omega_lambda_gt_683
plain-language theorem explainer
The dark-energy density parameter exceeds 0.683 under the RS formula Ω_Λ = 11/16 − α/π. Cosmologists comparing the vacuum-mode prediction to Planck bounds cite this lower edge. The proof subtracts a strict upper bound on the electromagnetic correction from the exact raw saturation 0.6875 and closes by linear arithmetic.
Claim. The cosmological dark-energy fraction $\Omega_\Lambda = \frac{11}{16} - \frac{\alpha}{\pi}$, with $\alpha$ the measured CODATA fine-structure constant, satisfies $\Omega_\Lambda > 0.683$.
background
In the Ω_Λ derivation module, the dark-energy fraction comes from phase-mode saturation on the eight-tick cycle. The raw saturated fraction is the ratio of Q₃-symmetric vacuum modes (11) to the 4-bit tick-addressing space (16), giving exactly 11/16 = 0.6875. An electromagnetic one-loop correction α/π is then subtracted, using measured CODATA α as the single external input.
Locally, Ω_Λ is defined as raw saturation minus that EM correction. Upstream, omega_raw_val pins the raw term at 0.6875 exactly, and em_correction_small asserts the correction is strictly less than 0.004 (from α < 0.0073 and π > 3.14). The module target is consistency with Planck 2018, Ω_Λ = 0.6889 ± 0.0056.
proof idea
Unfold Ω_Λ as raw saturation minus EM correction. Apply omega_raw_val to replace the raw term by the exact decimal 0.6875, and em_correction_small for the strict bound correction < 0.004. Linear arithmetic then yields 0.6875 − correction > 0.683.
why it matters
This lower bound pairs with the companion upper bound to form the closed interval theorem Ω_Λ ∈ (0.683, 0.686). That interval feeds the Planck-consistency certificate, which shows the RS prediction lies within 2σ of Planck 2018 (0.6889 ± 0.0056). The construction rests on the eight-tick octave (T7) and the 11-mode Q₃ count from the Gray-code asymmetry of the mode budget. The formula retains one measured input (CODATA α); the module notes that an earlier attempt to swap in a constructed α was reverted because the seed identification is not a derived coupling.
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