I still can't see a way for your calculations to end up with a negative number. Aside from how you stopped talking about reach and started talking about measure. Add to that, you still didn't address attacking from above, so at the end of it, while interesting, your long explanation didn't hit the target.
If you really can't understand how the design of a scythe shortens the reach, then I really don't know how else to explain it for you, but I'll try.
Let's take a quarterstaff as our baseline this time, since it's the weapon all polearms are built on; in this case, its length doesn't matter. If you thrust with it or strike with it, the very end of its haft is its "lethal" range, i.e., where a strike can do damage.
Because a thrust has longer range than a strike (which goes back to the concept of measure I explained), we'll set the former at a 100% value and the latter at a 95% value.
Now, a spear or glaive adds onto that, say, about 10-20% to each value, for a lethal range equal to 110-120% that of a quarterstaff's, depending on the length of the blade.
An axe, on the other hand, doesn't add to the staff's lethal range, so a thrust is still at the baseline 100% range, while a slash is still at the base 95% (axes aren't typically designed to thrust, but you're still hitting with the end of the haft, so it's no different from thrusting with a staff).
Now for the scythe. In the thrust, it has the same 100% baseline as the staff, but the slash gets more complicated depending on the curve of the blade.
Most scythes have roughly 1" of curvature (measured by making a straight line between the tip and where it meets the haft and then measuring the gap between that line and the edge at the deepest point of the curve).
This makes striking directly like you would with an axe difficult and shaves off a bit of range; probably about 2%. Cutting with the actual edge (instead of attacking with the tip like you would with a war pick) further shaves off a bit of range because you have to adjust the striking angle so much, for a total of, say, 5% loss over a strike with an axe or staff; this number also gets worse the longer the blade is, because the angle you have to swing at becomes more extreme (think of it like the turning radius of a pickup truck vs an 18-wheeler).
If we compare the ranges of a glaive in the slash vs. a scythe in the slash, we get, roughly, 105-115% vs 90% the range of the quarterstaff.
Is that easier to understand? If not, then I'm out of ideas.
As for attacking from above, I
did address it. I specifically said that flight would allow you to mitigate some of the issues of using a scythe as a weapon, but that it also presents its own issues.
And also, it would only
mitigate the issues a scythe poses, because the extra mobility would help you get in and out of measure faster, but it doesn't magically make your reach longer.
I wasn't saying a scythe would necessarily be better than a spear, just a weapon that actually works. I don't think it does normally. Unless maybe if you're a giant, but that's not "normally".
I would also assume that since they're creating one here specifically for the task, angles, length, and blade geometry would not be standard for a scythe, but similar enough to be identifiable as one. A normal scythe cuts at a sharp angle to the blade, not nearly perpendicular as most bladed weapons (other than heavily curved swords and such), so make it more efficient. For a combat-designed one, you'd have a different angle and blade shape.
The easiest way to make a "scythe" that would function better as a weapon would be to essentially make a Chinese
dagger-axe, where the blade is not only quite short, but also double edged and mostly straight; it also has typically has an L-shaped head with the edge running down the bottom of the L.
The much shorter blade turns it into a long-hafted spontoon axe, essentially, which all but eliminates most of the issues a scythe would face. But at that point, is it really a scythe anymore?
You could also adjust the angle, so that the blade is at approximately a 45 degree angle from the haft, which would negate pretty much all the issues a scythe would face... But at that point, why not go a little further and just put it fully in line with the haft and make a
war scythe?
Theoretically, you could just make a dagger-axe, but make the blade longer so it's more recognizably a scythe (and I've designed a weapon just like that in the past), but there's no telling how well it would actually work without making and then testing one. And, at the end of the day, would it be worth the effort?
That's a large part of what I've been getting at. Yes, you
can make a "scythe" that would function decently well as a weapon, but you need a lot more materials and testing to find the optimal design than "slap a pointy metal spike on the end of a straight stick".
To say nothing of the training. The angels are already familiar with how to fight with their lances and are quite good at it; switching to an entirely new weapon would require an almost complete retraining from the ground up.
