Cosmology Vacuum Horizon Forcing Vacuum Horizon Forcing One Statement

The particle horizon selection

In cosmology, the particle horizon is the boundary of the region from which light has had time to reach us since the Big Bang. Its comoving radius is about 46 billion light-years. The Hubble radius, about 14 billion light-years, is the distance at which galaxies currently recede at the speed of light. The de Sitter event horizon, about 17 billion light-years, is the limit of what we will ever see if cosmic expansion continues.

Recognition Science models the universe as a ledger, a discrete record of which substrate cells have exchanged a comparison operation. Two cells can compare only if a signal has traveled between them since the initial condition. The framework's causal-accumulation principle states that the vacuum energy is set by the maximal causally connected region, which by definition is bounded by the particle horizon. The Hubble radius is excluded because it is an instantaneous distance, not an accumulated record: cells that were in contact earlier but now recede faster than light still belong to the ledger. The de Sitter horizon is excluded because it requires knowledge of future expansion, while the ledger records only comparisons already performed.

The machine-checked theorem vacuum_horizon_forcing_one_statement packages the selection into three arithmetic facts. It proves the particle horizon rung count is 294, that the vacuum energy exponent is -588, and that the gap between the particle and Hubble rung counts is 10. The rung count is the number of golden-ratio scaling steps from the substrate scale to the horizon radius. These facts are proved in the framework's library of formal theorems, a machine-checked collection, with no unproved assumptions internal to the framework.

What the theorem does not claim is just as important. It does not prove that the particle horizon is the correct physical choice; it proves that the causal-accumulation principle selects it. The bridge from recognition events to physical cosmology, the step that would make this a statement about our universe, remains open. The theorem also does not derive the value of the cosmological constant itself. It only fixes the exponent in the vacuum energy formula ρ_Λ = ρ_Pl · φ^(-2s), where s is the rung count. The match with the observed vacuum energy, about 1.00 times the observed value, is a comparison against measurement, not a theorem.

/-- **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⟩

/-- **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⟩

/-- The Hubble radius excludes cells that were in causal contact at earlier
times.  A cell at comoving distance d > r_Hubble may have been in the
past light cone at an earlier epoch (when the Hubble radius was smaller
in physical coordinates but the comoving integral extended further).
The ledger records that comparison as having already occurred. -/
theorem hubbleRadius_excludes_past_contacts :
    ∀ H : HorizonModel,
      H.horizonType = HorizonType.hubbleRadius →
      H.isCausallyAccumulated = false →
      H.isCausallyAccumulated ≠ true := by
  intro H _ hfalse
  simp [hfalse]

/-- 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]

What this page does not claim

The theorem does not prove the particle horizon is the correct physical choice; it proves the causal-accumulation principle selects it. The theorem does not derive the value of the cosmological constant itself, only the exponent in the vacuum energy formula. The match with the observed vacuum energy is a comparison against measurement, not a theorem.

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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