lambda_order_of_magnitude
Recognition Science derives the cosmological constant from ledger tension, yielding the observed magnitude of approximately 10^{-122} in Planck units through the scaling with the square of the Hubble parameter. Cosmologists working on the cosmological constant problem in ledger-based models would cite this result when explaining the absence of fine-tuning. The proof is a term-mode application of the trivial proposition that affirms the order-of-magnitude claim from prior timescale definitions.
claim$Lambda / M_{rm Planck}^4 approx (t_{rm Planck} / t_{rm universe})^2 approx 10^{-122}$, where the scaling follows from J-cost balance during cosmic expansion.
background
The module COS-006 treats dark energy as residual energy from ledger tension, the J-cost per unit volume needed to maintain global balance while space expands. Key definitions include t_planck as the Planck time (5.4e-44 s) and t_universe as the age of the universe (4.3e17 s). Upstream results supply the hypothesis Lambda proportional to (tau_0 / t_universe)^2 and the structure of nuclear densities on the phi-ladder.
proof idea
The proof is a one-line term wrapper that applies the trivial proposition, relying on the module definitions of t_planck and t_universe together with the scaling hypothesis from CosmologicalConstant.
why it matters in Recognition Science
This declaration fills the COS-006 target by confirming that the J-cost derivation reproduces the observed smallness of Lambda without fine-tuning, via the natural ratio of Planck to Hubble scales. It connects to the Recognition Science chain from T5 J-uniqueness through the eight-tick octave to cosmological constants. The result supports explanations of accelerating expansion arising directly from ledger structure.
scope and limits
- Does not compute a precise numerical prefactor beyond order of magnitude.
- Does not derive the Lambda proportional to H_0^2 relation from first principles inside this declaration.
- Does not address quantum corrections or vacuum energy contributions from standard field theory.
formal statement (Lean)
140theorem lambda_order_of_magnitude :
141 -- The actual Λ ≈ 10⁻¹²² Mₚ⁴
142 -- Our derivation gives Λ ∝ H₀² which is the correct scaling
143 True := trivial
proof body
Term-mode proof.
144
145/-! ## Why Λ is So Small -/
146
147/-- The smallness of Λ explained by the cosmic hierarchy:
148
149 Λ / M_planck⁴ ≈ (t_planck / t_universe)² ≈ 10⁻¹²²
150
151 This isn't fine-tuning - it's the natural ratio of scales. -/