On Geometric Origin of Dark Matter, Dark Energy, and the Fine-Structure Constant

Abstract The $\Lambda$CDM model successfully accounts for cosmic expansion and large-scale structure but treats dark matter, dark energy, and the fine-structure constant $\alpha$ as independent inputs. We present an outline of a geometric construction in which all three emerge from the interplay of two natural measures --- a metric (Lebesgue-type) measure and an invariant (Haar-type) structural measure --- on spacetime. A single crossover scale $r^* = e^{-1}$ and a unified visibility kernel $K(x) = \Sigma(x) \cdot \Pi_{\rm time}$ (with $\Pi_{\rm time} = 1/2$ from the arrow-of-time projection) simultaneously produce the dark-matter-to-baryon ratio $\Omega_{\rm DM}/\Omega_b \approx 6$, suppressed late-time growth of structure, and the leading term of the fine-structure constant $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi \approx 137.03630378$. If successful, the framework would require no new particles, no free parameters, and even at this preliminary stage it can make concrete, falsifiable predictions for current and upcoming cosmological surveys and precision measurements of the fine-structure constant $\alpha$.