omega_lambda_bounded
The declaration establishes that the Recognition Science dark energy density parameter lies strictly between zero and one. Researchers modeling cosmic evolution under the RS ledger framework would reference this bound to ensure consistency with observed dark energy dominance without fine-tuning issues. The proof proceeds by direct invocation of the cosmological constant resolution theorem, which itself derives the bounds from ledger mode counting in the eight-tick cycle.
claimIn the Recognition Science framework, the dark energy density parameter satisfies $0 < Ω_Λ < 1$, where $Ω_Λ = 11/16 - α/π$ and $α$ is the RS fine-structure constant.
background
The module addresses D-006 on whether dark energy is constant or evolving, formalizing bounds on Ω_Λ from the ledger cosmological-constant resolution. The key definition is the RS prediction Ω_Λ = 11/16 - α/π, arising as the fraction of vacuum modes in the 8-tick cycle with a small correction from matter-coupled modes. Upstream, the cosmological constant resolution theorem states that the cosmological constant is the fraction of vacuum modes in the ledger, dissolving the 10^120 discrepancy because QFT vacuum energy is a misidentification and the actual Ω_Λ is a number from ledger mode counting: 11/16 − α/π.
proof idea
The proof is a one-line term wrapper that directly applies the cosmological constant resolution theorem from the Early Universe module.
why it matters in Recognition Science
This theorem supplies the positivity and sub-unity bounds required by the parent result dark_energy_evolution_structure, which asserts the dark energy evolution from ledger. It fills the D-006 registry item by confirming Ω_Λ bounds from the ledger resolution, tying into the eight-tick octave where the 11/16 fraction originates.
scope and limits
- Does not compute the numerical value of Ω_Λ.
- Does not model the time dependence of dark energy.
- Does not incorporate observational constraints from cosmology.
- Does not address potential evolution beyond the structural bounds.
Lean usage
theorem dark_energy_evolution_structure : dark_energy_evolution_from_ledger := omega_lambda_boundedformal statement (Lean)
22theorem omega_lambda_bounded :
23 0 < EarlyUniverse.omega_lambda ∧ EarlyUniverse.omega_lambda < 1 :=
proof body
Term-mode proof.
24 EarlyUniverse.cosmological_constant_resolution
25
26/-- Structural placeholder for effective equation-of-state evolution. -/