I have a question in regards to rifling twist rate versus bullet weight. Is there a way to determine the optimal bullet weight for a given twist rate and barrel length? In other words, the goldilocks bullet, when I say optimal I do not necessarily mean most accurate, I know that the best way to find the most accurate bullet out of a given rifle is to test and experiment with components in a real world setting and not on a computer. This still leaves me wondering if there is a way to determine an optimum bullet weight given several known factors such as rifling twist rate, barrel length and muzzle velocity.

The other reason I am interested in this is the fact that I recently started to Hand load for the for the 30-06 and I jumped into the preverbal deep end of bullet weights starting out with the 208 grain Hornady HPBT and I am curious how much heavier I can go while still maintaining stable flight characteristics of the projectile, when firing them through a 1-10” twist barrel. I know that most people recommend a 220 grain max for the 1 in 10 twist but I want to see if I can push it to 225 or 240 grains. If you have or know of a math formula that would be very helpful as I plan to hand load for more calibers in the future.

I think this is what you are looking for:

Twist rate and bullet stability

In 1879, George Greenhill, a professor of mathematics at the Royal Military Academy (RMA) at Woolwich, London, UK[10] developed a rule of thumb for calculating the optimal twist rate for lead-core bullets. This shortcut uses the bullet's length, needing no allowances for weight or nose shape.[11] The eponymous Greenhill Formula, still used today, is:



where:
C = 150 (use 180 for muzzle velocities higher than 2,800 f/s)
D = bullet's diameter in inches
L = bullet's length in inches
SG = bullet's specific gravity (10.9 for lead-core bullets, which cancels out the second half of the equation)

The original value of C was 150, which yields a twist rate in inches per turn, when given the diameter D and the length L of the bullet in inches. This works to velocities of about 840 m/s (2800 ft/s); above those velocities, a C of 180 should be used. For instance, with a velocity of 600 m/s (2000 ft/s), a diameter of 0.5 inches (13 mm) and a length of 1.5 inches (38 mm), the Greenhill formula would give a value of 25, which means 1 turn in 25 inches (640 mm).

Improved formulas for determining stability and twist rates include the Miller Twist Rule[12] and the McGyro program[13] developed by Bill Davis and Robert McCoy.




A Parrott rifle, used by both Confederate and Union forces in the American Civil War.
If an insufficient twist rate is used, the bullet will begin to yaw and then tumble; this is usually seen as "keyholing", where bullets leave elongated holes in the target as they strike at an angle. Once the bullet starts to yaw, any hope of accuracy is lost, as the bullet will begin to veer off in random directions as it precesses.

Conversely, too-high a rate of twist can also cause problems. The excessive twist can cause accelerated barrel wear, and also induce a very high spin rate which can cause projectile jacket ruptures causing high velocity spin stabilized projectiles to disintegrate in flight. A higher twist than needed can also cause more subtle problems with accuracy: Any inconsistency within the bullet, such as a void that causes an unequal distribution of mass, may be magnified by the spin. Undersized bullets also have problems, as they may not enter the rifling exactly concentric and coaxial to the bore, and excess twist will exacerbate the accuracy problems this causes. Lastly, a bullet which is "overstabilized" will maintain the orientation it was fired at; a ballistic trajectory requires the gun to be aimed above the target, thus an overstabilized bullet at long range will be pointing upward, even though it is moving downward, resulting in an oblique impact and aerodynamic inefficiency. Ideally, a spin stabilized bullet is stable enough to not tumble, but unstable enough to allow aerodynamic forces to cause the tip to always point in the direction of travel.

Note that the above formula does NOT use weight, only bullet [u]length[/u].
IOW a 150gr VLD being longer than a 220gr RN requires a faster twist also (even though it's lighter). Weight only comes into play due to it's effect on velocity while staying within sane pressure levels.
The "standard" twist of 1:10 that a 30-06 uses comes from the original bullet that the 1903 cartridge used IE the same 220gr RN that was used in the 30-40 Krag. With the later 150gr spitzer design, the twist was actually a little faster than needed.
Fast forward to the 7.62x51/308Win, that was designed around 145-165gr SBT, and you'll find the slower 1:12 twist being used