Finite Laplace Jets Do Not Determine Spectral Stability in Scalar Volterra Systems

For scalar Volterra equations, finite moment reductions provide natural finite-dimensional approximations of hereditary dynamics. Since kernel moments are precisely the derivatives of the Laplace transform at the origin, moments through order r retain the order-r Laplace jet of the kernel. This information determines the corresponding local Taylor model exactly and yields a pointwise approximation of the full memory operator under explicit regularity assumptions. Such local approximation, however, does not by itself control spectral behaviour. Moment information can provide rigorous stability certificates in restricted parameter regimes. In particular, existing mean-delay comparison results identify regions in which the mean alone guarantees stability uniformly over all admissible delay distributions. The present paper addresses the complementary classification problem: whether a finite moment vector can provide a complete stability verdict across the cone of nonnegative kernels. We show that it cannot. First, we exhibit two nonnegative Erlang kernels with identical mass and mean lag but opposite spectral stability. We then prove that, for every finite order r, there exists an r-dependent pair of nonnegative smooth kernels with identical moments through order r and opposite spectral stability. The witnessing fibres necessarily leave the region in which mean information already supplies a uniform stability certificate. The proof begins at a simple nonzero-frequency Hopf point and constructs compactly supported moment-annihilating perturbations that preserve nonnegativity while moving the critical characteristic pair transversely across the imaginary axis. Thus no finite moment map provides a complete spectral-stability classifier on the full cone of nonnegative kernels, even though low-order moments may certify stability on protected subregions. The qualitative influence of delay-distribution shape on stability is classical. The contribution here is an all-orders, cone-preserving separation theorem showing that adding finitely many higher moments does not restore complete classification once the system enters the regime where shape-dependent stability is possible.