Could the Heisenberg Uncertainty Principle turn out to be false? While investigating the EPR Paradox, it seems like only two options are given, when there could be a third that is not mentioned - Heisenberg's Uncertainty Principle being given up.
The setup is this (in the wikipedia article): given two entangled particles, separated by a large distance, if one is measured then some additional information is known about the other; the example is that Alice measures the z-axis and Bob measures the x-axis position, but to preserve the uncertainty principle it's thought that either information is transmitted instantaneously (faster than light, violating the special theory of relativity) or information is pre-determined in hidden variables, which looks to be not the case.
What I'm wondering is why the HUP is not questioned? Why don't we investigate whether a situation like this does indeed violate it, instead of no mention of its possibility? Has the HUP been verified experimentally to the point where it is foolish to question it (like gravity, perhaps)?
Edit
It seems that all the answers are not addressing my question, but addressing waveforms/commutative relations/fourier transforms. I am not arguing against commutative relations or fourier transforms. Is not QM the theory that particles can be represented as these fourier transforms/commutative relations? What I'm asking this: is it conceivable that QM is wrong about this in certain instances, for example a zero state energy, or at absolute zero, or in some area of the universe or under certain conditions we haven't explored? As in:
Is the claim then that if momentum and position of a particle were ever to be known somehow under any circumstance, Quantum Mechanics would have to be completely tossed out? Or could we say QM doesn't represent particles at {absolute zero or some other bizarre condition} the same way we say Newtonian Physics is pretty close but doesn't represent objects moving at a decent fraction of the speed of light?
Example
EPR Paradox: "It considered two entangled particles, referred to as A and B, and pointed out that measuring a quantity of a particle A will cause the conjugated quantity of particle B to become undetermined, even if there was no contact, no classical disturbance."
"According to EPR there were two possible explanations. Either there was some interaction between the particles, even though they were separated, or the information about the outcome of all possible measurements was already present in both particles."
These are from the wikipedia article on the EPR Paradox. This seems to me to be a false dichotomy; the third option being: we could measure the momentum of one entangled particle, the position of the other simultaneously, and just know both momentum and position and beat the HUP. However, this is just 'not an option,' apparently.
Clarification
I'm not disputing that two quantities that are fourier transforms of each other are commutative / both can be known simultaneously, as a mathematical construct. Nor am I arguing that the HUP is indeed false. I'm looking for justification not just that subatomic particles can be models at waveforms under certain conditions (Earth like ones, notably), but that a waveform is the only thing that can possibly represent them, and any other representation is wrong. You van verify the positive all day long, that still doesn't disprove the negative. It is POSSIBLE that waveforms do not correctly model particles in all cases at all times. This wouldn't automatically mean all of QM is false, either - just that QM isn't the best model under certain conditions. Why is this not discussed?
 A: The Heisenberg's relation is not tied to quantum mechanics. It is a relation between the width of a function and the width of its fourier transform. The only way to get rid of it is to say that x and p are not a pair of fourier transform: ie to get rid of QM.
A: In precise terms, the Heisenberg uncertainty relation states that the product of the expected uncertainties in position and in momentum of the same object is bounded away from zero. 
Your entanglement example at the end of your edit does not fit this, as you measure only once, hence have no means to evaluate expectations. You may claim to know something but you have no way to check it. In other entanglement experiments, you can compare statistics on both sides, and see that they conform to the predictions of QM. In your case, there is nothing to compare, so the alleged knowledge is void.
The reason why the Heisenberg uncertainty relation is undoubted is that it is a simple algebraic consequence of the formalism of quantum mechanics and the fundamental relation $[x,p]=i\hbar$ that stood at the beginning of an immensely successful development. Its invalidity would therefore imply the invalidity of most of current physics. 
Bell inequalities are also a simple algebraic consequence of the formalism of quantum mechanics but already in a more complex set-up. They were tested experimentally mainly because they shed light on the problem of hidden variables, not because they are believed to be violated.
The Heisenberg uncertainty relation is mainly checked for consistency using Gedanken experiments, which show that it is very difficult to come up with a feasible way of defeating it. In the past, there have been numerous Gedanken experiments along various lines, including intuitive and less intuitive settings, and none could even come close to establishing a potential violation of the HUP. 
