lambda_exponent
The declaration supplies the integer 583 as the exponent n such that phi to the minus n reproduces the observed 10^{-122} suppression of the cosmological constant relative to the Planck scale. Cosmologists examining fine-tuning in Recognition Science would cite this value when constructing vacuum-energy predictions from J-cost ground states. It is obtained by direct evaluation of the logarithmic ratio 122 * log(10) / log(phi) rounded to the nearest integer.
claimDefine the positive integer $n = 583$ such that the cosmological constant satisfies $Λ ∝ φ^{-n}$, chosen so that $φ^{-n} ≈ 10^{-122}$ bridges the gap between the observed vacuum energy and the Planck scale.
background
The Cosmology.CosmologicalConstant module derives the cosmological constant from the J-cost ground state of the vacuum in Recognition Science. The vacuum is not empty but carries a nonzero baseline cost due to phi-mismatch, expressed as J_vac = (φ + 1/φ)/2 - 1. Observed Lambda is approximately 10^{-52} m^{-2}, or 10^{-120} times smaller than naive QFT estimates, which the module treats as the worst fine-tuning problem in physics.
proof idea
The definition is a direct constant assignment of the integer 583. This integer is the rounded result of the explicit ratio 122 * log(10) / log(phi) required to produce the target suppression factor.
why it matters in Recognition Science
This supplies the numerical input to hypothesis3, which sets the vacuum energy scale to 1 / phi^lambda_exponent and thereby implements the COS-013 target of resolving the cosmological constant problem via phi-scaling of the J-cost ledger. The exponent connects directly to the phi-ladder and the self-similar fixed point in the forcing chain, yielding the concrete prediction Lambda ~ m_P^2 / l_P^2 * phi^{-583}.
scope and limits
- Does not derive the value of phi or the logarithmic ratio from first principles.
- Does not prove that the observed Lambda matches this scaling exactly.
- Does not address quantum corrections or higher-order terms in the J-cost expansion.
Lean usage
noncomputable def hypothesis3 : ℝ := 1 / phi^lambda_exponentformal statement (Lean)
94noncomputable def lambda_exponent : ℕ := 583
proof body
Definition body.
95