Appendix C: Determinant scope

C. Determinant scope and finite spectral regularization

The finite determinant is a product over the retained modes. In the infinite-dimensional statements of this paper, $\det_*$ is the Fredholm determinant on $I+\mathcal L^1$; Lemma 5.7 gives the precise dissipative criterion. If an increasing sequence of finite-dimensional projections converges strongly to the identity and the corresponding trace-ideal approximants converge in trace norm, Fredholm determinants converge. The finite sections must be extended by the identity on their complementary subspaces when using this statement. For unbounded elliptic operators whose eigenvalues tend to infinity, $f_T(\lambda_j)\to0$, so $f_T(K)-I$ is not compact and the uncut operator is outside this Fredholm class. Its finite retained sectors are nevertheless legitimate CSCF budget realizations. Zeta and heat-kernel counterterms, including any multiplicative anomaly, require a separate specification.