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So I was reading the recent Staff Thread about a clear standard for Low 1-C scaling, what does and doesn't qualify as such and more. Within that thread, I came across a transcript of a statement apparently made by Ultima in this post within a Kingdom Hearts CRT. To save you the trouble, the statement reads as the following:
Low 1-C is fine by me. We already assume that the space in which spacetimes are displaced is 5-D, at present. Add that to the space in question here being explicitly described as infinite and the worlds as "small" compared to it, and you have a fairly straightforward case.
I don't think the 2-A option is terribly logical either, after mulling it over: 2-A would imply that universes in KH are displaced over 4-D space, which doesn't work when parallelism of any two objects by definition requires an extra axis: For two line segments to be parallel, you'd have to set it so they wouldn't touch regardless of how far they are extended, which wouldn't be possible if they stood side-by-side in 1-D space as in here, meaning you would need them to be displaced over a plane. Same thing happens with planes: For them to be parallel, they shouldn't ever be able to meet, so you'd need them to be displaced over 3-D space. Generalizing that to the 4-D case, spacetimes would obviously have to be displaced over a 5-D region (This works by definition, too: If they're different spacetime continuums then obviously they can't share the same space, in the way 3-D objects exist around us for instance)
Now, while I admittedly do not have nearly as much knowledge and experience in Tier 1 scaling as Ultima does (nor am I nearly as familiar with the intricacies of our standards for such), I believe Ultima here made a blunder in his explanation which leads to several mathematical tautologies, especially if these are actually our standards as his statement seems to imply. Allow me to explain:
"which doesn't work when parallelism of any two objects by definition requires an extra axis"
This is not necessarily true always. For two objects to be held parallel to one another, they must simply be physically displaced in a sufficiently large space (ergo, at least larger than their size + whatever distance between them) of the same order. For example, two finite parallel line segments can be parallel to each other while being merely parts of two infinite and parallel line segments. If Ultima is trying to talk about infinitely sized objects instead, then it is accurate. Parallelism of two infinitely sized objects does warrant a higher order space. This distinction is important I believe as I'll explain further.
"For two line segments to be parallel, you'd have to set it so they wouldn't touch regardless of how far they are extended, which wouldn't be possible if they stood side-by-side in 1-D space as in here, meaning you would need them to be displaced over a plane."
I underlined that part specifically because it is the most important part of that statement and related directly to what I stated above. It is ONLY a requirement if the object in question is infinite in its size. But there is yet another flaw with this statement, in that last part Ultima mentions that accommodating two infinitely extending line segments warrants a plane, which is only partially true. Said planes however ALSO need to be infinitely sized in AT LEAST one axis. Try drawing an infinitely extending line segment (or even just a very long line segment) on normal sheet of paper, you'll see what I mean. You can only do this if the plane (in this case the sheet of paper) was infinitely large in either length of width. The reason I mention axes instead of just one axis is because of a simple fact about rotation being isometric [SO(2)]. This implies that by taking an arbitrary origin for an object and rotating it about that, properties such as Euclidean distance are invariant. In simpler words, two lines being displaced "horizontally" or "vertically" are the same thing, with the difference being a mere rotation matrix (this generally works for rotation of almost any kind). So the plane has to be infinite in size in at least one of these axes to be able to hold within it two infinitely extending line segments. I hope this is clear.
"Same thing happens with planes: For them to be parallel, they shouldn't ever be able to meet, so you'd need them to be displaced over 3-D space."
As I explained above already, this isn't necessarily true either if the planes aren't infinite in size (in either of the axes). If the infinitely sized, then not only do you need a 3-D space to hold them, but you in fact need an infinitely sized 3-D space to hold them for the exact same reasons I explained above. And yes, it has to be infinite in at least any one of the 3-dimensions, either in its length, width or height. The planes can be rotated within the space, as most of the important properties are invariant to SO(2) or SO(3) rotation (a theorem I've proven in linear algebra classes btw).
