Discrete Invariant Projection Spaces: Incidence Operators, Spectral Moments, and the Kinematic Obstruction

We introduce Discrete Invariant Projection Spaces (DIPS), a finite analogue of integral geometry on homogeneous varieties. The central object is a quintuple (G, X, G, I, μ) where X and G are finite homogeneous G-sets and I is an incidence relation. The projection operator Π = B and its dual yield a normal operator N = B^T M B that commutes with the group action. We prove a Structure Theorem for Cayley hypergraphs, and show that for the Johnson hypergraph N belongs to the Bose–Mesner algebra with explicit eigenvalues. A Cheeger–Crofton inequality h²/2 ≤ λ₁ ≤ 2h is established for regular simplicial complexes, linking the kinematic spectral gap to an expansion constant derived directly from the invariant projection. We introduce spectral moment invariants tr(N_F^m) and prove that they form a hierarchy strictly finer than the Laplacian spectrum: the first moment already separates certain Laplacian co‑spectral graphs, and higher moments distinguish graphs with identical subgraph counts but different intersection topologies. Finally, we formulate the kinematic obstruction to the existence of a regular measure via Farkas' lemma, clarifying the subtle relation between the spectral gap and the discrete analogue of the Kähler–Einstein condition.