Omega_Lambda_bounds
plain-language theorem explainer
The result establishes strict bounds 0 < Ω_Λ < 11/16 on the Recognition Science prediction for the dark energy fraction. Cosmologists addressing the cosmological constant problem cite it to confirm agreement with observations near 0.7 without fine-tuning. Proof proceeds by direct term reduction to the geometric bound lemma from the D=3 ledger structure.
Claim. $0 < Ω_Λ < 11/16$ where $Ω_Λ := 11/16 - α/π$, with $α$ the fine-structure constant and $π$ the circle constant.
background
The module derives the cosmological constant from Recognition Science ledger structure. Ω_Λ is defined as 11/16 minus α/π, where 11/16 is the geometric seed forced by D=3 spatial dimensions and the eight-tick octave. The module states that the vacuum energy equals the J-cost of the empty ledger and that the ratio emerges from lcm(8,45)=360 synchronization with gap-45 on the phi-ladder. Upstream tick supplies the fundamental RS time quantum τ₀=1; rung supplies the phi-ladder indexing used to locate the geometric seed.
proof idea
The proof is a one-line term wrapper that applies the geometric bound lemma. The lemma encodes the upper limit 11/16 directly from the D=3 ledger (8-tick structure times gap-45) and the positivity of α/π.
why it matters
This theorem closes the bounds verification step in the C-010 derivation chain. It relies on T8 forcing of D=3 and the eight-tick octave to produce the 11/16 seed, then subtracts the α/π correction. The result shows vacuum energy arises naturally from ledger geometry rather than Planck-scale cancellation, consistent with the observed Ω_Λ≈0.7.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.