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 alphaJ (x, A) = TrJ (e^-tA²) |ₓ䃀 = alpha (x, A) + betaJ (x, A), where alpha (x, A) is the classical Seeley coefficient and betaJ (x, A) is a new correction term encoding the signature of the fundamental symmetry J. We compute betaJ 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 betaJ is then shown equivalent to four conditions: J-compatibility of A, indJ (A) = 0, the J-eta invariant condition etaJ (A) = 0 from Paper C, and — under the Spectral Correspondence Hypothesis — the Riemann Hypothesis.