Omega_Lambda_bounds
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 in Recognition Science
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.
scope and limits
- Does not derive the exact numerical value of Ω_Λ beyond the stated interval.
- Does not address quantum-field-theory vacuum energy calculations.
- Does not prove the geometric seed 11/16 from first principles.
- Does not treat time evolution or clustering of dark energy.
formal statement (Lean)
107theorem Omega_Lambda_bounds : (0 : ℝ) < Omega_Lambda_RS ∧ Omega_Lambda_RS < (11/16 : ℝ) :=
proof body
Term-mode proof.
108 omega_lambda_bounds
109
110/-! ## C-010: Structural Origin -/
111
112/-- The geometric seed 11/16 from D=3 ledger structure.
113
114 11 = φ⁵ - 1 ≈ 10.09 (approximate, exact value forced by gap-45)
115 16 = 2⁴ = (2³) × 2 = 8 × 2 (from 8-tick structure)
116
117 The ratio 11/16 emerges from the lcm(8, 45) = 360 structure:
118 - 360°/16 = 22.5° (related to electron rung structure)
119 - 11 is the E_pass energy quantum (from cube geometry)
120 -/