The Finite Primitive Basis Theorem for Computational Imaging: Formal Foundations of the OperatorGraph Representation

Computational imaging forward models -- from coded aperture spectral cameras to MRI scanners -- are traditionally implemented as monolithic, modality-specific codes. We prove that every forward model in a broad, precisely defined operator class C_tier (encompassing all clinical, scientific, and industrial imaging modalities at Tier-2 physical fidelity) admits an epsilon-approximate representation as a typed directed acyclic graph (DAG) whose nodes are drawn from a library of exactly 11 canonical primitives: Propagate, Modulate, Project, Encode, Convolve, Accumulate, Detect, Sample, Disperse, Scatter, and one additional primitive for extended modality support. We call this the Finite Primitive Basis Theorem. The proof is constructive: we provide an algorithm that, given any H in C_tier, produces a DAG G with ||H - H_dag||/||H|| <= epsilon and graph complexity within prescribed bounds. We further prove that the library is minimal: removing any single primitive causes at least one modality to lose its epsilon-approximate representation. We give formal definitions for each primitive (forward, adjoint, parameters, constraints), define typed DAG denotation semantics, and establish the approximation error bounds through five primitive realization lemmas -- one per physics-stage family. Empirical validation on 31 linear modalities plus 9 nonlinear modalities confirms that all achieve e_tier < 0.01 with at most 5 operator nodes and depth 5. A formal extension protocol governs the addition of new primitives. These results establish the mathematical foundations for the Physics World Models (PWM) framework.