Backpacker - You keep missing the point.
Well, I have just read through all your posts on this subject again and the only point you seem to want to hammer home is that descents have to be done at 1500 fpm, regardless of the type of aircraft or its groundspeed. Or, at least, that the 3X rule is somehow linked to a ROD of 1500 fpm.
Obviously with a 1500 fpm descent rate and a groundspeed of, say, 100 knots, you are not going to have a three degree glidepath but something way steeper, and thus the 3X rule does not apply.
After all, the only thing the 3X rule will calculate for you is the distance required to descend x feet, based on a three degree glidepath. How to achieve that flightpath, powersettings, ROD etc., will differ from aircraft to aircraft. (If you use five times the groundspeed as your ROD you will get a reasonably accurate three degree glideslope though, as others have pointed out.)
And indeed, if your desire is not a three degree flightpath, but to be x feet lower in y nm, there are other formula that you can use. We've already established that long ago.
But I have asked before, and will ask once again: why do small aircraft need a 1500 fpm descent rate, leading to a glide path of approximately nine degrees? What's wrong with a three degree glide path at 500 fpm? Particularly if they are able to plan their descent beforehand?
Unless the point you're trying to make is different, or until you are able to answer that question, I'm just going to enjoy watching you dig an even bigger hole for yourself...
(Must admit that alongside the enjoyment, I am a bit worried too. You are an FI and you fly RHS in an airliner. There's four people here on this forum, one of which a long time LHS and FI, two other airline pilots, and a PPL with a bachelors degree in math, all trying to tell you you are wrong. Yet you maintain you are right and claim that the math prove without a shred of doubt that you are right - without giving the actual proof. Then someone runs the numbers for you, proves that your math is incorrect and that the formula lead to plausible numbers for all relevant scenarios. And you try to debunk that with a simple "you are missing the point" rebuttal. You sound very reluctant to admit you're wrong, and in your position, that worries me.)