"Generalizing that to the 4-D case, spacetimes would obviously have to be displaced over a 5-D region (This works by definition, too: If they're different spacetime continuums then obviously they can't share the same space, in the way 3-D objects exist around us for instance)"
And lastly this statement. Now one might already imagine that it has the exact same flaws as the statements before it, and before I even talk about it I need to point out something else. Specifically, the "generalizing" part. Where does this generalization come from, assuming everything about it is accurate (which, if you've been paying attention, I've pointed out that it isn't)? How can Ultima prove that this generalization is accurate? It seems quite hand-wavy, a generalization like that needs proof, or some kind of standard. He didn't prove that this "fact" can be extended to the 4th dimensional case, nor did he prove the ACTUAL generalization that this holds true for any number of N or N+1 dimensions. By the way, Ultima didn't clarify whether he is talking about 4D Euclidean space or Minkowski space, I assume he meant Minkowski space as he mentions "space-time", but if that's the case, his "generalization" is immediately invalid as the examples he's been citing of 2D and 3D spaces are that of Euclidean space, which is different to Minkowski space as the former only has spacelike dimensions while the latter also possesses timelike dimensions and, at least to my knowledge, there is no 2D or 3D analogue of Minkowskian space that he could've been talking about anyway. Math nerd rant aside, this still has the same problems as I described above already, a 4D space would ONLY need to be displaced across a 5D space IFF it is infinite in at least one of its dimensions, in which case, the 5D space must ALSO be infinite in at least the same number of dimensions (notice I didn't say the same "kind" of dimensions, that's precisely because, again, a rotation can solve that problem, we use quarterionic and octionionic rotations in these cases which also happen to be isometric, a theorem I proved in Topology).
I am aware that this is my very first attempt at writing such a huge post, so it is very much possible I may have misunderstood or misinterpreted some of what Ultima had said, especially since this is just one of his countless other statements on how Tier 1 functions, which may offer further context than this single isolated statement does. That being said, as I pointed out before, it seems to me that what Ultima is describing is actually our current standards for Tier 1 (though that might change soon), and in that sense, I am more so pointing out the problems in the standards. The main problem is about infinite size, the secondary problem being about that hasty and (possibly wrong) generalization. On that note, I should point out that I do have proof of a generalization like that in my notes, where I prove how certain properties are invariant regardless of however many dimensions we are talking about. I would be happy to send that here HOWEVER that proof is ONLY for Euclidean spaces, not Minowski space, which is how the wiki treats 4D space as (space-times).
Edit: I am not a huge fan of arbitrary flip-flopping between mathematical logic and whatever other weird (and arbitrary) standard is being used here, and this may very well be a huge reason why Tier 1 is easily the most messed up tier on the wiki currently. Some mods may not make exactly the same kind of arbitrary jumps in logic that Ultima does, and thus would end up with vastly different definitions and standards of Low 1-C and the original thread linked above demonstrates this fact well.
Low 1-C is fine by me. We already assume that the space in which spacetimes are displaced is 5-D, at present. Add that to the space in question here being explicitly described as infinite and the worlds as "small" compared to it, and you have a fairly straightforward case.
I don't think the 2-A option is terribly logical either, after mulling it over: 2-A would imply that universes in KH are displaced over 4-D space, which doesn't work when parallelism of any two objects by definition requires an extra axis: For two line segments to be parallel, you'd have to set it so they wouldn't touch regardless of how far they are extended, which wouldn't be possible if they stood side-by-side in 1-D space as in here, meaning you would need them to be displaced over a plane. Same thing happens with planes: For them to be parallel, they shouldn't ever be able to meet, so you'd need them to be displaced over 3-D space. Generalizing that to the 4-D case, spacetimes would obviously have to be displaced over a 5-D region (This works by definition, too: If they're different spacetime continuums then obviously they can't share the same space, in the way 3-D objects exist around us for instance)
Now, while I admittedly do not have nearly as much knowledge and experience in Tier 1 scaling as Ultima does (nor am I nearly as familiar with the intricacies of our standards for such), I believe Ultima here made a blunder in his explanation which leads to several mathematical tautologies, especially if these are actually our standards as his statement seems to imply. Allow me to explain:
"which doesn't work when parallelism of any two objects by definition requires an extra axis"
This is not necessarily true always. For two objects to be held parallel to one another, they must simply be physically displaced in a sufficiently large space (ergo, at least larger than their size + whatever distance between them) of the same order. For example, two finite parallel line segments can be parallel to each other while being merely parts of two infinite and parallel line segments. If Ultima is trying to talk about infinitely sized objects instead, then it is accurate. Parallelism of two infinitely sized objects does warrant a higher order space. This distinction is important I believe as I'll explain further.
"For two line segments to be parallel, you'd have to set it so they wouldn't touch regardless of how far they are extended, which wouldn't be possible if they stood side-by-side in 1-D space as in here, meaning you would need them to be displaced over a plane."
