We develop a geometric and spectral framework for studying structural fluctuations in discrete arithmetic sequences, operating entirely within real-variable analysis and without appeal to the Riemann zeta function, Euler products, explicit formulae, or analytic continuation. An arithmetic lattice is mapped onto a transcendental logarithmic manifold via an area-filling projection; the resulting Moiré-type interference field serves as the primary object of study. The principal contributions are as follows. (i) We prove, with explicit constants, that the projection map is injective and that successive images are uniformly separated (Propo-sitions 2.3-2.4). (ii) We establish a uniform L 2 near-orthogonality theorem for finite phase families, yielding a computable operator-norm bound on the associated Gram matrix via Gershgorin's circle theorem [5] (Theorem 3.3). (iii) We introduce an explicit Moiré di-lation operator T σ = −i(x ∂ x + σ) on L 2 ([1, ∞)), show that the geometric phasor family {x −(σ+it) } t∈R constitutes its formal eigenfunction family (Lemma 3.12), and prove by integration by parts that T σ is formally self-adjoint if and only if σ = 1 2 (Theorem 3.14). (iv) We derive quantitative variance bounds for node-count statistics, controlled by the pairwise phase-correlation bounds (Theorem 3.19). Complete exponential-sum estimates with explicit constants (Appendix C) and Abel-smoothing remainder formulae including an incomplete-Gamma bound (Appendix B) are provided. All theoretical claims are either proved using the above tools or explicitly designated as conjectural.