Can a mathematical proof replace experimentation? I know that this is very similar to How important is mathematical proof in physics? as well as Is physics rigorous in the mathematical sense? and The Role of Rigor. However, none of the answers to those questions really resolved my own question : 

Is there a case where mathematical proof can replace experimentation? 

Most of the answers I read seem to be saying that you can mathematically prove facts about a model, but not that reality corresponds to the model. You have to experimentally validate the assumptions of the proof which demand the conclusion as true. But what if the assumptions have already been experimentally validated? 
For example, if I show that if certain physical laws or accepted theories are true, a model must be (I'm not aware of such a proof, or if one exists), since the assumptions have been validated, do I still need to go through the trouble of experimentation? If we've shown it would be logically inconsistent for a conclusion to be false, and we take data that seems to be contradicting it, what's more likely to be false or mistaken - our logic, or our tools/experiment? I imagine that if scientists ever claimed to have found a right triangle in nature that violates Pythagorean's theorem, it would be more logical to assume they made a mistake.
The reason I ask this is because most, if not just about all of the ToEs in theoretical physics pretty much only have their mathematics going for them. The one most infamous for this is string theory. If string theory could be mathematically proven in the way I presented, and this proof was independently replicated and stood the test of time in the same way the Pythagorean theorem has, do we need to go through all the trouble of actually making an experiment?
 A: Building upon the mathematics of previously-tested models is indeed a thing that is done.  It's called engineering.  We do it literally all the time.
The difference is that in engineering, we are trying to make the best product we can within some constraints, while a scientist is theoretically seeking the truth.  Thus in engineering there's a whole slew of cost/benefit analysis that go on, which includes "what happens when our assumptions fail."
Want a great example of it failing?  The Tacoma Narrows bridge.  We used our math, we did the rigor, and our math was simply not in line with reality.
In engineering, we have a process called Verification and Validation (V&V).  Verification can be done mathematically, as you strive to make sure your model doesn't try to make right triangles that violate Pythagoreas' theorem (i.e. Is the equation being solved correctly?).  Validation is more about figuring out whether the model is indeed answering the questions desired correctly (i.e. Is the right equation being solved?).  We can never know for sure what reality has under the next rock unless we look under it. 
Any good pure-math science, like the ToE efforts underway right now, eventually are "graded" on their ability to make interesting predictions to go out and test.
Practically speaking, we find pure math and experimentation are entwined in a complicated dance like yin and yang.  There are aspects of science which are far more math than experimentation (like string theory), and other aspects which are more experimentation than math.  But they're always a mix
A: 
Is there a case where mathematical proof can replace experimentation?

Yes.
Every time you can prove that some proposition $P$ is implied by a premise $A$, and $A$ is experimentally verifiable, then you never need to experimentally verify $P$. Verifying $A$ is good enough.
As an example, Gauss' Law, $\oint E \cdot dA = \frac{Q}{\epsilon_0}$, can be proven by Coulomb's law: $E = \frac{Q \hat{r}}{4 \pi \epsilon_0 r^2}$, and vice versa. They are equivalent statements. Gauss' law is hard to verify in a lab (it's hard to measure the flux of electric field over an entire surface), but Coulomb's law is pretty easy to verify (it's easy to observe the inverse square law from a charge). Because it is proven that Coulomb's law implies Gauss' law, in principle, you need never verify Gauss' law directly; you may only ever verify Coulomb's law, and you will be exactly as confident in Gauss' law as you are in Coulomb's, because you have proof that it's implied by Coulomb's law.
Now, an objection to this example might be:

But this is just trivial. Since Gauss' law and Coulomb's law are equivalent, experimental verification of Coulomb's law is experimental verification of Gauss' law.

Okay, that's true, but without the proof of equivalence, it's not obvious at all. If we didn't have the proof that Coulomb's law implies Gauss' law, we would need to do experimental verification on both of them separately. And, because it's harder to verify Gauss' law than Coulomb's law, we would probably be less confident in the former than the latter. This is an example of a mathematical proof replacing experimental verification.
Now, my example takes a scenario where two statements are mutually implying, but this is true in general for when the implication of a proposition by a premise is one-sided, and you only need to experimentally verify the premise. Though I'm not sure if there are many examples of that.

