Static Geometric Origin of Dark Energy Density and the Resolution of Its Coincidence Problem —— $\Omega_\Lambda = 0.7$ as an Inevitable Output of the Eigen-Spectrum in the Current Universe

Abstract: This paper accomplishes two core tasks. First, within the axiomatic system of the Self-Construction Theory (SCT), it rigorously proves that the dark energy density parameter $\Omega_\Lambda$ of the current universe is not a free parameter for cosmological fitting, but a geometric invariant statically locked by the pure combinatorial relations of the eigen-spectrum $\{1,2,3,4,5,120\}$—its theoretical value is $\Omega_\Lambda = 0.7$. The matter fraction is determined by the information share of the frozen low-dimensional quanta (dimensions 0–2): $\Omega_m = (Q_0+Q_1+Q_2)/(Q_3\times Q_4) = 6/20 = 0.3$; the dark energy fraction is given by its geometric complement: $\Omega_\Lambda = 1 - \Omega_m = 0.7$. This derivation belongs to the L2 theorem level of SCT (confidence class A), independent of any dynamical assumptions or free parameters.

Second, based on the above derivation, it systematically resolves the dark energy coincidence problem that has troubled the cosmological community for more than twenty years. This paper presents a three‑fold argument: (1) Geometric locking only holds after time‑body freezing—before the three‑dimensional spatial extension is determined, the geometric dark energy fraction is not yet stable; time‑body freezing is an inevitable step in the layer‑by‑layer freezing process, so the appearance of 0.7 is a necessary consequence of constructive renormalization, not an accident. (2) The dramatic decrease of absolute time flow after time‑body freezing provides a window for the evolution of complex structures—only after time‑body freezing does the universe have sufficiently long physical time to evolve complex life and consciousness. (3) “Necessary coincidence”—observers, as products of evolution, necessarily appear after time‑body freezing, i.e., at the epoch when the dark energy fraction is locked to 0.7. Hence, our observation $\Omega_\Lambda\approx0.7$ is not an accident that needs explanation, but a natural outcome of the co‑action between cosmic evolution and observer conditions. The coincidence problem is completely dissolved in SCT.

This paper further clarifies the essential differences between this resolution and mainstream responses (anthropic principle, dynamical dark energy models), and points out that SCT provides a stronger and more testable explanation than the anthropic principle.

Keywords: dark energy; cosmological constant; eigen-spectrum; static geometry; coincidence problem; constructive renormalization; time flow; Self-Construction Theory