dark_energy_evolution_from_ledger
The declaration defines the proposition that the RS-predicted dark energy density parameter satisfies strict bounds between zero and one. Researchers deriving equation-of-state evolution from the ledger cosmological-constant resolution would cite this when establishing non-degenerate behavior for dark energy. The definition is a direct conjunction of the positivity and upper-bound inequalities taken from the upstream omega_lambda expression.
claimLet $Ω_Λ$ denote the dark energy density parameter. The dark energy evolution from ledger is the proposition $0 < Ω_Λ < 1$.
background
In the Recognition Science framework the dark energy density parameter is given by the upstream definition $Ω_Λ = 11/16 - α/π$, where the fraction 11/16 arises from the proportion of vacuum modes in the 8-tick cycle and the correction accounts for matter-coupled perturbations. The module addresses D-006, which asks whether dark energy is constant or evolving, by formalizing structural bounds on $Ω_Λ$ derived from the ledger cosmological-constant resolution. This definition supplies the central proposition that captures the effective equation-of-state evolution bounds.
proof idea
The definition directly equates the proposition to the conjunction of the strict inequalities $0 < omega_lambda$ and $omega_lambda < 1$, where omega_lambda is the upstream definition from EarlyUniverse.
why it matters in Recognition Science
This definition underpins the dark energy evolution structure theorem, which proves the proposition from omega_lambda_bounded, and enables the family of implication theorems that exclude the degenerate cases $Ω_Λ = 0$ and $Ω_Λ = 1$ while confirming positivity and the subunit bound. It fills the D-006 registry item by providing the structural placeholder for equation-of-state evolution, drawing on the eight-tick octave and ledger resolution for the cosmological constant.
scope and limits
- Does not derive the explicit numerical value of $Ω_Λ$.
- Does not specify the functional form of the equation-of-state parameter $w(a)$.
- Does not address time-dependent evolution of dark energy.
- Does not incorporate observational constraints or data fits.
formal statement (Lean)
27def dark_energy_evolution_from_ledger : Prop :=
proof body
Definition body.
28 0 < EarlyUniverse.omega_lambda ∧ EarlyUniverse.omega_lambda < 1
29