Causal Depth and Spectral Observability in Finite Path-Sum Propagators

A finite strict causal order defines a nilpotent matrix whose height limits which moments of a hidden positive operator can appear in a prescribed path-sum response. We give a self-contained reduction of this model to a finite spectral-moment problem. For the specified calibrated, complete source/readout channel, a causal order of height h exposes the moment prefix m_0, ..., m_(h-2). When the observable cyclic spectral measure has r distinct support points, exact rank certification requires moments through m_(2r); accordingly h >= 2r+2 is a sufficient and uniformly sharp height threshold for certifying that observable cyclic spectrum. A positive Jacobi-matrix family shows that one additional observable spectral point can remain indistinguishable at every smaller height. We identify the limitations imposed by invisible direct-sum sectors, uncalibrated scale, restricted detector access, and finite-precision moments. The moment and Krylov ingredients are classical; the candidate contribution is the explicit causal-height interface and its sharp observability statement within a declared finite response model. The manuscript makes no claim to reconstruct physical particle content or continuum quantum field theory.