This paper reinterprets Fourier analysis within the framework of Finite–Closure Geometry (FCG), challenging the widespread assumption that Fourier transforms constitute a neutral or representation-invariant tool. The central claim is that Fourier transforms, as used in physical and computational practice, are always applied after finite projection and truncation, and therefore encode prior assumptions about closure, resolution, and domain extension. Formally, the work emphasizes the non-commutativity between finite projection and ideal limit operations, expressed schematically as [𝒫ₗ, lim] ≠ 0 From this perspective, commonly treated artefacts such as spectral leakage, aliasing, and Gibbs oscillations are not numerical defects or noise, but structural residues of non-closure. The paper develops this interpretation across classical and quantum settings. In particular, Simon’s algorithm is analyzed as a paradigmatic case where quantum advantage arises not from the Quantum Fourier Transform (QFT) in isolation, but from its non-commutative interplay with physically enforced projection (measurement onto a fiber of the oracle function). The Fourier transform is thus recast as a diagnostic applied to a projected structure, rather than a global analyzer of an idealized system. A continuous analogue is presented via truncated Gaussian wave packets, illustrating how projection in one domain necessarily induces structured residue in the conjugate domain. More broadly, the paper argues that physically meaningful Fourier analysis consists not of a single ideal transform, but of a family of finite transforms conditioned by projection. This document is part of a coherent set of works developing Finite–Closure Geometry (FCG) as a structural framework. In particular, it should be read in conjunction with: Categorical Structure of Renormalization (A), which formalizes projection, truncation, and flow using categorical tools, Myth of Fundamentality (B), which critiques the notion of fundamental descriptions and situates closure as a structural, not ontological, assumption. Together, these works treat Fourier analysis, renormalization, and fundamentality as interconnected manifestations of the same underlying non-commutativity between projection and ideal closure, each explored from a complementary angle within FCG. This work does not propose a new mathematical formalism. Instead, it offers a structural reading of existing tools, positioning Fourier analysis as a probe of non-closure rather than a transparent window onto infinite systems. The framework aims to unify phenomena across signal processing, quantum algorithms, and measurement theory under a single conceptual principle. A global conceptual map, epistemological positioning, and guided reading of the foundational FCG papers is provided in the foundational Paper 0: https://zenodo.org/records/18001440