Omega_Lambda_positive
plain-language theorem explainer
The theorem establishes that the Recognition Science value for the dark energy density parameter exceeds zero. Cosmologists applying the RS framework cite this to confirm positive vacuum energy arising from ledger geometry rather than fine-tuning. The proof is a direct term application of the registry lemma omega_lambda_positive that encodes the inequality 11/16 > alpha/pi.
Claim. $11/16 - alpha/pi > 0$, where the left-hand side is the RS expression for the dark energy density parameter Omega_Lambda derived from the D=3 ledger seed.
background
The Cosmological Constant Derivation module resolves the vacuum energy problem by expressing Omega_Lambda as the difference between a geometric seed 11/16 and the fine-structure correction alpha/pi. The seed 11/16 originates from the eight-tick octave (period 2^3) combined with gap-45 synchronization in three spatial dimensions, as detailed in the module documentation. The referenced definition Omega_Lambda_RS supplies the explicit formula noncomputable def Omega_Lambda_RS : Real := 11/16 - (alpha / Real.pi). Upstream results include the consistent predicate from SAT backprop, which enforces semantic agreement between partial assignments and total solutions in the underlying combinatorial structures.
proof idea
The proof is a one-line wrapper that applies the lemma omega_lambda_positive from Unification.RegistryPredictionsProved. That lemma discharges the strict inequality by invoking alpha < 1/2 together with pi > 1, which together guarantee that the geometric seed exceeds the correction term.
why it matters
This result supplies the C-010.3 step in the cosmological constant derivation chain, confirming that dark energy is positive as required by observation. It rests on the T8 forcing that fixes D=3 and produces the eight-tick structure whose LCM with 45 yields the 11/16 seed. The theorem supports the claim that Lambda emerges from phi-cancellation without fine-tuning and precedes the sibling bounds Omega_Lambda_bounds and Omega_Lambda_lt_upper_bound.
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