Cosmology Dark Energy Evolution Structure Dark Energy Evolution From Ledger

The bounded density

Dark energy is the name cosmologists give to the unknown force that appears to accelerate the expansion of the universe. Standard cosmology measures its share of the universe's total energy as a positive number less than one, roughly 0.7. The Recognition Science framework, which derives physical structure from a discrete record of recognition events, formalizes exactly this boundedness as a theorem about its own model of the early universe.

The declaration dark_energy_evolution_from_ledger is a definition, not a measurement. It states that the framework's early-universe dark energy density, written ΩΛ, is greater than zero and less than one. The library proves this as a theorem: the density is positive, it is below one, and it is neither zero nor one. These are the two endpoints the framework excludes. A zero density would mean dark energy plays no role at all; a density of one would mean dark energy is the entire universe.

In Recognition Science, this boundedness follows from the framework's resolution of the cosmological constant problem, the puzzle of why the observed vacuum energy is so much smaller than quantum field theory predicts. The theorem omega_lambda_bounded derives the two inequalities directly from that resolution. The definition and its three corollary theorems are all machine-checked in the framework's library of formal theorems.

What the declaration does not do is model how dark energy changes over time. The name suggests evolution, but the definition is static: it only pins the density to the open interval between zero and one. The framework marks the question of whether dark energy is constant or evolving as started, not finished. The structural placeholder for effective equation-of-state evolution remains a placeholder, a target for future work rather than a claim.

/-- Structural placeholder for effective equation-of-state evolution. -/
def dark_energy_evolution_from_ledger : Prop :=
  0 < EarlyUniverse.omega_lambda ∧ EarlyUniverse.omega_lambda < 1

/-- Baseline RS dark-energy density is positive and subunitary. -/
theorem omega_lambda_bounded :
    0 < EarlyUniverse.omega_lambda ∧ EarlyUniverse.omega_lambda < 1 :=
  EarlyUniverse.cosmological_constant_resolution

/-- Dark-energy evolution structure enforces positivity of `Ω_Λ`. -/
theorem dark_energy_implies_positive (h : dark_energy_evolution_from_ledger) :
    0 < EarlyUniverse.omega_lambda :=
  h.1

/-- Dark-energy evolution structure enforces the subunit upper bound. -/
theorem dark_energy_implies_subunit (h : dark_energy_evolution_from_ledger) :
    EarlyUniverse.omega_lambda < 1 :=
  h.2

/-- Dark-energy structure excludes the degenerate `Ω_Λ = 0` endpoint. -/
theorem dark_energy_implies_ne_zero (h : dark_energy_evolution_from_ledger) :
    EarlyUniverse.omega_lambda ≠ 0 :=
  ne_of_gt h.1

/-- Dark-energy structure excludes the degenerate `Ω_Λ = 1` endpoint. -/
theorem dark_energy_implies_ne_one (h : dark_energy_evolution_from_ledger) :
    EarlyUniverse.omega_lambda ≠ 1 :=
  ne_of_lt h.2

/-- Baseline RS dark-energy density is positive and subunitary. -/
theorem omega_lambda_bounded :
    0 < EarlyUniverse.omega_lambda ∧ EarlyUniverse.omega_lambda < 1 :=
  EarlyUniverse.cosmological_constant_resolution

What this page does not claim

The declaration does not model how dark energy changes over time. The declaration does not give a numerical value for ΩΛ. The declaration does not claim dark energy is constant, only that it is bounded.

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/DarkEnergyEvolutionStructure.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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