This equation

The  Kuramoto-Sivashinsky equation for chaos boundaries.

Solve the time independent version with u as a tanh function. We get a fourth order polynomial in tanh and we exploit symmetry in the roots.  Look for an equilibriums locally that yields the best fractional approximation. It would be nice to assume superposition among the roots, guessing that we need superposition to have a conversion between vector (symmetry space)  and index space, Just looking at a quantum mechanical form. We are using a curved space that tends toward flat everywhere,  We approximate the implied five color planar separation.

Vector space seems a bit is named to me, it is really the space in which symmetries are laid bare, the minimum fundamental variable set. In this case, it is root space, the behavior of the two or three roots that determine the local conditions to meet the equation. We have root space because we have exploited symmetries, found balance points.

The solutions I describe are both the theory of nothing and the theory of everything. Theory simply gives you great quants (digits) to use if your aggregate does fractional approximations. It is the data geek giving you a general solution for nothing in particular, but it identifies the proper finite element orientation so finite summations meet Ito conditions.