In general, however, mathematical proofs cannot completely replace experimental validation; you always need experimental validation for any theory, and lots of it.
I just wanted to add this addendum, to justify why my answer is basically the opposite of  all other, well written, and highly upvoted answers. I think it's because they are more generally trying to address a misunderstanding of the OP, which is illustrated well in this quote:

The reason I ask this is because most, if not just about all of the ToEs in theoretical physics pretty much only have their mathematics going for them. The one most infamous for this is string theory. If string theory could be mathematically proven in the way I presented, and this proof was independently replicated and stood the test of time in the same way the Pythagorean theorem has, do we need to go through all the trouble of actually making an experiment?

Okay, so let me unpack this a little. You will never mathematically prove a physical theory. You can only prove a theorem, and theorems are simply maps between propositions: if proposition $A$ is true, then proposition $B$ is true. You cannot use a proof to create a proposition out of thin air. All physical theories must start with propositions (we call them "axioms" or "postulates"). A model is constructed by starting with postulates, then mathematically proving lots of consequences of those postulates. Generally you cannot prove the postulates. If you do, then they are no longer postulates, and you needed new postulates to do so anyway. (This generally happens when we move to a more general theory whose postulates are either simpler or have more explanatory power; for example, Maxwell's equations are postulates for classical electrodynamics, but quantum electrodynamics has broader postulates from which you can derive Maxwell's equations.)
For this reason, you will always need experimental verification. And, usually, it doesn't have the clean transitive power of implication that I described above. Usually postulates are very difficult to experimentally verify, and their consequences are much easier to verify. Above, I stated that if premise $A$ proves proposition $B$ and you can experimentally verify $A$, then you don't need to verify $B.$ But quite often (especially if $A$ is a postulate), $B$ is easier to verify than $A$. But verification of $B$ does not equivalently verify $A$. Rather, failure to verify $B$, or verifying that $B$ is false does verify that $A$ is false because of the implied contrapositive. (This is how, for example, the Aether theory of light was discredited by Michelson and Morley's experiment, by showing that one of its consequences is false.) 
Verifying $B$ alone doesn't necessarily verify $A$ because, there could be some other postulate, $C$ that also implies $B$. The only exception to this is if $A$ and $B$ are mutually implied, and thus, equivalent, like my Coulomb/Gauss law example. But, generally, to help build our "confidence" in some premise $A$, assuming we can't verify $A$ directly in the lab, we want to verify many of consequences of $A.$ Though we will never gain as much confidence in $A$ as we do in any of its consequences because, for every consequence $B$ of $A$, there could be some other set of premises that imply $B$. This is what makes verification of a scientific theory very difficult.
A: With your experiment you can only say that your experiments support the theory, not that it proves it. Experiments may agree with theories for many reasons, not always because the assumptions are correct. In areas in which lots of experiments give the predicted results you can have  large confidence that a theorem of your theory will agree with the experiment, but you can never be 100% sure. If you find a triangle that violates Pythagoras, it might be cause space is curved and non euclidean, like general relativity predicts.
A: A physical theory or model is based on  assumptions. By mathematical methods you the make predictions. Even if the mathematical part is sound, you will still need experimental results to verify or challenge your assumptions.
A: Perhaps logic cannot be used to confirm a theory but it can be used to refute it. 
Given and assumed a certain model of physical reality logic can be used to find a contradiction in that model leading to the conclusion that one should refuse the theory. In fact a logical contradiction, given the assumptions, corresponds to a physical contradiction. 
We are witnessing a similar state of affairs with the black hole information paradox. Although in that case we do not have a complete theory (gravity+quantum) to completely understand the phenomenon.
A: No.  Physics remains an experimental science and so it is not possible to replace experiment by a proof.  Descartes tried this when he proposed his theory of propagation of light - very elegant - but it predicted incorrectly that the angle would increase for light passing into an optically denser medium.   Indeed the story goes he refused to attend a demonstration that showed him wrong 
A rigorous proof is essential to properly understand and extend some aspects (and possibly some limits) of a theory, and to shed light on how phenomena can be linked and explained, but has no physical applications if it predicts something that contradicts experiment. 
————— 
Edit: There is a related discussion in this paper by David Mermin:

Mermin ND. What’s bad about this habit. Physics today. 2009 May 1;62(5):8-9.

