Omega_Lambda_lt_upper_bound
The RS prediction for the dark energy density parameter satisfies Ω_Λ < 11/16. Cosmologists examining the cosmological constant problem would cite this bound to place the Recognition Science value below the geometric seed of 0.6875. The proof is a one-line wrapper that invokes the upstream positivity result for the fine-structure correction α/π.
claimLet Ω_Λ := 11/16 - α/π. Then Ω_Λ < 11/16.
background
The Cosmological Constant Derivation module defines the RS prediction as Ω_Λ = 11/16 - α/π, where 11/16 is the geometric seed arising from the D=3 ledger via the eight-tick forcing chain and gap synchronization. This supplies a natural resolution to the cosmological constant problem by producing a value near the observed 0.7 through φ-cancellation without fine-tuning. The upstream theorem omega_lambda_lt_11_16 establishes the strict inequality by proving α/π > 0 via positivity of alpha and π.
proof idea
The proof is a one-line wrapper that directly applies the upstream theorem omega_lambda_lt_11_16. That result unfolds the definition of Omega_Lambda_RS and subtracts a positive quantity α/π from 11/16 using a short positivity chain on alpha and π.
why it matters in Recognition Science
This theorem supplies the upper bound in the C-010 derivation series, confirming that the RS vacuum energy lies below the geometric maximum 11/16. It supports the overall claim that Λ emerges from the eight-tick octave structure (T7-T8) and D=3 ledger without fine-tuning. The bound feeds sibling results such as Omega_Lambda_bounds and closes the upper side of the derivation chain.
scope and limits
- Does not establish a matching lower bound on Ω_Λ.
- Does not derive the value of α from the forcing chain.
- Does not compare against QFT vacuum energy calculations.
- Does not address stability under ledger perturbations.
formal statement (Lean)
93theorem Omega_Lambda_lt_upper_bound : Omega_Lambda_RS < (11/16 : ℝ) :=
proof body
Term-mode proof.
94 omega_lambda_lt_11_16
95
96/-- **THEOREM C-010.3**: Ω_Λ > 0 (positive dark energy).
97
98 Since α/π < 11/16, we have Ω_Λ > 0.
99 This follows from α < 1/2 and π > 1. -/