Einstein Field Equations Operator Partition and the Dark Energy Fraction: A Fixed-Point Hypothesis for ΩΛ from the Nonlinear Self-Coupling Structure

The Einstein field equations decompose exactly into linear and nonlinear operator parts. For a system of compactness C = 2GM/rc², this decomposition defines a parameter-free binary partition between the matter-source fraction and the gravitational self-energy fraction. The Thales altitude of this partition — the geometric mean of the two fractions — satisfies a cubic self-consistency condition (C³ + 2C² + C − 1 = 0) whose unique positive root is C_crit = 0.4656. At this critical compactness, the coupling mediator equals the coupling parameter, defining a fixed point. We hypothesize that the present-epoch dark energy fraction Ω_Λ is determined by the condition that the cosmic matter–dark energy partition sits at this fixed point. The hypothesis predicts Ω_Λ = 0.6823, within 0.3σ of the Planck 2018 measurement (Ω_Λ = 0.6847 ± 0.0073). The chain of inference is: Einstein field equations → operator partition → cubic fixed point → predicted Ω_Λ. Steps 1–6 are exact mathematical consequences of the EFE with no free parameters; Step 7 (the identification with the cosmic partition) is the hypothesis. Partition sheet diagnostics computed from Planck data show the cosmic asymmetry χ = 0.369 ≈ 1/e (0.4%), the sheet sound speed v_s = 0.616 ≈ 1/φ (0.4%), and the coupling efficiency η = 0.863 ≈ √3/2 (0.3%). Falsifiable predictions are provided: the partition altitude should track C_crit across redshifts, the local expansion rate should correlate with local matter environment, and the Hubble tension reactive fraction (w ≈ 0.149) should correlate with the density of strongly coupled gravitational nodes. If correct, the cosmological coincidence problem is dissolved: Ω_m and Ω_Λ are comparable today because the cosmic partition is transiting its self-coupling threshold.