I'll bite (because few things would please me better than to see you learn what this stuff means and why the questions you've asked don't make sense and/or are trivial).

steve waterman wrote:Does x' = x -d, when restricted to a mathematical Cartesian setting?

Maybe. For a given value of d and a cartesian co-ordiante system, x'=x-d (I'm assuming y'=y, z'=z. w'=w ... for any other co-ordinates) specifies a single cartesian co-ordinate system which satisfies that relationship i.e. the one whose origin sits at (d,0,0,0,...) in and has the same orientation and handedness as the given system. If we let d range over all real values, we get the set of all cartesian co-ordinate systems of the same handedness and orientation which label every point with identical y, z, w, ... co-ordinates.

steve waterman wrote:Given coincident Cartesian coordinates systems S(x,y,z) and S'(x',y',z'), where x = x', y = y', z = z', d = 0.

*d = distance of separation between S and S' along the common x/x' axis

Your first line here assumes that d is 0 (by assuming the trivial identity transformation). As such, all you're considering is a special case of x'=x-d where d=0 and pretending you're including the whole set by still using d is either a glaring error or deliberate obfuscation.

steve waterman wrote:Allow one of the two systems to be repositioned by distance d > 0.

Except you previously assumed that d=0. As such, you've assumed two contradictory statements and so the

principle of explosion applies and no conclusion you reach has any relevance to anything. It is worth noting that, the statement 2 = -1 is true in the logical system as your assumptions define it. This is in fact true of any proposition. If we work in this world where d = 0 and d != 0, it is true that "badgers and blue than has".

steve waterman wrote:1 x' = x-d, if mathematically allowed, would equate unequal coordinate ( lengths ), like 2 = -1.

Only because you assumed d = 0 in your first line and are now assuming that d != 0. In the system as you have defined it, it is indeed correct that 2 = -1 and no contradiction lies in

this. The contradiction lies in the assumption that d = 0 on the first line.

steve waterman wrote:2 whereas, S(x,y,z) = S'(x',y',z') equates the given coordinate ( lengths ), like 2 = 2.

It does no such thing. Go back and read Schrollini's posts about co-ordinate transformations in the other thread. S(x,y,z)=S'(x',y',z') says that the points S(x,y,z) and S'(x',y',z') are the same. It says absolutely nothing about any lengths (because no such concept need be introduced in order to produce such statements as demonstrated by Schrollini in the other thread) and it says nothing about the relations between the co-ordinates themselves. Without knowing the co-ordinate systems that S and S' refer to, the only thing this tells us is that S(x,y,z) is the same point as S'(x',y',z').

steve waterman wrote:Observation, given x = x', we cannot do any kind of spiffy math and wind up with x ≠ x'.

No, because you assumed it at the beginning.

If you had managed to derive it, your logic would by definition be circular and not useful to anything.

It's clear that you have not understood what Schrollini seemed to have taught you. Go back and read the post when he first introduced co-ordinate transformations.

Edit: wrote this reply when this was another thread so a few things might be worded slightly confusingly.