omega_lambda_lt_one
Establishes that the Recognition Science dark energy density parameter satisfies Ω_Λ < 1. Early-universe cosmologists working in the RS ledger framework cite it when bounding the vacuum-mode fraction. The proof is a direct linarith reduction after unfolding the definition and invoking positivity of the locked fine-structure correction.
claim$Ω_Λ < 1$ where $Ω_Λ = 11/16 - α_{lock}/π$ and $α_{lock} = (1 - φ^{-1})/2$.
background
The EarlyUniverse module derives early-universe conditions and the dark sector from ledger mode counting under EU-001, D-002, and D-003. The definition of omega_lambda supplies the explicit expression Ω_Λ = 11/16 − α_lock / π, where the leading term is the vacuum fraction of ledger modes in the 8-tick cycle and the correction accounts for matter-coupled perturbations. Upstream results include alphaLock_pos, which proves 0 < α_lock, together with the positivity of π.
proof idea
Unfolds the definition of omega_lambda to obtain the expression 11/16 − alphaLock / Real.pi. Invokes alphaLock_pos and Real.pi_pos to produce a strictly positive quantity, then applies linarith to conclude the difference lies below one.
why it matters in Recognition Science
Supplies the upper bound used in cosmological_constant_resolution to obtain 0 < omega_lambda ∧ omega_lambda < 1, completing the D-003 claim that the cosmological constant equals the ledger vacuum-mode fraction rather than QFT vacuum energy. The result rests on the eight-tick octave (T7) and the locked fine-structure constant from the forcing chain.
scope and limits
- Does not establish positivity of Ω_Λ.
- Does not match the predicted value against observational data.
- Does not address the full 10^120 discrepancy beyond ledger counting.
- Does not incorporate QFT vacuum fluctuations.
Lean usage
theorem cosmological_bounds : 0 < omega_lambda ∧ omega_lambda < 1 := ⟨omega_lambda_pos, omega_lambda_lt_one⟩formal statement (Lean)
58theorem omega_lambda_lt_one : omega_lambda < 1 := by
proof body
Term-mode proof.
59 unfold omega_lambda
60 have h_alpha := alphaLock_pos
61 have h_pi := Real.pi_pos
62 linarith [show (0 : ℝ) < alphaLock / Real.pi from div_pos h_alpha h_pi]
63
64/-! ## Cosmological Constant Problem Resolution -/
65
66/-- **D-003 Resolution**: The cosmological constant is NOT the vacuum energy
67 of QFT. It is the fraction of vacuum modes in the ledger.
68
69 The "10^120 discrepancy" dissolves because:
70 1. QFT vacuum energy is a misidentification (not a physical observable)
71 2. The actual Ω_Λ comes from ledger mode counting: 11/16 − α/π
72 3. This is a NUMBER, not an energy density requiring renormalization
73
74 There is no fine-tuning because there is no parameter to tune. -/