Topology of D-Operator Space. Algorithm for Checking the Physical Admissibility of Linear Differential Operators (Admissibility Criteria for Free Kinetic Operators)

We present a structured admissibility workflow for linear differential operators, complete for standard strict low‑derivative free kinetic operators in a local, relativistic, unitary field theory. The workflow formalizes the notion of the D‑operator space $\mathfrak{D}_{\text{phys}}$ and provides explicit computational criteria for locality, hyperbolicity, gauge invariance, degree of freedom counting, and positivity of energy. Each step is illustrated with fully worked examples (Klein–Gordon, Dirac, Maxwell, Proca, linearized Einstein, and counterexamples such as tachyons and higher‑derivative operators). The presentation is self‑contained and mathematically disciplined, while honestly acknowledging the limitations and the precise scope of the admissibility conditions.