Sorry, I know I'm missing a big opportunity here but I've got to continue and delay re-phrasing the questions to some future date.

So when Bob's line of present at 3 yrs intersects Alice = 1.8, what does it really mean? Possibilities:

1. Is Alice's past concurrent with Bob's present as Prof Brian Greene has stated in his relativity course? His take is that since Alice's line of present is slanted, she can intersect the past or future of the participant she is leaving or going towards which means past, present and future exist concurrently. I will show why this is not true because option 3 is the only correct answer.

2. Is it an illusion of Bob's perspective of Alice? Is reciprocal time dilation real or an illusion of perspective? Is it valid to say that when Bob=3, Alice =1.8 in his present it cancels out the reciprocal fact that when Alice=3, Bob = 1.8 in her present? SR states both perspectives are independent realities that only need to be resolved when they either co-locate or share 0 relative velocity. If that's the only time it matters, I see no problem about making definitive statements of who is actually aging slower when it doesn't matter so long as the age results are the same when resolution matters.

3. The answer is this: Alice =1.8 is not a fair comparison to bob=3. Since the motion is relative, these values must always be equal so Alice's total time is being diminished by her motion through space relative to Bob's network of clocks. Alice doesn't have a network of clocks to care about what her perspective of Bob is.

If you look at the STD, when Bob =3, Alice = 1.8 forms a pythagorean triangle where ct is the y-axis, x is the x-axis and the hypotenuse ct/Y ends up being shorter than the y-axis. I'm not familiar with the math terminology but it's some sort of hperbolean pythagorean formula like this:

(ct/Y)

^{2} = (ct)

^{2} - x

^{2}It's not a sum of squares but a subtraction. Bob stays in one spot so his time of 3 is not diminished by travel through space. Alice actually travels the same time but it is partly expressed as her travel through space. So however you draw the STD, the times from both perspectives are equal but a mix of time through time and time through space (I'm not sure of the SR terminology).

So I feel this makes you free to choose only 1 perspective to determine age difference and when they are relatively stopped, the time through space component becomes automatically converted to all time through time. This final result is not affected by ignoring the space through time component throughout the analysis. So, in effect, whether reciprocal time dilation and age difference are the same or not doesn't really matter to the end result. Choosing the earth perspective or the LHC perspective for particles and assuming the time dilation from only this perspective will yield age difference. I guess the other questions don't really matter anymore.

Just as a side note, the math works out the correct formula for Y:

(ct/Y)

^{2} = (ct)

^{2} - x

^{2}(ct)

^{2} * (1-1/Y

^{2}) = x

^{2}(ct)

^{2} * (Y

^{2} - 1)/Y

^{2} = x

^{2}c

^{2} * (Y

^{2} - 1) = Y

^{2} * (x

^{2}/t

^{2})

c

^{2}Y

^{2} - c

^{2} = v

^{2}Y

^{2}Y

^{2} =c

^{2}/ (c

^{2} - v

^{2})

Y=c/sqrt(c

^{2}-v

^{2})