A: I think (emphasis to be explained below) that the most important thing to realize in thinking about this question is that it is a question about physicists rather than about physics.  In other words, it's a question about the philosophy and practice of how we as humans do and think about science, rather than about the natural world itself.  
The philosophy of science is pretty well established nowadays.  We call it "the scientific method".  While it's important to realize that it is an unprovable philosophy, with potential pitfalls (scientists are known to make mistakes), you won't get far in trying to convince anyone to abandon it, because by and large it has proven pretty successful.  Though the scientific method takes many forms (see the Wikipedia page for discussion of this), it is pretty unambiguous about the need for experimental proof.  The scientific method is considered superior to alternative forms of acquiring knowledge (such as the pure reason of the ancient Greeks) essentially entirely because of its strict accountability to experiment.  There are many attractive theories which appeal to logic, but are considered false because they don't conform to reality.  There are many reasons for this: human fallibility, incomplete information, ego, politics, etc. 
As an illustrative anecdote, I've been enjoying an audiobook recently about the development of a field called "behavioral economics".  In the mid-20th century, economists had developed models of economic phenomena which assumed that all the people involved were perfectly rational and had perfect information.  This was extremely convenient, and the mathematical formalism led to many elegant results.  There was just one problem: Many of these elegant results were false.  For example, a prediction of standard economic theory was that bubbles could not exist in the stock market, which was dramatically disproven with the 2007-2008 market crash.  Curiously, not all economists were believers in this classical economic theory, and the dissidents had even done quite a lot of laboratory experiments to prove that people do not act as rationally as conventional economic models assumed.  The dissidents were called behavioral economists, and they borrowed tools from psychology to ascertain how people actually behaved in economic situations, rather than how they should behave to be mathematically logical and convenient.  To my surprise as a physicist, the majority of economists totally ignored behavioral economics for many decades, for no particularly good reason.  The data clearly showed that behavioral economics was experimentally solid, but due to mixtures of ego and the desire for a certain theory to be true, it was not until recently that such experimental data came to be considered mainstream. 
Such incidents as the above are not peculiar to economics.  Newton's endorsement of the particular theory of light famously led many physicists astray for decades until Fresnel and Young definitively proved the wave theory right.  Boltzmann cited this example when defending his own unpopular opinions about statistical mechanics, and in retrospect he was spot on.  Incidents like these are why almost all physicists would agree that all theory needs to be confirmed through experiments.  Humans are fallible, no matter how smart they think they are.  History is littered with instances of unproven-but-attractive hypotheses leading people astray for decades before being disproven (aether, static universe, maybe WIMPs, etc.)
Even when a proof seems watertight, it may be the case that there are factors we simply didn't account for in our proof.  Another story tells of Euler calculating how to do plumbing for his patron.  He worked out the math perfectly of course, but the contraption he designed didn't work at all.  The non-ideal factors Euler had left out of his model proved important.  
The fact that so many such errors continue to happen shows that in practice people are not living up to the standard they preach.  Especially for well-established theories, a mathy argument why something should be a certain way is often taken as gospel.  I think this is especially true of impossibility or "no-go" theorems.  E.g. the concept behind the "Levitron" toy was thought impossible by some--until somebody made one that worked.  The problem with no-go theorems is that, if you're trying to prove that something like levitation is impossible, it's really, really hard to consider every possible set of circumstances.  There are innumerable complicated effects in physics (like, in the case of the Levitron, gyroscopic precession) which can subtly effect your analysis, and to be really sure that you've accounted for every such effect is absurdly hard. 
To be clear about the previous point: The problem with a mathematical proof is not that nature will somehow prove the math wrong.  It is always that there may be a way to invalidate the hypotheses under which the proof was carried out.  So if we never tried to extend known mathematical results beyond the hypotheses under which we knew they were true, we would have no problem.  But being humans, we like to use intuition about things we understand to learn about things we don't understand.  This is also part of science.  And under such circumstances, until someone demonstrates otherwise (not "proves otherwise!"), there is no substitute for experiment. 
(Of course, this is just my thought on the matter.  To prove that I am right, you would have to do an experiment where e.g. two groups of randomly chosen researchers did research, one group using experiments to verify everything, and the other not.  Then see who ends up with the better theory.)
A: If you only make assumptions that have been experimentally verified (up to a high degree of precision) then a purely mathematical proof might be fine. However there are two problems with this:
1) Most of the time not all the assumptions can be experimentally verified (for example the axioms of Newtonian mechanics)
2) If you can only perform measurements up to some degree of precision then you are never really sure that it is correct.
A: Welcome to our flock! Your question touches on one of the fundamental problems in life, I think: How do we know that what we experience is real? 
I believe it is customery to distinguish between Mathematics on one side and Science (that is to say: Empirical Science) on the other:


