  If h is a ring-valued function on a simplicial complex G we can define two
matrices L and g, where the matrix entries are the h energy of homoclinic
intersections. We know that the sum over all h values on G is equal to the sum
of the Green matrix entries g(x,y). We also have already seen that that the
determinants of L or g are both the product of the h(x). In the case where h(x)
is the parity of dimension, the sum of the energy values was the standard Euler
characteristic and the determinant was a unit. If h(x) was the unit in the ring
then L,g are integral quadratic forms which are isospectral and inverse
matrices of each other. We prove here that the quadratic energy expression
summing over all pairs h(x)^* h(y) of intersecting sets is a signed sum of
squares of Green function entries. The quadratic energy expression is Wu
characteristic in the case when h is dimension parity. For general h, the
quadratic energy expression resembles an Ising Heisenberg type interaction. The
conjugate of g is the inverse of L if h takes unit values in a normed ring or
in the group of unitary operators in an operator algebra.
