Convergence of the Developmental Euler Product

Paper 4 in the Developmental Geometry Theorem Series. This paper establishes the absolute convergence of the developmental Euler product associated with the spectral decomposition of the developmental attractor. Building on the global attractor structure (Paper 2) and the spectral coherence theorem (Paper 3), we define the spectral weights arising from the Koopman semigroup and prove that the associated Euler product converges absolutely in a right half‑plane. The proof uses a polynomial counting bound on the spectral weights and a summation‑by‑parts argument to control the corresponding Dirichlet series. The result provides the analytic foundation for the developmental zeta function and completes the initial four‑paper analytic spine of the developmental geometry program: Benford’s Law as a Labeling Artifact (Paper 1) A Global Attractor for the Developmental Flow (Paper 2) Spectral Coherence on the Developmental Attractor (Paper 3) Convergence of the Developmental Euler Product (Paper 4, this paper) Together, these papers establish the phase space, long‑time structure, spectral organization, and analytic behavior of the developmental system, forming the basis for the forthcoming developmental zeta function and the monograph sequence.