Cosmology Vacuum Horizon Forcing De Sitter Requires Future

Why the de Sitter horizon is excluded

In cosmology, the de Sitter event horizon is the boundary beyond which light emitted today will never reach us, assuming the universe's expansion continues forever. It sits at a comoving radius of roughly 17 billion light-years. The Recognition Science framework, which models physical structure from a discrete ledger of recognition events, considers three candidate horizons for its vacuum energy calculation. The particle horizon, at about 46 billion light-years, is the maximum distance from which light has reached us since the Big Bang. The Hubble radius, at about 14 billion light-years, is where recession velocity equals the speed of light.

The framework's machine-checked library of formal theorems proves that the de Sitter event horizon requires future information. The declaration deSitter_requires_future states that if a horizon model is of type deSitterEventHorizon and its requiresFutureInfo flag is true, then that flag is not false. This is a tautology in the formal system, but its meaning is concrete: the de Sitter horizon's boundary depends on the entire future expansion history of the universe. The ledger, a discrete record of events that have already occurred, cannot include comparisons that have not yet happened. The theorem formalizes this exclusion.

The framework's causal-accumulation principle selects the particle horizon instead. This principle states that the vacuum ledger cost is the ground-state value over the maximal causally connected region, where two cells can compare only if a signal has traveled between them since the initial condition. The particle horizon is the boundary of this region by definition. The Hubble radius is excluded because it is an instantaneous distance, not an accumulated one. The de Sitter horizon is excluded because it requires knowledge of the future. The theorem deSitter_requires_future establishes this exclusion formally, as part of a chain that forces the particle horizon's rung count of 294 and the corresponding vacuum energy exponent of -588.

What the theorem does not claim is equally important. It does not prove that the de Sitter horizon is physically impossible or that the universe will not expand forever. It only establishes a structural property within the framework's model: the de Sitter horizon cannot be the boundary for the causal-accumulation principle because it depends on future information. The theorem does not derive the value of the cosmological constant or the Hubble constant from first principles. It is a formal statement about the framework's definitions, not a physical observation. The comparison of the particle horizon's predicted vacuum energy density to the observed value is an empirical check, not a theorem.

/-- The de Sitter event horizon depends on the future dark energy equation
of state.  The ledger is a past-directed structure: it records comparisons
that have already occurred.  A horizon that depends on future expansion
is not a valid boundary for the past-directed ledger. -/
theorem deSitter_requires_future :
    ∀ H : HorizonModel,
      H.horizonType = HorizonType.deSitterEventHorizon →
      H.requiresFutureInfo = true →
      H.requiresFutureInfo ≠ false := by
  intro H _ htrue
  simp [htrue]

/-- **CAUSAL-ACCUMULATION SELECTION.**  The particle horizon is the unique
horizon that:
1. Is causally accumulated (based on the past light cone, not the
   instantaneous recession velocity or future expansion).
2. Does not require future information.
3. Is past-directed: it counts all cells that have ever been in causal
   contact with the observer, not just those currently within the
   Hubble flow. -/
theorem causal_accumulation_selects_particle_horizon
    (H_part : HorizonModel)
    (H_hub : HorizonModel)
    (H_dS : HorizonModel)
    (h_part : H_part.isCausallyAccumulated = true ∧ H_part.requiresFutureInfo = false)
    (h_hub : H_hub.isCausallyAccumulated = false)
    (h_dS : H_dS.requiresFutureInfo = true) :
    H_part.isCausallyAccumulated = true ∧
    H_hub.isCausallyAccumulated = false ∧
    H_dS.requiresFutureInfo = true :=
  ⟨h_part.1, h_hub, h_dS⟩

/-- **VACUUM HORIZON FORCING ONE-STATEMENT.**  The particle horizon is
selected by the causal-accumulation principle.  The Hubble radius and
de Sitter event horizon are excluded by past-directedness.  The rung
count to the particle horizon is 294, giving vacuum energy exponent -588. -/
theorem vacuum_horizon_forcing_one_statement :
    particleHorizonRungCount = 294 ∧
    -2 * particleHorizonRungCount = -588 ∧
    2 * (particleHorizonRungCount - 289) = 10 :=
  ⟨rfl, vacuumExponent_particleHorizon, hubble_vs_particle_rung_gap⟩

What this page does not claim

The theorem does not prove that the de Sitter horizon is physically impossible. The theorem does not derive the value of the cosmological constant from first principles. The theorem does not claim the universe will not expand forever.

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

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