Having handled thousands of shotgun over the years, the job demands it, I wanted to figure out what makes a "lively" gun, as opposed to one that "balances" at some point, ie the cross pin.

Balancing at a certain point along the length says nothing about that live feel in good shotguns.

Take apart the (break open) shotgun to its three parts, stock and receiver, barrels, forend. Now hold each part from the point you would normally hold, ie the grip for the stock and receiver unit. In lively guns the part has a positive weight concentration towards the center of the gun. The more pronounced this tendency the livelier the gun.

It is possible to alter the feel by adding or removing weight. Adding weight to the center, or removing weight from the extremities (muzzle, under the butt pad) can alter the feel. Even a little weight, in the form of a lead sheet at the breech end of the barrels can have a dramatic effect on the feel of a shotgun.

As for fittig, like Rocketman above, I have been trying to get a rational system, and made my own trygun in which the shape of the stock is figured out separately from the line of sight. But that is for another thread.