Geometric origin of the cosmological constant from Einstein-Chern-Simons gravity compactified to four dimensions

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 $\mathrm{\Lambda }$ depending on the compactification radius $r_c$, the coupling parameter $l$, and the trace $\tilde{h}$ of the compactified field $h^a$, 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, $\mathrm{\Lambda }\propto l^2\tilde{h}\mathrm{/}{r}_c^3$, whose sign is controlled by $l^2\tilde{h}$, requiring fine-tuning to reproduce ${\mathrm{\Lambda }}_{\text{obs}}\approx {10}^{-52}{\text{m}}^{-2}$. In the strong-field regime, dependence on $l$ and $\tilde{h}$ cancels algebraically, yielding $\mathrm{\Lambda }\approx 3\mathrm{/}(4r_c^2)$ independently of the Chern-Simons coupling. This regime naturally reproduces ${\mathrm{\Lambda }}_{\mathrm{o}\mathrm{b}\mathrm{s}}$ for $r_c\approx 0.78H_0^{-1}\approx 8.2\times {10}^{25}\text{m}$, without fine-tuning. The Bekenstein-Hawking entropy of the cosmological horizon gives $S_{\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{m}}=4\pi k_B{r}_c^2\mathrm{/}{l}_{\mathrm{P}\mathrm{l}}^2\sim {10}^{122}{k}_B$, 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 $\mathrm{\Lambda }$ is small, one asks why $r_c$ is large – a reformulation compatible with a large extra dimension without violating established gravitational tests.