You are setting up a fallacy here. You are saying here that because GR is supported BY equal amounts of empirical evidence as QM is, then they are both equally valid. In fact you even somewhat undermine QM by saying that because it is based more on mathematical formalism than measurement and observation and GR is based on hard empirical observation and measurement, GR is somehow is more credible. I reply, that no amount of empirical evidence and successful trials can ever prove a theory correct, but it requires only one single unsuccessful trial to falsify the theory.
As this part is perhaps the most important part of your last post, and goes to the heart of your argument in many ways, I hope you don't mind that I place it first, rather than address your points/responses in order.
I'm not saying that relativity and QM are supported by equal amounts of evidence. The issue is much more complex than this. But I also wonder why you think that the experimental and empirical evidence behind QM somehow falsifies relativity because the measurements in these experiments (those like Aspect's and Gisin's) are less exact and more problematic than the experimental evidence behind GR.
Additionally, I don't think you understand (or perhaps I was too unclear) what I meant by the problems created because of the relationship between the formalism and theoretical framework behind QM, and the methods used in experiments. So I'll use a clearer (I hope) example for illustration.
The wavefunction is a probability function. But obviously there are all sorts of probability functions. So I'll use probability and measurement (which are the foundations for the experimental results of Aspect and others) to demonstrate what I meant. For the sake of clarity and simplicity, I'll sacrifice accuracy (mathematical rigor/precision) and approximate the technical details.
In a number of fields, ranging from anthropology to various disciplines within physics (and basically all the social sciences), the most important probability function is that which is behind the "normal" distribution or normal probability curve (that common bell-shaped graph which appears everywhere). Thanks largely to Laplace and Gauss, certain things about this probability function and its relationship to others have been mathematically proven. Two results will suffice here.
The first is that, given some normally distributed variable X, we can determine with exact precision the probability that any given
x from the population (i.e.,
x is an element of X) will be within some arbitrarily small interval. This is the basis for a great deal of hypothesis testing, because it means that I know that the probablity that a randomly selected
x is an element of X will be within just two standard deviations from the mean is 95%, which is huge.
Even better (here's were the work by Gauss and Laplace becomes extremely important, along with the work by Karl Pearson), is how the normal distribution relates to any probability function/probability curve describing some population. Although the probability of sampling single values/elements/members from this population and getting one within 2 standard deviations from the mean may not be anywhere near 95% (i.e., if the curve is very skewed, or has multiple "humps", unless I know the probability curve in advance calculating the probability of a single selection is impossible), things change if I increase my sample size. It has been mathematically proven that, with any probability curve, even one heavily skewed or one in which almost all the population rests at both of the "tails" of the curve, by sampling sufficiently large samples from this population the
mean (average) of the samples will approximate a normal distribution. Just about any intro to statistics textbook out there will not only state this (and usually alongside an example), but will proceed to show over several chapters (perhaps the rest of the textbook) how this can be used to confirm hypotheses with samples of merely 20 people when the target population is millions (note: I use people here only because I have found that it is easier conceptually to explain using people rather than arbitrary values/measurement/etc. when talking about populations).
How wonderful! This means that armed with my probability function of a normal distribution, I can sample from any population and know that my samples will approximate the normal distribution
such that my mathematical models will tell me about things I can't observe directly (as in QM).
Except there's these two little problems. First, although the examples in textbooks of samples of 40 being quite adequate are all well and good, that's not real life. In reality, however, even arbitrarily departures from normality (which are probably the norm) can drastically change the validity of one's results. Secondly, assuming a symmetric probability curve (even a normal one), all we know really know about the mean (average) is that it's in the middle. In other words, if we could
directly observe the population, an infinite number of observations would show us the population mean (average) is at the center, and thus the probability that finite samples from this
unobservable population have a .5 chance of being below (smaller, less, etc.) than that mean,
if and only if we could only if we already took sample an infinite number of samples from this population. Which means that while mathematical models tell us certain things about the probability that random samples of certain sizes are likely to be adequate enough to approximate some sort of probability concerning how the samples are related to the actual unobservable population, they rely on using sample means, and the probability that these sample means, no matter how large the sample size (even if the almost the entire population is the sample) or how many samples are taken, the probability that the the
observed sample means are less than (or greater than) the population mean can be arbitrarily close to zero.
A simple illustration of why (in part, anyway) this is true: suppose my population is the people working for some company. Let's say
N (the number of people in the company) is 2,000. Certainly a measurable population, and it's easy enough to get a few random samples of salaries of 40 employees/workers/whatever or so in order to estimate the salary distribution (i.e., the curve representing the amount of money employees receive, including the average amount, the one or several "clusters" where lots of employees receive the same amount, etc.) But imagine I'm a thorough researcher. So I use random samples of 200 over and over again. In fact, although I don't realize it, I've done so well that I've ended up with the salaries of everyone in the company, and for most I used their salary multiple times (this is a case where double counting the same datum is a good thing). So I plug these all into my models and get what seems certain to be perhaps an exact probability curve of salaries at the company.
Unfortunately, My model is completely wrong. Why? Because a single person whose salary was sampled only once has completely distorted all of my data. Most workers at this company make 50,000 or below. A minority make between 50,000 and 150,000. The boss, however, makes hundreds of millions. As a result, the entire curve is distorted and heavily skewed toward the higher end. The average is a joke compared to what most people there actually make.
This may seem like an unlikely an extreme example, but it pops up all the time in actual measurements of all sorts of populations in various sciences. Mathematical models tell us one thing, when it turns out that they are distorting reality or that they are correctly modeling something that we interpret as something else.
There are similar problems with physics, only greater. Here, it's not a matter of a population of people, their salaries, or some other theoretically measurable population. With other sciences, there are ways to make the models more robust, avoid the problems outlined above, detect outliers, and so forth. This is because the measurements which are used to construct the models are independent of the samples and population in question.
This is not the case in QM. The reason that the probabilities in QM are so hard to interpret isn't that the mathematical formalisms may be wrong in this way or inadequate in that way, but because both the math and the measurements are too much a part of the observation to know what is going on. The wavefunction is a probability function. But how does it correspond to reality? Is the photon or some other particle in question located somewhere in the probability space determined by the wavefunction? Or is the wavefunction actually the physically superpositioned particle itself (or rather, a notational device used to represent it)?