*

*In Maths, we are dealing with absolute truth, but with a caveat: We always start with a set of axioms, and all logical conclusions from those are true if the axioms are true. So mathematical truth is absolute, when understood as the complex statement "If [axioms] then [conclusions]"

*In Science, on the other hand, we know that we can never prove the truth of a theory; the best we can do is disprove it: A theory is a (very educated) guess, which predicts something, that we can test in an experiment. If the experiment does not confirm the prediction, then we have proved that the theory was wrong, roughly speaking. So, the scientific method is a tool to filter out untruths, and the hope is that what we are left with after a long time, is something that is reasonably close to truth for all practical purposes.


So, to answer your question with a sweeping generalisation: No, maths can never replace empirical science, although the two complement and inform each other.
A: In the view of philosopher of science, Karl Popper, it is fundamentally impossible to prove/confirm any assumption or hypothesis about physics or the world in general. Starting from a set of assumptions (i.e. Newton's laws) a scientist can prove that IF this set of assumptions is valid, THEN certain outcomes should occur in the real world. 
If an experiment is conducted with a negative result, the prediction has been disproven and one or more of the assumptions from which it was derived must be incorrect. If a positive outcome "fails to disprove" the prediction, our confidence in the set of assumptions is increased relative to competing sets. Obviously, the confidence gain scales with how specific (and thus how "easy to disprove") the predictions are. However, no amount of positive outcomes can ever "prove" that the assumptions are correct. There could always be some other physical system, where the model fails. 
For example, starting from Newton's law of gravity, Victorian astronomers could predict the motion of the moon and other planets very accurately. Their models constantly "failed to be disproven". However, they eventually noticed that the orbit of Mercury did not behave as Newton would have predicted. Einstein's General Relativity provided a new set of assumptions, from which one can derive a highly specific prediction about Mercury's orbit (different from the Newtonian one) which "failed to be disproven" by the data. Using Einstein's assumptions, one can derive descriptions of the planetary orbits, which are almost equivalent (but slightly better) that the Newtonian ones. It even makes additional, novel predictions such as gravitational lensing - another highly specific, easily disproven claim, which nevertheless fails to be refuted by the data. None of this fundamentally proves the assumptions of general relativity to be correct/complete and its inability to describe the inside of black holes could be a sign that a better set of assumptions is needed. 
In short: Starting from a set of assumptions, one can derive predictions about the real world. If experiments prove the predictions (and thus the assumptions) wrong, one can pick a different (and hopefully better) set. Positive outcomes do not prove/experimentally validate the assumptions, but should inspire a good scientist to derive even more specific predictions from them and put them to increasingly stringent tests.
A: Science is very much about model-making. A model is a set of ideas within which mathematical proof may be possible, and used to show how one feature implies another within the model. But it is not possible to prove by mathematics that the model describes the physical world correctly. 
Your example of the right-angled triangle is a good one. Within the set of ideas of plane geometry, Pythagarus' theorem undoubtedly holds. But plane geometry does not describe spacetime.
Even if we had a most elegant and sophisticated theoretical model, one that appeared to be capable of capturing the nature of all physical phenomena, it would not be possible to prove that that appearance is reliable.
A related issue is the one in the foundations of logic---Godel's theorems. We can't even prove that mathematics itself is consistent! This is not the same issue as the one about model-making in science, but it illustrates the fact that here we rely on trust not proof. That is, we trust that mathematics is consistent.
At the level of basic physics further subtleties come into play. How do we know that the nature of the physical world can be captured in full by mathematics? We do not know that. This is not to say we should waste our time on idle speculation, but it is to encourage us to keep some awareness of the limits of what we know. 
A: 
Is there a case where mathematical proof can replace experimentation ?

