Since when is a mathematical definition necessary for this? Finding the "exact line of separation" between a couple of individuals and a powerful megacorp sounds like a fun mental puzzle, but it is not something that comes up in practice. What is the thought experiment in which finding the exact delineation is of importance? On the other hand, it is pretty easy to distinguish the limiting cases. This is like worrying about how large an asteroid has to be in order to be called a planet.

Profit motive, the ability for newcomers in your market-space to challenge your position in your market, and the form of your organization.

Two people in their house,

a large, transparent non-profit advocacy organization,

a dark money lobbying org, and

a hundred-billion-dollar for-profit incumbent that acts as a pillar of human interaction for news, education and entertainment across the globe

are very different situations. It isn't as simple as scaling up numbers.

At the point where a collective asks for the special privilege of forming a limited liability corporation?