As I said, the point basically is, there aren't rational interpretations. If 'chi' is used, we're essentially using a 'random number' based on technobabble on how space is already shown to NOT exist in order to explain WF inconsistances. That path's already screwed.
So we're 'realistically' left with options. Find a curve that makes sense to most of the datapoints and ignore the outliers.. .IE, take a scientific approach. Unfortunately, we're still pretty vague, as we see that only WF1 and WF6 are used with any consistancy. (See above).
We're basically trying to apply a mathematical formula to 'Hollywood pulled numbers out of its ass'. Good luck with that, it's been attempted for 40 years now. We're not any closer.. because it's a bit of folly to BEGIN with. Strict canon cannot happen because of the difference between reality and Hollywood's dramatic interpretation thereof.
Sorry I missed so much of this interchange...
I find myself agreeing with Vance here.
We simply cannot accept, without "mental retconning," every single thing seen on-screen for Trek (or for any other show, sci-fi or otherwise!) Some of it simply doesn't work. Some of that may be because the writers and/or producers didn't fully understand their topic. Some of it may be because they simply didn't CARE, and instead wrote the most interesting story that they could, without any regard for "scientific rationality."
The FUN part of all of this... for most of us in here, anyway... isn't "taking everything ever seen on-screen and insisting that it is inviolable and must be accepted without deviation, AMEN!" It's trying to find the "real" explanation which best fits what we've seen on-screen.
For me, the easiest explanation is to assume that many of the cruise lengths we saw were actually quite a bit lengthier than we were informed (either through inference or through direct dialog), for the simple reason that it would be extraordinarily TEDIOUS for the audience to have done so.
As a former soldier (well, technically you're never "former" in this regard, so I suppose I'm still a soldier!), one of the first things you realize is that the life of a soldier is basically long, extended periods of tedium, punctuated by short periods of intense activity. I can't imagine that life on a starship would be any LESS so. But I can easily see the audience not wanting to watch a show like that, any more than I can imagine an audience wanting to watch a "war movie" where the bulk of the film involves soldiers waiting for something to happen.
As far as I'm concerned... there is no "chi factor." As far as I'm concerned, the "WF^3" scale is what was "really" happening in TOS. I accept that because it makes a certain degree of sense from a mathematical standpoint.
Among engineers, we have a phrase we use to describe what's sometimes referred to as the "chi" factor here. We call it a "fudge factor." With "fudge" having scatalogical connotations... basically, the use of "fudge factors" is what's done when you don't really understand the math, but you have an equation you've invested your personal time into and don't want to have to rethink. So you insert that sort of a "constant" in there to compensate for the fact that you don't really know why your equation doesn't fit.
Ok, I'll grant, this isn't always a horrible approach. Sometimes it gives you the next step in developing your equations... since you can look at the curves well beyond your current range of interest, and determine how your supposed "constant" varies outside of the range you were looking at... and sometimes, that leads to some REAL MATH that does, in fact, reflect some factors in the equation you hadn't originally thought of.
The point is that the "fudge factor" is a stop-gap measure, not a solution endpoint.
It makes sense to me that as you step downwards from "real space" into "subspace," you'll see a mathematical shift in distances... and you need it to be in a form which allows both positive and negative values (which means you need an odd-number exponent, not an even one). WF^3.
Take another "step down" below the "surface of space/time"... or rather, a single "step down" below the "surface of subspace"... and you'd be looking at the next odd-numbered exponent. Essentially, you're "warping again from within subspace"... and that's what I treat as "transwarp"... TWF^5.
As for TNG-WF, well... the whole WF10=infinity thing is something I reject, for the most part, because it was supposed to "simplify things" but actually created a far, far worse problem. "Set course, warp factor 9.9999944323432523531242983572983751073... which is twice as fast as warp factor 9.9999944323432523531242983572983751072.
If I really need to think of that, "in-universe," we were given a cheat... it was only the "TNG-WF10 = infinity" warp drive system which was damaging the fabric of space/time."
It's 100% true that many elements from TOS, TAS, TMP-era, and TNG-era don't fit with this in any way. But there's NOTHING you can do to make them fit. NOTHING.
So, you either continue to beat your head against the wall, or you can "mentally retcon" some travel times and assume we just weren't exposed to the between-destinations tedium. I choose the latter.