Algebraic Tomography of Non-Hermitian Floquet Systems from Observable Traces

We formulate a framework of Floquet algebraic tomography for finite-dimensional non-Hermitian monodromy matrices from observable trace sequences $\zeta_n^{(O)}={\rm Tr}(OM^n)$. Since these sequences are constrained by the characteristic polynomial of $M$, the inverse problem is a finite-dimensional algebraic reconstruction problem rather than a generic exponential fit. We organize the reconstruction through the observable resolvent, spectral determinant, and Dirichlet spectral data, separating the common spectral skeleton from observable-dependent dressing. Cayley--Hamilton and Hankel methods recover the similarity-invariant spectral data, while multi-observable and Liouville-space extensions connect the construction to realization theory and tomography reconstruction. We further clarify the limits of identifiability from restricted observable algebras: the data determine a visible representative, micromotion can enlarge the sampled visible operator space, and exact symmetries impose residual invisible sectors. Two examples, a driven transmon qutrit and a finite non-Hermitian Floquet SSH chain, demonstrate leakage-induced visibility expansion, observable-dependent phase response, EP-accessible branch geometry, and disorder/probe-dependent observable-dimension readouts.