Garcia, I admit that this stuff can be a bit tricky.
It is, however, possible for your two observers to have the exact same speed through space whilst still moving differently relative to Earth.
For example, let's say that our Earth is moving to the right through space at 50 mph (50 miles per hour absolute speed through space). This of course means that the observer who is at rest on Earth is also moving at 50 mph to the right through space.
Now let the train move to the left at 100 mph relative to Earth. This means that the train is moving at 50 mph relative to space, just as is the other observer.
I would post a diagram of this, but I have no idea how to control it in this forum. Perhaps someone can give me a hint?
As for your claim that your observers did not look at each others' clocks, they did not really have to actually do this in order for the reciprocal time dilation to be involved. But if you think about the train observer looking at a light beam that is moving straight down in the other observer's frame, then you can see that the train observer would see the same diagonal as did the other observer see for the train's light.
Since your observers could be moving at equal speeds through space (as I showed above), this means that their clocks would really be running at the same rate, so all we have left is the fake slowing of special relativity - its reciprocal "time dilation" where each observer sees the other clock running slow.
By the way, it is much, much easier to show this fake time dilation by simply letting a single clock in one frame pass two clocks in another frame. This will reveal that the true cause of this reciprocal "time slowing" is simply Einstein's definition of clock synchronization, which, of course, does not truly synchronize clocks but instead leaves them asynchronous.
At the risk of making another mess, I will try to show this diagrammatically.
[4]-->passing clock
[4]--------------------[5]
-----------------------[6]-->
[6]--------------------[7]
Note that all 3 clocks moved up by 2 hours, so they are actually all running at the same rate; however, since the two clocks in the bottom frame are not absolutely or truly synchronous, the observers in this frame "see" the passing clock as "running slow." (It "fell behind" by 1 hour.) This is clearly a ridiculous and not-worthy-of-science situation of no more importance than the fact that two departing people see each other "shrink."
You wrote:
"Your 'Triplet example' cannot have A & C stationary because all objects are in motion. Thus, for space travelers to be a certain age as one passes by another and then reaches a third object requires dfferent speeds of all of them. SR clearly shows the rate of the passage of time for objects depends on their speed (not velocity). From that, I draw the conclusion that time is a property of matter and passes inversely proportional to any discrete object's speed."
I did not say that A & C were stationary, so I do not know why you said that. (Also, it is not necessarily true that "all objects are in motion" - any number of objects could very well be at absolute rest in space for all we know. Certainly, nothing prohibits this!)
As for SR showing that the passage of time depends upon clock or object speed, this is not really the case because SR did not predict physical clock slowing - it merely incorporated it when it accepted the null result of the Michelson-Morley experiment, which, by extension, says that clocks must physically slow. The only kind of "slowing" given by SR per se is its silly and trivial reciprocal "slowing."
Let's look a little closer at this sentence of yours:
"SR clearly shows the rate of the passage of time for objects depends on their speed (not velocity)."
If you are speaking of a real, physical, intrinsic passage of time (the only passage that counts), then you cannot be speaking of SR because this would mean that SR admits to an effect of absolute motion because only such motion can have any effect on a clock's physical or a person's age.
Let's also look a little closer at this sentence of yours:
"From that, I draw the conclusion that time is a property of matter and passes inversely proportional to any discrete object's speed."
When you say "speed," are you referring to "absolute speed" or "relative speed"? You do not make this clear in your essay. Also, it time were solely a property of matter, then why does the speed of the matter also come into play? (And this time, I am speaking of an absolute speed through space.)
I would rather say that a person's aging is slowed by his absolute motion through space. (But we do not yet know the why of this.) The faster a person moves through space, the slower he ages. Taken to the extreme, one triplet could still be a mere baby whilst his triplet brother could be a great-grandpa or long dead.