I underlined that part specifically because it is the most important part of that statement and related directly to what I stated above. It is ONLY a requirement if the object in question is infinite in its size. But there is yet another flaw with this statement, in that last part Ultima mentions that accommodating two infinitely extending line segments warrants a plane, which is only partially true. Said planes however ALSO need to be infinitely sized in AT LEAST one axis. Try drawing an infinitely extending line segment (or even just a very long line segment) on normal sheet of paper, you'll see what I mean. You can only do this if the plane (in this case the sheet of paper) was infinitely large in either length of width. The reason I mention axes instead of just one axis is because of a simple fact about rotation being isometric [SO(2)]. This implies that by taking an arbitrary origin for an object and rotating it about that, properties such as Euclidean distance are invariant. In simpler words, two lines being displaced "horizontally" or "vertically" are the same thing, with the difference being a mere rotation matrix (this generally works for rotation of almost any kind). So the plane has to be infinite in size in at least one of these axes to be able to hold within it two infinitely extending line segments. I hope this is clear.
"Same thing happens with planes: For them to be parallel, they shouldn't ever be able to meet, so you'd need them to be displaced over 3-D space."
As I explained above already, this isn't necessarily true either if the planes aren't infinite in size (in either of the axes). If the infinitely sized, then not only do you need a 3-D space to hold them, but you in fact need an infinitely sized 3-D space to hold them for the exact same reasons I explained above. And yes, it has to be infinite in at least any one of the 3-dimensions, either in its length, width or height. The planes can be rotated within the space, as most of the important properties are invariant to SO(2) or SO(3) rotation (a theorem I've proven in linear algebra classes btw).
"Generalizing that to the 4-D case, spacetimes would obviously have to be displaced over a 5-D region (This works by definition, too: If they're different spacetime continuums then obviously they can't share the same space, in the way 3-D objects exist around us for instance)"
And lastly this statement. Now one might already imagine that it has the exact same flaws as the statements before it, and before I even talk about it I need to point out something else. Specifically, the "generalizing" part. Where does this generalization come from, assuming everything about it is accurate (which, if you've been paying attention, I've pointed out that it isn't)? How can Ultima prove that this generalization is accurate? It seems quite hand-wavy, a generalization like that needs proof, or some kind of standard. He didn't prove that this "fact" can be extended to the 4th dimensional case, nor did he prove the ACTUAL generalization that this holds true for any number of N or N+1 dimensions. By the way, Ultima didn't clarify whether he is talking about 4D Euclidean space or Minkowski space, I assume he meant Minkowski space as he mentions "space-time", but if that's the case, his "generalization" is immediately invalid as the examples he's been citing of 2D and 3D spaces are that of Euclidean space, which is different to Minkowski space as the former only has spacelike dimensions while the latter also possesses timelike dimensions and, at least to my knowledge, there is no 2D or 3D analogue of Minkowskian space that he could've been talking about anyway. Math nerd rant aside, this still has the same problems as I described above already, a 4D space would ONLY need to be displaced across a 5D space IFF it is infinite in at least one of its dimensions, in which case, the 5D space must ALSO be infinite in at least the same number of dimensions (notice I didn't say the same "kind" of dimensions, that's precisely because, again, a rotation can solve that problem, we use quarterionic and octionionic rotations in these cases which also happen to be isometric, a theorem I proved in Topology).
I am aware that this is my very first attempt at writing such a huge post, so it is very much possible I may have misunderstood or misinterpreted some of what Ultima had said, especially since this is just one of his countless other statements on how Tier 1 functions, which may offer further context than this single isolated statement does. That being said, as I pointed out before, it seems to me that what Ultima is describing is actually our current standards for Tier 1 (though that might change soon), and in that sense, I am more so pointing out the problems in the standards. The main problem is about infinite size, the secondary problem being about that hasty and (possibly wrong) generalization. On that note, I should point out that I do have proof of a generalization like that in my notes, where I prove how certain properties are invariant regardless of however many dimensions we are talking about. I would be happy to send that here HOWEVER that proof is ONLY for Euclidean spaces, not Minowski space, which is how the wiki treats 4D space as (space-times).
Edit: I am not a huge fan of arbitrary flip-flopping between mathematical logic and whatever other weird (and arbitrary) standard is being used here, and this may very well be a huge reason why Tier 1 is easily the most messed up tier on the wiki currently. Some mods may not make exactly the same kind of arbitrary jumps in logic that Ultima does, and thus would end up with vastly different definitions and standards of Low 1-C and the original thread linked above demonstrates this fact well.
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