Paper D: The Local J-Index on Kreĭn Spaces: Seeley Coefficients and Geometric Invariants

We develop the local index theory for J-self-adjoint elliptic operators on Kreĭn spaces, computing the heat-kernel Seeley-DeWitt coefficients in the presence of an indefinite metric. The local J-index density is alpha_J(x,A) = Tr_J(e-tA^2)|t^0 = alpha(x,A) + beta_J(x,A), where alpha(x,A) is the classical Seeley coefficient and beta_J(x,A) is a new correction term encoding the signature of the fundamental symmetry J. We compute beta_J explicitly for the Eisenstein scattering operator on Gamma, showing it vanishes modulo two whenever the operator is J-compatible (that is, when J commutes with the principal symbol of A). The vanishing of beta_J is then shown equivalent to four conditions: J-compatibility of A, ind_J(A) = 0, the J-eta invariant condition eta_J(A) = 0 from Paper C, and — under the Spectral Correspondence Hypothesis — the Riemann Hypothesis.