going back to my comment:
"You get effectively a meeting of t and x in the "space" y,z influence."
You could say that if you can vary just "t" in fixed x against the space "y,z" influence; would build up a kind of "t" bulking up say of an x,t cell in "y,z" influence? The room to move that t had in x in "y,z" influence could be called a "fourier transform" of t in x (As the math description of "fourier transform" seems to translate into the English: "room to move that one variable has in another variable".
If you also had room to vary just "x" in fixed t against the space "y,z" influence; you would have fourier transform of x in t say.
If the room to move "x" in fixed "t" was at different parts of "t" than the room to move "t" was in "x"; you would stretch out the cell "x,t" into a linear shape (like a string?)
Perhaps you would reach a limit and get cellular division: two cells of "x,t" meeting in the space "y,z" influence?
Eventually you might get so many cells from further divisions that the space "y,z" influence was itself about to reach its limit here of non-contradicting room to move for x and t?
The final break up of the space "y,z" influence into "x,t" cells would occur at the lowest "energy" ("energy" as "alternatives") level; at the local definition of absolute zero.
Perhaps you would call this collection of cells "a Bose-Einstein condensate"...
Of course the local absolute zero might be much lower if your sample covered a longer trajectory as more alternatives would allow more cells to divide. The cells might even group together to make up optional "Bose-Einstein condensate groups"?
A Bose-Einstein condensate of x meets t cells in the space "y,z" influence would exhaust the possibilities of "y,z" influence so be one with that space. If the whole trajectory from start to the present was "one" with its space; it would have maximum freedom to negotiate any new space......?
Returning to Dr. Dick's thought experiment:
I wrote that he (then) says let us examine the paths of those entities of interest, each by itself, in the absence of the others. "The differential path length along the trajectory is exactly what is referred to as Einstein's invariant interval along the path of the thing being represented by that path. This fact is commonly used in high energy physics to determine the expected apparent path lengths (in x,y,z space) of particles with short half lives."
"differential path length" I took to be "component of path length"; as it describes a final value of one factor in the path length (say net change in x); I guess he calls it "invariant interval" as it is implying a net change in x relative to y,z,t so effectively freezes y,z,t as a non-varying group view of x-only change for the examined sample?
So an x-change that is an interval of x that is only in x-space; so a conservation of y,z,t here as an "invariant interval" where the y,z,t interactions are frozen along this x-only differentiated component?
This interval of x is against a non-varying y,z,t from x perspective as there is movement only of x re: itself (making it self-referent referent so time-like?) against relatively fixed y,z,t background here?
Dr. Dick says this interval is "imaginary" or "time-like"; and since a time-defined phenomenon like "half-life" is constant in its own rest-frame (of time-axis); then taking a time-axis component so holding the space x,y,z unchanging; can be used to find path length of high energy particle with short half-life?
Need to think about that.
If I take a simple graph just with x axis and y axis:
If I draw a line segment sloping from say co-ordinate (x,y) of (2,3) to (10,12); if I look only at the vertical y component I get 12-3= 9 units vertical line segment. For this y component the value of x is frozen at 2 spot along x.
So the invariant interval is x =2 for y-component of x,y here?
Lot more to write but short of time here.