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Given the widespread acceptance that “Einstein’s Relativity” reasonably proves the existence of Space-‘Time’, and thus “Time”... Can any member of the Physics StackExchange please show precisely where, in "On the Electrodynamics of Moving Bodies", (or related core Relativity) papers, “Time” is actually, in any reasonable way shown to exist, as opposed to just, or only being (unscientifically) “assumed”? (added) or, why it is legitimate for the paper to make this assumption or axiom?

Many individuals and publications refer to Relativity as our best theory of Space, and “Time”, and thus imply it is a given that time exists... because Relativity intricately employs the concept of time.

But wherever such claims are made, they are rarely accompanied by any clear justification or specific reference to a specific section of SR or GR. At best, it is claimed that SR proves “Time dilation”, and thus the existence of “time” as a genuine dimension.

Therefore unless it can be shown that Relativity, ("On the Electrodynamics of Moving Bodies" etc), actually incorporates a reasonable proof of times existence, or cites such a proof, or even just gives a valid reason to “suspect” a thing called “time” might exist, then the “time dilation” shown may in fact prove only that moving things “are” changing “slower” (i.e just at a dilated rate), than stationary things, and not that a “temporal past”, and/or “temporal future”, or thing called “time” also exists. (As per space-“time”, block-“time”, growing block-“time” etc).

Precisely where Relativity is valid in assuming a thing called time exists, is a very important question, given the large number of theories based on the belief Relativity does proves time’s existence. And, the large number of fundamentally conflicting theories about time, problems resolving quantum and classical “time”, and even disagreement about time’s actual existence or not.

Therefore, if anyone here on the stack exchange can show just where "on the electrodynamics of moving bodies" actually, legitimately validates its use of "time", or why they accept its assumptions in the specific area of "Time", please clarify this not just for me, but I assume many others..

MattMars
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First of all, physics does not ever talk about the question of existence, but about useful descriptions and predictions of observations. No physicist will ever prove to you he is not just a figment of your imagination but he can prove to you that Newton's law works pretty well for what you see.

In the scientific method, a theory is indeed used until it becomes useless because it does not describe facts properly. But relativity does describe facts properly and gives sound predictions. Hence in the eyes of the scientific method, there is no reason to throw it away unless a better proposal is put forward.

To sum it up, nobody in science cares whether time exists or not - unless you show hard facts which distinguish between the statement of the existence/non-existence of time. Obviously, this requires that the statement of "existence" is given a specific interpretation. However, if the given interpretation does not show a factual distinction between the statements, the question of their validity is non-scientific. So what factual distinction does the statement of non-existence of time represent?

Einstein does not prove time to exist and it is safe to assume he did not really aim to discuss it's "existence" in a philosophical sense. The work is a "mere correction" to the Newtonian paradigm with the Galilean transform where a parameter $t$ commonly called "time" is assumed. The need of such a parameter is assumed implicitly, the same way that it is commonly assumed that the world is not just a figment of your imagination.

So what does this parameter $t$ mean? Citing from Einstein's article on electrodynamics of moving bodies:

It might appear possible to overcome all the difficulties attending the definition of “time” by substituting “the position of the small hand of my watch” for “time.” And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch.

That is, for Einstein "time" is just the number you read off from a clock. Nothing more, nothing less, no discussion of existence of some abstract "time" entity is needed. A part of the body of the article then concerns itself with deriving how you can relate what does your clock show to things that other clocks show under certain synchronization procedures. (I described one such procedure in this answer.) You could eliminate "real time" and just talk about "change" and no-one can stop you. But the tacit fact still holds that you need a certain number $t$ to unambiguously label events.

I can already picture the comment which says things such as "But this is no proof the label $t$ is not redundant...". So ask yourself "How would I prove my dog exists?". You would say you see him in your living room. But you see an image in the shape of a dog in your head. Which means there are neurons flashing in your head somehow. How is this proof such a thing as a "dog" exists? And we could go on like this for a long time.

A truly scientific approach is to propose specific hypotheses associated with the fact of the existence and non-existence of your dog. E.g. "My dog exists if I hear him, see him, touch him, there is no logical inconsistency in these phenomena, and my psychiatrist is not telling me I am mad." But in the end these are only necessary, not sufficient conditions for the existence of your dog. You never gain sufficient conditions for the existence of anything from science.

The need of a fourth label "t" has a specific form and the proposal of non-existence of time would have to reformulate science in a specific and quantitative way to prove itself at least equally true as the proposal of need of $t$. The scientific method is open, it verifies only necessary conditions and this is also the case of the $t$ label. I.e. the necessary conditions for the "essentiality" of $t$ are given by it being a part of working scientific theories. As far as I know, this is not the case of "non-time" theories. (Even Barbour has a fourth parameter which is just not called $t$.)

Community
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Void
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The aim of special relativity and of spacetime (in particular: the Minkowski space time) is not to know about what time is. Spacetime is showing a relation between space and time from an observer's view only - and this whatever time is in reality (including the question if time exists or not). The result is that time (i.e. the value measured by clocks) may be different from one observer to the other.

In special relativity "time" does not mean anything else than that a clock is running.

Moonraker
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The question seems to imply that the OP has a fundamental problem with existential questions. Those are for psychology and philosophy to ponder. Physics simply observes that one can build very precise clocks that agree with each other reasonably well under certain circumstances and not at all under others. The theory of special relativity clarifies when these clocks will agree and by how much they will disagree, when they don't.

CuriousOne
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