There are a few issues people are hinting at but stop slightly short of mentioning.
1) It's the length that matters (this one was mentioned by one poster) not the weight. All other things being equal it does not matter, but plastic ballistic tips, hollow cores, steel penetrators, comparatively light weight tracer compounds, long match type point designs and designs using a longer than normal ogive can all add extra length to a round of a given weight.
2) when the powers that be were adopting a barrel twist for the M16A2 they recognized that the 1-9 twist was ideal for the SS109/M855 round (and was generally ideal for a lead core jacketed bullet in the 62 grain weight range.). However the M856 tracer round was substantially longer and would not stabilize out to the required range, so a faster twist was adopted to stabilize that round - even though it is seldom used in M16s or M4s.
3) velocity matters. What stabilizes the bullet is the rotational velocity / rpms. That is a product of both barrel twist and muzzle velocity, so a long bullet may stabilize in a slow twist barrel at a near maximum muzzle velocity and not stabize at a slower velocity.
4) applying #3, barrel length matters. You will note lots of college about x bullet staining in a 20" but not a 16" barrel. It's not a product of the length of the barrel in terms of more rifling, but rather a product of shorter barrels imparting less initial velocity and thus fewer revolutions per minute to the bullet.
To make this concept easier to grasp you can demonstrate it with small numbers and a 1 in 1 ft twist rate. If I fire a bullet at 1 fps through a 1-12 barrel, it will leave the barrel rotating 1 revolution per second (60 rpm). If I increase the velocity to 2 fps, it will exit at 2 revolutions per second (120 rpm) and continuing this increase, 3 fps equals 180 rpm, and 4 fps equals 240 rpm. I can get the same 240 rpm however using a 1-24 twist barrel and a velocity of 8 fps.
The same thing happens with 2000-3000 fps muzzle velocities, the numbers just get much larger. If I launch a 75 gr bullet at 2600 fps out of a 20" 1-9 barrel it will be spinning at about 208,000 rpm. If I shoot the same round out of a 16" 1-9 twist barrel and only get an MV of 2200 fps, then the rpm falls to 176,000 rpm - a 16% decrease in spin. The former round will be well stabilized while the latter is extremely marginal with a long 75 gr bullet and if it stabilizes, it may not stay that way beyond 100-200m.
5) range matters. bullets don't lose much spin down range, but they do lose some. And more importantly if a marginally stable bullet is yawing slightly around it's axis, the aerodynamic forces on the nose will cause it to precess - moving the bullet in a direction 90 degrees off the direction where it was pushed, and that is what quickly degrades accuracy.
6) quality matters. If a bullet is perfectly made, the center of form and the center of mass share the same point in the center of the bullet. However if those two points differ, the bullet will exit the barrel rotating around the center of form and then must transition to rotating around the center of mass. That transition before the bullet settles down causes the bullet to yaw slightly, and if it is only marginally stable, it will not dampen the yaw moments, they will amplify and the bullet will destabilize.
(The same issues effect low quality, lightweight bullets that are fired in 1-7 barrels with excess spin resulting. While accuracy with good quality bullets remains good, poor quality bullets experience greater yaw and precession after exiting the barrel and demonstrate accuracy that is worse than would have been the case in a 1-9 or 1-12 barrel. However in this case the lower velocity in carbine barrels reduces the spin and improve accuracy. Differences in barrel length and bullet quality account for much of the disagreement in the 55 gr / 1-7 twist argument.)
- longer bullets require more twist than shorter bullets regardless of weight,
- slowing a bullet down will not improve the stability situation if it is already only marginally stable,
- shorter barrels present greater stability challenges due to lower velocities,
- higher quality bullets are easier to stabilize than lower quality bullets
- a bullet that makes shooter "A" happy by stabilizing out to 100-300m may not please shooter "B" who is shooting at 600m