A Global Attractor for the Developmental Flow

A global attractor existence theorem for the discrete developmental flow of Developmental Geometry (DG). The configuration space is the longitudinal–transverse decomposition of the developmental manifold (Book 5) restricted to the discrete substrate (Book 9), with phase coordinate from gateway phenomena (Book 6) and geometric-time offset (Book 5). The developmental flow operator combines axis-field drift with gateway-driven phase update, generating a discrete semigroup. Monotone decrease of the total developmental potential Ξ=Φ+Ψ\Xi = \Phi + \Psi Ξ=Φ+Ψ along trajectories establishes dissipativity; the discrete structure of the substrate gives asymptotic compactness. The abstract global attractor theorem yields a unique stratified attractor Adev\mathcal{A}_{\text{dev}} Adev that is compact, strictly invariant, and attracts every bounded subset of the configuration space. The attractor inherits the shell foliation of the discrete substrate and is the canonical limiting object underlying the spectral results of Paper 3 (Spectral Coherence) and the Euler-product convergence of Paper 4, which use the analytic machinery of the Bochner Laplacian D=∇∗∇\mathcal{D} = \nabla^*\nabla D=∇∗∇ (Book 11). Companion to DG Books 0, 5, 6, 7, 8, 9, 11 and to Papers 3 and 4.