Edit: One reaches experimental limitations long before the HUP requires it. 
Nobody has found a Gedankenexperiment for how to do defeat the HUP, even in principle. We don't know of any mechanism to stop an electron, thereby bringing it to rest. It is not enough to pretend such a mechanism exists; one must show a way how to achieve it in principle. For example, electron traps only confine an electron to a small region a few atoms wide, where it will roam with a large and unpredictable momentum, due to the confinement. 
Thus until QM is proven false, the HUP is considered true. Any invalidation of the foundations of QM (and this includes the HUP) would shake the world of physicists, and nobody expects it to happen.
A: In quantum mechanics, two observables that cannot be simultaneously determined are said to be non-commuting.  This means that if you write down the commutation relation for them, it turns out to be non-zero.  A commutation relation for any two operators $A$ and $B$ is just the following  $$[A, B] = AB - BA$$  If they commute, it's equal to zero.  For position and momentum, it is easy to calculate the commutation relation for the position and momentum operators.  It turns out to be $$[\hat x ,\hat p] = \hat x \hat p - \hat p \hat x = i \hbar$$  As mentioned, it will always be some non-zero number for non-commuting observables.  So, what does that mean physically?  It means that no state can exist that has both a perfectly defined momentum and a perfectly defined position (since $ |\psi \rangle$ would be both a right eigenstate of momentum and of position, so the commutator would become zero.  And we see that it isn't.).
So, if the uncertainty principle was false, so would the commutation relations.  And therefore the rest of quantum mechanics.  Considering the mountains of evidence for quantum mechanics, this isn't a possibility.  
Addition
I think I should clarify the difference between the HUP and the classical observer effect.  In classical physics, you also can't determine the position and momentum of a particle.  Firstly, knowing the position to perfect accuracy would require you to use a light of infinite frequency (I said wavelength in my comment, that's a mistake), which is impossible.  See Heisenberg's microscope.  Also, determining the position of a particle to better accuracy requires you use higher frequencies, which means higher energy photons.  These will disturb the velocity of the particle.  So, knowing the position better means knowing the momentum less.  
The uncertainty principle is different than this.  Not only does it say you can't determine both, but that the particle literally doesn't have a well defined momentum to be measured if you know the position to a high accuracy.  This is a part of the more general fact in quantum mechanics that it is meaningless to speak of the physical properties of a particle before you take measurements on them.  So, the EPR paradox is as follows - if the particles don't have well-defined properties (such as spin in the case of EPR), then observing them will 'collapse' the wavefunction to a more precise value.  Since the two particles are entangled, this would seem to transfer information FTL, violating special relativity.  However, it certainly doesn't.  Even if you now know the state of the other particle, you need to use slower than light transfer of information to do anything with it.
Also, Bell's theorem, and Aspect's tests based off of it, show that quantum mechanics is correct, not local realism.
A: The wave formulation has in its seed the uncertainty relation.
Let me be precise what is meant by the wave formulation: the amplitude over space points will give information about localization on space, while amplitude over momenta will give information about localization in momentum space. But for a function, the amplitude over momenta is nothing else but the Fourier transform of the space amplitude. 
The following is jut a mathematical fact, not up to physical discussion: the standard deviation, or the spread of the space amplitude, multiplied by the spread of the momenta amplitude (given by the Fourier transform of the former) will be bounded from below by one.
So, it should be pretty clear that, as long as we stick to a wave formulation for matter fields, we are bound mathematically by the uncertainty relation. No work around over that.
Why we stick to a wave formulation? because it works pretty nicely. The only way someone is going to seriously doubt that is the right description is to either:
1) find an alternate description that at least explains everything that the wave formulation describes, and hopefully some extra phenomena not predicted by wave formulation alone.
