Aside from the SA hyperbole, do you think there is any evidence that particle physics is reaching critical mass in terms of convoluted explanations?
Of course, it could be the case that nature is just damn complex at the particle level.
I don't think a shift like this is precipitated by exceeding some limit on free parameters. It's precipitated by a new discovery not fitting with current models. When the Higgs was discovered, it was just a confirmation of what the SM already predicted; it filled in another free parameter in the SM (the Higgs mass, which had only been roughly estimated before), but it didn't point the way to any new, more fundamental model. This new evidence, if it pans out, does point the way to some new, hopefully more fundamental model, because it's something that isn't included in our current models. That is true even if it has no effect at all on the number of free parameters in our theories.
(Btw, the 26 free parameters in the SM sounds like a lot, but since it includes all of the fundamental interactions except gravity, those 26 are really just a collection of all the free parameters from previous theories of those same interactions. For example, the mass and charge of the electron were already free parameters in electrodynamics, so the SM didn't add them on out of nowhere.)
Also, as I've commented already, the SM has been understood to be incomplete for several decades. Everybody already expects that there will have to be a significant shift in our understanding when we go to the next level down. We just don't know what that shift will be, because up to now we've had no experimental data on the question. Now we might finally be getting some.
I agree that the number of parameters needed is less than the number of experimental measurements though, I confused that.
This is true, in practice we can't do all of the derivations from first principles. But AFAIK this is an issue of, as you say, computational power; it is not an issue of theoretical insufficiency (at least within the domain that the SM covers, i.e., the non-gravitational interactions in flat spacetime with gravity being negligible). We know what equations we would have to solve; we just can't solve them analytically and we don't have the computing power to solve them numerically in all cases.
"Originally theorists thought that the SM would be an approximation of a more fundamental theory that would be quickly discovered. This is what has always happened in the past."
No, what has always happened in the past is that it's taken a long time for the next more fundamental theory to arise. The article gives Newton's theory of gravity as an example. Newton published the Principia in 1687 (and discovered the key elements of the theory a couple of decades before that). Einstein published General Relativity in 1915. That's not a "quick" discovery of the next more fundamental theory.
As another example, Maxwell's Equations were published in the early 1860s. Quantum electrodynamics, the next more fundamental theory, was published in the late 1940s. Also not a "quick" discovery.
Can SM not represent integers?
No. The point of physical models is not to be able to answer all possible questions that you can ask about the models, that as you point out, is not possible.
However, the point of a physical model is to simulate reality. This does not entail being able to answer arbitrary questions (in finite time) about it.
Consider building something like a Nintendo Emulator. As a first pass, you might simply say "I just need to specify the registers and assembly instructions." You might miss out some assembly instructions, and when you pass in a program that uses them, your emulator would crash. You then add that in, and might find out that in order to do faithful reproduction, you need to model instruction execution time and so forth.
In short, your building an emulator requires you to figure out
1) What the state of the Nintendo system needs to be (this may include a time variable).
2) How to correctly handle state transitions.
However, even after you get a complete model of the Nintendo, this obviously doesn't let you answer questions like "If I were to pass in an arbitrary program to my emulator, would it halt on arbitrary inputs?"
Physics is the same. We want to figure out how to specify the "state" of the system, and how that state evolves with time. When we say that a model is incomplete, it means that we're missing some assembler instructions, a register, or something else that doesn't allow us to faithfully simulate reality.
Discovering a new particle is like discovering that there are a bunch of new registers and associated instructions that we need to account for.
The person I responded to said that SM was incomplete, but that must be the case, unless the model is self-contradictory (assuming it's not, as it would not qualify as a scientific theory if it were), or, the model must not have enough arithmetical complexity to fall under Gödel's rubric.