No. Every physical model must be validated experimentally.  Besides those two things are not related at all. Pure mathematical theories proves something and physical experiments verifies theoretical models.
If you let me to make a joke: 
Does proving that you are not hungry, eliminates a test with food ?

if scientists ever claimed to have found a right triangle in nature that violates Pythagorean's theorem, it would be more logical to assume they made a mistake

No. It just means that this triangle is on non-Euclidean surface. For example, Pythagorean theorem fails on spherical triangles, however for that case one can find Pythagorean theorem analog :

$$ \cos c=\cos a\cos b\ $$
Here $C = \pi/2$
And one can easily draw such triangle on spherical surface, so that ALL it's angles would be $\pi/2$ ! :

Such triangle $A'B'C'$
And for your pleasure : Given that mathematically there exists infinite number of topologies,- there exists infinite number of Pythagorean theorem exceptions.
To raise your curiosity, one example of "creative" exception- triangle on wave-like surface :

Sorry, I'm not a good drawer, so my picture is not ideal, but you get the idea.
A: Yes. This is exactly what Einstein did in his 1905 paper on special relativity. He considered how coordinates are defined operationally and from this he derived results mathematically. Admittedly, Einstein used the empirical assumption of the constancy of the speed of light, but the logical (mathematical) argument does not depend on the physical properties of light, but only on the existence of a maximum speed in nature. The operational definition of coordinates depends on the maximum speed. We do not have to measure the maximum speed, because measurement of speed depend on the definition of coordinates. Consequently, all speeds can be expressed as fractions of that maximum speed.
More generally, we can derive all of relativity, special and general, from the general principle of relativity, that local laws of physics are the same irrespective of the reference matter which a particular observer uses to quantify them.
It is less well known, or understood, that von Neumann did the same for quantum mechanics. The mathematical structure of quantum mechanics depends only on the assumption that we can give probabilities for the results of measurements, given initial conditions, together with the observation that the results of measurements are not consistent with determinism.
Dirac put qm and sr together, in relativistic quantum mechanics and derived actual properties of electrons and photons. The argument can be extended to the standard model of particle physics.
In short, the general form of the whole of modern physics can be mathematically derived from general principles. Experiment is still needed but only to exclude discrete alternatives and to determine the values of fundamental parameters. I have shown how to do this in conceptual and mathematical detail in my books (see profile).
A: There is the case of the mathematical proof that gravity has to exist in the light of string theory:

String theory predicts the existence of gravitons and their well-defined interactions. A graviton in perturbative string theory is a closed string in a very particular low-energy vibrational state