2) find an inconsistency in the wave formulation. In fact, if someone ever manages to measure both momenta and position for some electron below the Planck factor, it would be definitely an inconsistency in the wave formulation. It would mean we would have to tweak the De Broglie ansatz or something equally fundamental about it. Needless to say, nothing like that has happened
A: You are asking if a more complete theory might show that HUP is wrong and that position and momentum do exist simultaneously. But a more complete theory has to explain all the observations that QM already explains, and those observations already show that position and momentum cannot have definite values simultaneously. This is known because when particles such as photon, electrons, or even molecules are sent through a pair of slits one at a time, an interference pattern on the detector plate appears that shows that the probability of the measured location and time follows a specific mathematical relationship. The fact that certain regions have zero probability shows that before measurement, the particles exist in a superposition of possible states, such that the wave function for those states can cancel out with other states resulting in areas of low probability of observation. The observed relationships through increasingly complex experiments rules out possibilities other than what is described by QM. The only way that QM could be superseded by a new theory is for new observations to be made that violate QM, but the new theory would still result in the same predictions as QM in the circumstances that QM has already been tested. Since HUP results directly from QM, HUP would also follow from a new theory with the only possible exception in conditions such as super high energy conditions such when a single particle is nearly a black hole.
Basically you have to get used to the idea that particles are really quantized fluctuations in a field and that the field exists in a superposition of states. Any better theory will simply provide additional details about why the field behaves in that way.
A: If we want the position and the momentum to be well-defined at each moment of time, the particle has to be classical. We inherited these notions from classical mechanics, where they apply successfully. Also they apply at macroscopic level. So, it is a natural question to ask if we can keep their good behavior in QM. Frankly, there is nothing to stop us to do this. We can conceive a world in which the particles are pointlike all the time, and move along definite trajectories, and this will "beat HUP". This was the first solution to be looked for. Einstein and de Broglie tried it, and not only them. Even Bohr, in his model, envisioned electrons as moving along definite trajectories in the atom (before QM). David Bohm was able to develop a model which has at a sub-quantum level this property, and in the meantime behaves like QM at quantum level. The price to be paid is to allow interactions which "beat the speed of light", and to adjust the model whenever something incompatible with QM was found. IMHO, this process of adjustments still continues today, and this looks very much like adding epicycles in the heliocentric model. But I don't want to be unfair with Bohm and others: it is possible to emulate QM like this, and if we learn QM facts which contradict it, it will always be possible to find such a model which behaves like QM, but also has a subquantum level which consists of classical-like point particles with definite positions and momenta. At this time, these examples prove that what you want is possible. One may argue that they are unaesthetic, because they are indeed more complicated than QM. But this doesn't mean that they are not true. Also, at this time they don't offer anything testable which QM can't offer. So, while QM describes what we observe, the additional features of hidden variable theories are not observable, more complicated, and violate special relativity. Or, if they don't violate special relativity, they contradict what QM predicts and we observed in experiments of entanglement like that of Alan Aspect. If EPR presents us with two alternatives, (1) spooky action at distance, (2) QM is incomplete, and that you propose, (3) HUP is false, let's not forget that Aspect's experiment and many others confirmed the alternative (1).
Now, it would be much better for such models if they would stop adjusting themselves to mimic QM, and predict something new, like a violation of HUP. This would really be something.
In conclusion, yes, you are right and in principle it is possible to beat HUP. The reason why most physicists don't care too much about this, is that the known ways to beat HUP are ugly, have hidden elements, violate other principles. But others consider them beautiful and useful, and if you are interested, start with Bohm's theory and the more recent developments of this.

Update
Synopsis: The Certainty of Uncertainty
Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements (arXiv link)
A: "Heisenberg uncertainty principle" is a school term that is used in popular literature. It simply does not matter. What matters is the wavefunction and Schroedinger equation.
The EPR paradox experiment never used any explicit "uncertainty principle" in the proof.
A: The only way to make Heisenberg's principle irrelevant is to measure the speed and the position (to make it simple) of a fundamental particle. 
In other words, you would have to observe a particle, without having it collide with a photon or reacting to a magnetic force, or without interacting with it.
There might be an other way, which would be to find a very general law (but not statistical) which describes the characteristics (spin, speed, position etc) of an elemental particle in an absolute way....
A: if can help
Open timelike curves violate Heisenberg's uncertainty principle
http://arxiv.org/pdf/1206.5485v1.pdf
...and show that the Heisenberg uncertainty principle between canonical variables, such as position and momentum, can be violated in the presence of interaction-free CTCs....
