omega_lambda_positive
plain-language theorem explainer
The RS dark-energy fraction Ω_Λ = 11/16 − α/π is strictly positive. Cosmologists and anyone citing C-010 (Λ derivation) use this as the lower half of the structural bound 0 < Ω_Λ < 11/16. The proof unfolds the definition, uses π > 1 to get α/π < α, then closes by linear arithmetic from α < 1/2.
Claim. Let $\Omega_\Lambda^{\mathrm{RS}} := 11/16 - \alpha/\pi$, where $\alpha$ is the measured fine-structure constant (CODATA anchor). Then $\Omega_\Lambda^{\mathrm{RS}} > 0$.
background
Module C-010 addresses the cosmological-constant problem: QFT vacuum energy overshoots observation by ~10^120, while data give Ω_Λ ≈ 0.7. Recognition Science replaces that with a forced geometric seed minus an EM correction,
$$\Omega_\Lambda^{\mathrm{RS}} = 11/16 - \alpha/\pi.$$
The seed 11/16 comes from the D=3 ledger (T8 eight-tick structure and gap-45 synchronization; LCM(8,45)=360 yields the 11/16 fraction). α is the single measured input (CODATA fine-structure constant); π is the circle constant. Sibling facts already on hand: α > 0 and α < 1/2 (both by direct numeric unfold of the CODATA anchor).
Positivity is the first structural sanity check: without Ω_Λ > 0 the formula would not describe dark-energy domination, and the later claim that the α/π correction is smaller than the geometric seed would fail.
proof idea
Unfold Ω_Λ^{RS} to 11/16 − α/π. From Real.pi_gt_three get 1 < π, hence α/π < α by div_lt_self using α > 0. Then linarith with the sibling α < 1/2 yields α/π < 1/2 < 11/16, so the difference is positive. Pure real arithmetic; no cosmology-specific lemmas beyond the local α bounds.
why it matters
Feeds the public theorem Ω_Λ > 0 (C-010.3), the two-sided bound 0 < Ω_Λ < 11/16 (C-010.4), and the structural smallness statement α/π < 11/16 (C-010.6), which notes it "follows directly from Ω_Λ > 0." The same positivity is re-exported in RegistryPredictionsProved (omega_lambda_positive, omega_lambda_bounds) and is one of the witnesses inhabiting registry_predictions_cert_exists.
In the RS chain this is the lower edge of the Λ resolution: T8 forces D=3 and the eight-tick octave, the geometric seed is 11/16, and α/π is a small IR correction. Establishing Ω_Λ > 0 without fine-tuning is what lets the framework claim the vacuum energy is just the residual J-cost of the empty ledger rather than a Planck-scale catastrophe.
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