One operator to rule them all: Unifying connectome harmonics, turbulence and complex harmonics in brain dynamics

Brain dynamics can be described in three different convenient mathematical languages, namely connectome harmonics, turbulence and complex harmonics (CHARM). Here we demonstrate that these theoretical frameworks can be rigorously unified, under the functional calculus, as one self-adjoint operator and its single spectral measure. The connectome Laplacian carries that measure; the harmonics are its spectral projections, the turbulence smoothing kernel is its resolvent, and the CHARM form is its unitary propagator. The bridge that makes this exact is a textbook fact: The exponential distance rule, which is the empirical kernel of the turbulence model, is the Green's function of a screened Lapla