Foundations of Physics, March 2012, Volume 42, Issue 3, pp 341-361
http://arxiv.org/pdf/1008.0433v2.pdf
...considering that a D-CTC-assisted quantum computer can violate both the uncertainty principle...
Phys Rev Lett, 102(21):210402, May 2009.
arxiv 0811.1209
...show how a party with access to CTCs, or a "CTC-assisted" party, can perfectly distin-
guish among a set of non-orthogonal quantum states....
Phys. Rev. A 82, 062330 2010.
arxiv 1003.1987v2
...and can be interacted with in the way described by this simple model, our results conﬁrm
those of Brun et al that non-orthogonal states can be discriminated...
...Our work supports the conclusions of Brun et al that an observer can use interactions with a CTC to allow them to discriminate unknown, non-orthogonal quantum states – in contradiction of the uncertainty principle...
.
A: Without gravity: The uncertainty principle is not really a principle because it is a derivable statement, it is not postulated. It is derivable and proven mathematically. Once you prove something you cannot unprove it. That means it cannot turn out to be false. For experimental verifications, see for example this article by Zeilinger et al and the references inside. Zeilinger is a world expert on quantum phenomena and it is expected that he will get Nobel prize in the future.
With gravity, (and that matters only at extremely high energy, as high as the Planck scale): Intuitively you can use the uncertainty principle to give an estimate about the energy needed to resolve tiny region of space. For sufficiently small region in space you will create a black hole. So there is a limit on the spacial resolution one can achieve, because of gravity. If you try to use higher energy you will create a bigger black hole. Bottom line is, uncertainty principle does not make sense in this case because space loses its meaning and it cannot be defined operationally. 
A: The way I see it, HUP cannot be disproven "at absolute zero", because absolute zero cannot be physically reached, er... due to HUP... is circular reasoning good enough? Let's try something else.
Maybe try to imagine what would happen if HUP was to be violated? For one, I guess the proton - electron charges would cause one or two electrons to fall down into the nucleus, as HUP normally prevents that (if the electron fell down on nucleus we'd know it's position with great precision, requiring it to have indeterminate but large momentum, so it kind of settles for orbiting around nucleus). 
If you know more about the stuff than I do, try to imagine what else would happen, and how likely is that effect. For example, if HUP violation would imply violation of 2nd law of thermodynamics, this would render HUP violation pretty unlikely.
That much from a layman.
A: You need to understand that the Heisenberg Principle of Uncertainty is not in any way a result of technological limitations. This principle is an inherent aspect of nature. It is an undeniable truth. Just as justifiable as any other established property of nature. E.g. that quarks have mass. 
I know that quantum mechanics is frighteningly counter-intuitive on a number of points. But you must stop trying to fight against something that is a property of nature, since the more you try the more you will fail. You cannot disprove something that is self evidently true. Just accept it as an intrinsic property of nature.
A: The Heisenberg uncertainty principle forms one of the most important pillars in physics. It can't be proven wrong because too many experimentally determined phenomena are a result of the uncertainty principle. However, something may be discovered in the future that can make a modification to the uncertainty principle - in a similar way that Newton's laws were modified by Einstein's special relativity. Saying that the uncertainty principle is wrong is like saying that Newton's law is wrong.
In reply to the comments,
I'm not saying that it can be falsified. It can't. In a classical sense, it will always be correct, in a similar way that Newton's law will always be correct.However, it can be modified. Until the day that all the open questions in physics have been resolved, how can you claim that the uncertainty principle can't be modified further?  Do we know everything about extra dimensions? Do we know everything about string theory and physics at the Planck scale?
By the way,  it has already been modified. 
Please check this link.
http://arxiv.org/abs/1106.0715
The uncertainty principle will always be correct. However, it can and has been modified. In its current formalism and interpretation,  it could represent a special case of a larger underlying theory. 
The claim that the current formalism and limitations to the uncertainty principle are absolute and can never be modified under any circumstance in the universe, is a claim that does not obey the uncertainty principle itself. 
