The Spectral Existence Theorem for Developmental Geometry

PaperE of the Developmental Geometry (DG) program. Proves the Spectral Existence Theorem: there exists a developmental manifold (M, g_D) = (ℝ × S¹, dx² + α(x)²dθ²) satisfying the five DG axioms whose developmental operator D, restricted to the minimal expression class F_min, has spectrum {1, 2, 3, ...} with countably infinite multiplicity at each positive integer. This spectrum matches the Collatz packet-size function k(n) = v₂(3n+1), making the discrete embedding ι : ℤ_odd → F_min with D(ι(n)) = k(n)·ι(n) a well-defined construction rather than a definition by assertion. The construction uses the full-line inverse Sturm-Liouville theorem in the Gel'fand-Levitan-Marchenko formulation to produce the unique weight function α realizing the prescribed spectrum. As a consequence, the discrete-geometric bridge of the DG program is unconditional, and the Collatz conjecture is equivalent to the Unified Density Conjecture of Book 9 (in any of its three canonical formulations: density, mean, or spectral) without additional hypothesis. Companion to DG Books 9–12 and to the Arc 3–5 substrate identification chain.