Cosmology Dark Energy Cosmic Ratio Large

The cosmic ratio

The age of the universe is about 13.8 billion years, or 4.3 × 10^17 seconds. The Planck time, roughly 5.4 × 10^-44 seconds, is the smallest meaningful interval in modern physics, the scale where quantum effects of gravity become unavoidable. Dividing one by the other gives a pure number, the cosmic ratio, that measures how many Planck-time ticks fit into the lifetime of the cosmos. That ratio is enormous, on the order of 10^60.

A machine-checked library of formal theorems contains a proof that this ratio is greater than 10^60. The proof does not rely on observation; it computes directly from the two defined constants. The declaration cosmic_ratio_large states that the ratio of the universe's age to the Planck time exceeds 10^60. The library defines the universe's age as 4.3 × 10^17 seconds and the Planck time as 5.4 × 10^-44 seconds, then verifies the inequality by arithmetic. This is a theorem about the definitions, not a measurement of the sky.

In Recognition Science, this ratio gains a role. The framework models reality as a ledger, a discrete record of recognition events, and proposes that dark energy emerges from the tension between the ledger's need to balance and the universe's expansion creating new volume. Within that account, the cosmological constant Λ is defined as 3 × H₀², where H₀ is the Hubble constant set to 2.2 × 10^-18 per second. The framework's library proves that this Λ is positive and that the dark energy density parameter exceeds 0.5, meaning dark energy dominates the universe's energy budget. These are consequences of the definitions, not derived from first principles in the machine-checked library.

The cosmic ratio itself is not a prediction of dark energy. It is a statement about the size of the universe's age relative to the Planck time. The framework's larger story, that this ratio explains the smallness of Λ, remains a hypothesis. The library contains a structure for a falsifier, an observation that would disprove the framework's account, but no such observation is recorded. The ratio being large is proved; the reason it is large, and the connection to dark energy, is not.

/-- The cosmic ratio is enormous (Gap-45 scale). -/
theorem cosmic_ratio_large : cosmicRatio > 1e60 := by
  unfold cosmicRatio t_universe t_planck
  norm_num

/-- The age of the universe (in seconds). -/
noncomputable def t_universe : ℝ := 4.3e17  -- ~13.8 billion years

/-- The Planck time (in seconds). -/
noncomputable def t_planck : ℝ := 5.4e-44

/-- The tension energy density is the cosmological constant. -/
noncomputable def cosmologicalConstant : ℝ :=
  -- Λ ≈ (energy to maintain ledger balance) / volume
  -- This scales as H₀² due to the expansion rate
  3 * H0^2  -- In natural units with c = 1

/-- The Hubble parameter today (in natural units, H₀ ≈ 2.2 × 10⁻¹⁸ s⁻¹). -/
noncomputable def H0 : ℝ := 2.2e-18

/-- **THEOREM**: The cosmological constant is positive (repulsive). -/
theorem lambda_positive : cosmologicalConstant > 0 := by
  unfold cosmologicalConstant H0
  norm_num

/-- **THEOREM**: Dark energy dominates the universe today. -/
theorem dark_energy_dominates : omegaLambda > 0.5 := by
  unfold omegaLambda
  norm_num

What this page does not claim

The theorem does not derive the value of Λ or the dark energy density from the ledger structure. The framework does not prove that the cosmic ratio explains the smallness of Λ; that connection is a hypothesis. No measured value of the dark energy density is compared against a framework prediction in this pack.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cosmology/DarkEnergy.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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