Why brain models mysteriously simplify to the Lorentz distribution

When modeling thousands of coupled neurons, scientists use a mathematical trick called the Lorentzian Ansatz to collapse the system into manageable equations. This paper proves the trick isn't arbitrary: the Lorentz distribution is the *only* two-dimensional probability shape that stays invariant under the system's dynamics, a property rooted in projective geometry rather than computational convenience. This explains why simpler approaches like Gaussian approximations fail and validates reduction methods used across computational neuroscience.