Cosmology Dark Energy Evolution Structure Dark Energy Implies Subunit

The subunit bound

In cosmology, the density of dark energy is usually written as Ω_Λ, a number that compares the actual dark energy density to the critical density needed to halt cosmic expansion. A value of 1 would mean dark energy alone closes the universe; a value of 0 would mean there is no dark energy at all. The Recognition Science framework, which builds physical structure from a discrete record of recognition events, formalizes a structural claim about this number.

The declaration dark_energy_implies_subunit, a theorem in the framework's machine-checked library of formal theorems, states that if the framework's dark-energy evolution structure holds, then Ω_Λ is less than 1. The same structure also forces Ω_Λ to be greater than 0, so the full picture is a strict inequality: 0 < Ω_Λ < 1. The two endpoints are excluded as degenerate cases. This is not a numerical prediction with a fitted value; it is a structural bound, derived from the framework's ledger-based resolution of the cosmological constant.

What the declaration does not claim matters as much as what it proves. It does not say whether dark energy is constant or evolving over time. The framework labels that question D-006 and marks it as STARTED, meaning only the density bounds are formalized so far. The theorem also does not produce a specific number for Ω_Λ, and it does not claim that the framework's structure is the only way to obtain such bounds. The bound is a consequence of the framework's own assumptions, not an independent measurement.

In plain terms, the result narrows the allowed range for dark energy density within the framework: it must be positive but not enough to close the universe on its own. That leaves room for a wide family of possible behaviors. The practical consequence is that the framework can rule out the degenerate endpoints while remaining silent on the dynamics. A reader should not take this as evidence that dark energy evolves, only that the framework's current formal structure places it strictly between zero and one.

/-- 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

/-- 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

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

What this page does not claim

The theorem does not claim dark energy is constant or evolving over time. The theorem does not produce a specific numerical value for Ω_Λ. The theorem does not claim the framework's structure is the only way to obtain such bounds.

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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