See for more information this article.
Gravity was already observed before this "afterdiction". Did the observation of gravity come before this (still hypothetical) prediction of gravity? No. But of course, without other observations, the whole of string theory wouldn't exist.
That's one of the basic rules of the sciences. Theories have to be based on observation.
It could be that gravity somehow got into string theory a priori (which I don't know). 
I'm sure that there are examples of theories that predict new (in contrast to the example of gravity in string theory) unseen measurable physical processes, effects, constants, laws, etc. 
For example, in statistical thermodynamics, the fact that heat flows from hot to cold is predicted. This was already verified before the advent of statistical thermodynamics, but it could just as well have been the other way around, I guess.
You can question if the base of the statistical approach (the existence of atoms) could have been made without all of the physics that came before. But I think it could.
We will never be sure because physics developed the way it did, which isn't to say we can't make an educated guess.
A: Can it be that the basic assumptions of a mathematical theory have been experimentally validated, but the prediction of a derivation in that theory could be experimentally INvalidated?  Yes - besides all reasons already given, there is always the possibility that there is something in Nature which is NOT being modeled by the theory.  This is why, for example, some scientists keep looking for a so-called 'fifth force'.
On a more prosaic level: It is possible that some astronomical phenomenon is predicted by a mathematical derivation starting with General Relativity, but actual observations deviate from that prediction due to the presence of electromagnetic effects [which GR could not account for as it only addresses gravitation].  I mention this because most astronomers believe electromagnetism is negligible on a galactic scale, but at least one researcher has disputed this assumption.
A: I can think of at least one instance where scientific laws have been proven mathematically.
In mathematics you prove theorems by applying logic to the axioms. Axioms are facts which are assumed to be correct, without requiring any proof. Examples of axioms used by Peano include:
0 is a natural number.
For every natural number x, x = x.
For all natural numbers x and y, if x = y, then y = x

In nearly all cases, science does not use axioms. Instead, it uses observations to see how nature behaves.
There are however, a few instances where science comes close to using axioms, and in that case mathematics can be used to prove other facts. That proof then relies heavily on the fact that the assumptions (axioms) are correct.
A famous example is the law of conservation of energy, the first law of thermodynamics. Ever since the investigations of steam engines started by Carnot, it has been realised that the energy in an isolated system can be converted into other forms, but it cannot be created. This law was put on a much stronger footing by Emmy Noether when she published her conservation theorem in 1918. In this theorem, she proved mathematically that, as long as the laws of physics do not vary with time, the conservation of energy follows. The same goes for conservation of momentum (which follows if physical laws do not change with location) and other conserved quantities. For every symmetry there is a corresponding conservation.
In other words, here we treat very basic facts, like the invariance of physical law with respect to time or place, as axioms, and use maths to deduct our laws from that. 
A: Yes, you can. How much is the sum of all positive integers? -1/12, right? Is that true? Yes. Can you prove it empirically? no. Empirically you would get a larger number than the one before, theoretically you get a smaller number than the first positive integer.
A: 
Is there a case where mathematical proof can replace experimentation?

Yes, this happens all the time in real scientific settings. 
(Now, for those who disagree, please give this a full read before you downvote me.)
There are certain principles that are "dogma" among physicists. Some classic examples include:


*

*Causality

*Conservation of Energy


If someone was showing you some theory they came up with (maybe trying to resolve some known problems in a particular field), and you showed them that their math implies that conservation of energy is violated - this is equivalent among physicists as you telling the other person that they are wrong. 
If, in this example, the math is clear that "conservation of energy" is violated, physicists WOULD insist that doing an experiment to test this theory is a waste of time. 
Now I'm not saying these things as a criticism to physicists, but the reality is that there are some ideas that are so entrenched, that it would take an extraordinary amount of evidence to change physicists minds. And this is usually for good reason. Some concepts are so well-established (such as the "dogma" I mentioned above) that it is indeed a waste of time to second-guess them. 
So yes, in the end (as time goes to infinity), experiments and empirical evidence will always override mathematical theory. But in a practical setting, there are a lot of exceptions.
It is in some way a reason why some think that major changes only happen in giant "paradigm shifts" where the younger, more open-minded generation outlives the older generation's stubbornness. (I have heard it said that) the birth of quantum mechanics had this type of shift as much of the older generation refused to accept the new models that matched the experimental work.
