dark_energy_implies_ne_zero
The theorem shows that the ledger-derived dark energy evolution structure forces the density parameter Ω_Λ away from zero. Recognition Science cosmologists cite it when confirming that dark energy must be present rather than absent under the structural bounds. The proof is a one-line wrapper applying ne_of_gt to the positivity conjunct of the hypothesis.
claimIf $0 < Ω_Λ < 1$, then $Ω_Λ ≠ 0$, where $Ω_Λ = 11/16 - α_{lock}/π$ is the RS dark energy density parameter.
background
Module D-006 formalizes the RS framework for dark-energy equation-of-state behavior, starting from the ledger resolution of the cosmological constant. The hypothesis dark_energy_evolution_from_ledger is the proposition $0 < Ω_Λ < 1$. Upstream, omega_lambda is defined as the noncomputable real $Ω_Λ := 11/16 - α_{lock}/π$, whose positivity follows from the fraction of unexcited ledger modes in the eight-tick cycle.
proof idea
The proof is a one-line wrapper that applies ne_of_gt to the first conjunct of the hypothesis, which asserts that omega_lambda is strictly positive.
why it matters in Recognition Science
This result excludes the degenerate Ω_Λ = 0 endpoint, reinforcing that dark energy exists under the ledger assumption and supporting the D-006 registry item on constant versus evolving dark energy. It aligns with the eight-tick octave (T7) in the forcing chain. No downstream uses are recorded, leaving its role in larger cosmological derivations open.
scope and limits
- Does not derive the explicit value of Ω_Λ from the full forcing chain.
- Does not establish the upper bound Ω_Λ < 1 independently of the hypothesis.
- Does not address time dependence or equation-of-state evolution.
- Does not connect to the Recognition Composition Law or J-uniqueness.
formal statement (Lean)
43theorem dark_energy_implies_ne_zero (h : dark_energy_evolution_from_ledger) :
44 EarlyUniverse.omega_lambda ≠ 0 :=
proof body
Term-mode proof.
45 ne_of_gt h.1
46
47/-- Dark-energy structure excludes the degenerate `Ω_Λ = 1` endpoint. -/