omega_lambda_positive
The declaration proves that eleven sixteenths minus alpha over pi is positive. Cosmologists working on the smallness of the cosmological constant in Recognition Science would cite it to fix the sign of dark energy density. The proof first extracts alpha less than one half from the upstream structural bound alphaInv greater than two, then uses pi greater than one to place the ratio strictly below eleven sixteenths via linear arithmetic.
claim$11/16 - (α / π) > 0$, where α is the fine-structure constant defined by reciprocal of the exponential resummation of the geometric seed 4π·11.
background
In the Registry Predictions module the theorem supplies a calculated lower bound for the cosmological constant density parameter. The fine-structure constant α is defined as the reciprocal of alphaInv, itself the product of the seed 4π·11 and an exponential correction exp(−f_gap/alpha_seed). The upstream lemma alphaInv_gt_2 supplies the key inequality alphaInv > 2 that drives the chain.
proof idea
The tactic proof first obtains positivity of alpha and alphaInv by unfolding definitions and applying positivity. It derives alpha < 1/2 from alphaInv_gt_2 via field_simp and div_lt_div_iff₀, then invokes Real.pi_gt_three to obtain pi > 1 and shows alpha/pi < alpha by div_lt_self. The final linarith step closes the comparison against 11/16.
why it matters in Recognition Science
The result is invoked directly by Omega_Lambda_positive and omega_lambda_bounds in the CosmologicalConstantDerivation module, completing the calculated proof for registry item C-010. It supplies the positivity half of the bound 0 < Ω_Λ < 11/16 that resolves the sign of dark energy density from the alpha derivation. The theorem therefore closes one concrete prediction in the unification registry without introducing free parameters.
scope and limits
- Does not derive the numerical value of alpha beyond the inequality alpha < 1/2.
- Does not discharge the sorry inside the upstream alphaInv_gt_2 lemma for the gap bound.
- Does not establish the expression for Ω_Λ itself.
- Does not address higher-order corrections or precision refinements to alpha.
Lean usage
theorem Omega_Lambda_positive : Omega_Lambda_RS > 0 := omega_lambda_positiveformal statement (Lean)
117theorem omega_lambda_positive : 11/16 - (alpha / Real.pi) > 0 := by
proof body
Tactic-mode proof.
118 have h_alphaInv_pos : alphaInv > 0 := by unfold alphaInv alpha_seed; positivity
119 have h_alpha_pos : alpha > 0 := by unfold alpha; positivity
120 -- alpha < 1/2 since alphaInv > 2
121 have h_alpha_lt_half : alpha < 1/2 := by
122 have h_eq : alpha = 1/alphaInv := by unfold alpha; field_simp
123 rw [h_eq]
124 rw [div_lt_div_iff₀ h_alphaInv_pos (by norm_num : (0:ℝ) < 2)]
125 linarith [alphaInv_gt_2]
126 -- alpha/pi < alpha (since pi > 1)
127 have h_pi_gt_1 : Real.pi > 1 := by linarith [Real.pi_gt_three]
128 have h_ratio : alpha / Real.pi < alpha := div_lt_self h_alpha_pos h_pi_gt_1
129 linarith
130
131/-- **BOUNDS**: 0 < Ω_Λ < 11/16 -/