What proof does Vedanta show that experience is non-dual?
Vedanta defines non-duality as that which is free from subject-object duality. In other words, there is no separation between you as awareness and the objects perceived as “out there.” Non-duality simply means everything comes out of, and revolves back into awareness.
It takes a cognitive shift to see that which goes against our basic understanding of how experience works. This cognitive shift is similar to learning for the first time that the Earth revolves around the Sun, and not the other way around. Even after gaining this knowledge, we might still feel that the Sun revolves around the Earth due to it rising in the east and setting in the west. This is why knowledge is paramount, because, as is so often the case with nature, what you see isn't always what you get!
Empirical reality shows us that all physical objects when divided into finer and finer parts, are insubstantial and can be reduced to empty space. Empty space, for both science and Vedanta, isn’t no-thing. Einstein showed that space has matter-like characteristics and in Vedanta, space is considered on of the five basic elements. Still, Vedanta takes it a step further by saying that all objects—even space—can be reduced to awareness.
Awareness is what you arrive at when an object is no longer divisible. It’s the ultimate end-point to everything in the universe. Even space is consumed by awareness with nothing coming after awareness because awareness is that which has no cause; it is the essential is-ness of all experience—even space.
Vedanta can never empirically prove that reality is non-dual and that all objects only exist within consciousness. But neither can science ever verify that all objects reside outside of consciousness...
We take it for granted that awareness is essential to experience just like we take it for granted that light is essential to seeing. For example, if I raise my hand and ask you what you see, you'll say "your hand" and probably never mention the light that makes seeing my hand possible. Light is a common metaphor for awareness in Vedanta, because similar to light, awareness reveals all objects without any action.
Vedanta shows that all objects are actually just a thought—which should be obvious. When you look at a table, is the table located "out there" or in your mind? And if the table ultimately resides in your mind, how far away is the table from you? This makes sense when you consider that the way we interpret objects is by data coming in through the sense instruments (eyes, nose, ears, mouth and skin) and sense organs (functions of the mind that interpret sense objects). We interpret objects via certain attributes, such as shape, color, smell, taste or touch. The mind then applies name and form to the composite of data, as well as memories of past experience, in order to relate to it. A table doesn't "enter" an observer, nor does the observer actually see a table. Instead, the observer sees shapes and colors ("stimuli") that when constructed by the mind, resemble a table. Thus, Vedanta takes the common belief that objects lie outside of us and turns it on its head.
Can Vedanta show proof that reality is non-dual?
No, because awareness (the subject) isn't an object we can point to or ever know with our senses. We are like the camera looking for proof of itself in the photograph. The subject cannot become the object unless given a medium to reflect it, and even then, the reflection will only be a distortion of the original and not the actual subject.
So, Vedanta can never empirically prove that reality is non-dual and that all objects only exist within consciousness. But neither can science ever verify that all objects reside outside of consciousness because we are all using the same sense instruments to measure experience and don't have an independent means of knowledge to compare them with. Thus, trying to scientifically show proof of non-duality is a non-starter because non-duality isn't something that can be tested. That said, if we apply the principle of Occam's razor to both duality and non-duality (the idea that given one or more hypotheses consistent with the available data, the hypothesis with the fewest new assumptions will be the preferred), non-duality certainly looks interesting and explains a lot about nature, including that “hard problem”—how consciousness evolved from inert matter (it didn't).
Vedanta can only ever prove that experience is non-dual through the use of words and logic, and our own experience (i.e. nobody needs to tell you that your are conscious). Scripture can only ever be “the finger pointing at the moon and not the moon itself.” That said, Vedanta isn't really about proving anything but instead, disproving that which obscures the truth. It indirectly shows with teachings such as the three states of existence (waking, dreaming, sleeping) that awareness is the only true constant and that everything else is superimposed. The Mandukya Upanishad with Karika is one of the principal Vedantic texts that delves deeply into such philosophical matters.
Lastly, the ultimate aim of Vedanta isn't to prove that reality is non-dual but that the essence of what you are is unchanging, unbounded, eternal awareness and not the limited body-mind you believe yourself to be. By itself, the topic of non-duality amounts to little more than a philosophical exercise. In other words, non-duality isn’t the end, but only a means to reducing our sense of limitation.