We present a model in which the cosmological constant emerges as a purely geometric effect from the four-dimensional compactification of five-dimensional Einstein-Chern-Simons gravity. The compactification of the extra dimension generates an effective cosmological constant Λ depending on the compactification radius r_c, the coupling parameter l, and the trace h̃ of the compactified field hᵃ, rather than being introduced as a free parameter. The resulting field equations are structurally equivalent to those of General Relativity with a cosmological constant, so all known vacuum solutions – Schwarzschild–de Sitter, Kerr–de Sitter, and FLRW spacetimes – remain valid. As a concrete application, we derive the Kottler (Schwarzschild–de Sitter) black hole solution. We identify two dynamical regimes. In the weak-field regime, Λ∝ l²h̃/r_c³, whose sign is controlled by l²h̃, requiring fine-tuning to reproduce Λ_obs ≈ 10⁻⁵² m⁻². In the strong-field regime, dependence on l and h̃ cancels algebraically, yielding Λ≈ 3/(4r_c²) independently of the Chern-Simons coupling. This regime naturally reproduces Λ_(rm obs) for r_c ≈ 0.78 H₀⁻¹ ≈ 8.2 × 10²⁵ m, without fine-tuning. The Bekenstein-Hawking entropy of the cosmological horizon gives S_(rm cosm) = 4πk_B r_c²/l_(rm Pl)² ∼ 10¹²² k_B, consistent with the Gibbons-Hawking result and admitting a direct geometric interpretation in terms of r_c. This framework geometrically reframes the cosmological constant problem: rather than asking why Λ is small, one asks why r_c is large – a reformulation compatible with a large extra dimension without violating established gravitational tests.