  We introduce an integrable Hamiltonian system which Lax deforms the Dirac
operator D=d+d* on a finite simple graph or compact Riemannian manifold. We
show that the nonlinear isospectral deformation always leads to an expansion of
the original space, featuring a fast inflationary start. The nonlinear
evolution leaves the Laplacian L=D^2 invariant so that linear Schroedinger or
wave dynamics is not affected. The expansion has the following effects: a
complex structure can develop and the nonlinear quantum mechanics
asymptotically becomes the linear relativistic Dirac wave equation u''=Lu.
While the later is not aware of the expansion of space and does not see the
emerged complex structure, nor the larger non-commutative geometric setup, the
nonlinear flow is affected by it. The natural Noether symmetries of quantum
mechanics introduced here force to consider space as part of a larger complex
geometry. The nonlinear evolution equation is a symmetry of quantum mechanics
which still features super-symmetry, but it becomes clear why it is invisible:
while the McKean-Singer formulas str(exp(i D(t) t)) = str(exp(-L t))=chi(G)
still hold, the super-partners f,Df are orthogonal only at t=0 and become
parallel or anti-parallel for |t| to infinity.
