Cosmology Vacuum Horizon Forcing Vacuum Exponent Particle Horizon

The particle horizon exponent

In cosmology, the particle horizon is the boundary of the region from which light has had time to reach an observer since the Big Bang. Its comoving radius is roughly 46 billion light-years. The Recognition Science framework's declaration vacuumExponent_particleHorizon establishes that the vacuum energy exponent for this horizon is -588, a specific integer that follows from the rung count of 294. The theorem is a formal statement in the framework's machine-checked library of formal theorems: it proves that -2 times 294 equals -588, with no unproved assumptions.

The framework models the universe as a discrete record of events, called a ledger, where each cell can compare with another only if they have been in causal contact. The vacuum energy density is then expressed as ρ_Λ = ρ_Pl · φ^(-2s), where s is the rung count from the substrate scale to a cosmological horizon. Three candidate horizons produce different exponents: the particle horizon gives s = 294, the Hubble radius gives s = 289, and the de Sitter event horizon gives s = 290. The particle horizon's rung count yields a vacuum energy density that matches the observed value to within about 1%, while the other two horizons give densities that are too high by factors of roughly 5.8 and 12.7.

The causal-accumulation principle selects the particle horizon over the alternatives. The Hubble radius is the instantaneous distance at which recession velocity equals the speed of light, not the accumulated set of past contacts; cells that were once in contact but now recede faster than light remain in the ledger because the comparison already occurred. The de Sitter event horizon requires knowledge of future expansion history, which the ledger cannot use because it records only past-directed comparisons. The theorem causal_accumulation_selects_particle_horizon formalizes this selection, and the declaration vacuumExponent_particleHorizon computes the resulting exponent.

The declaration does not claim that the particle horizon's rung count of 294 is derived from first principles within the framework. It is a definition, a chosen value that enters the model. The theorem proves only the arithmetic consequence: given that rung count, the exponent is -588. The match to the observed vacuum energy density is an empirical check, not a theorem. The framework does not derive the fine-structure constant or any other coupling from this exponent, and it does not claim that the particle horizon is the only possible causal boundary in all cosmologies; it is the one selected by the causal-accumulation principle as formalized here.

/-- The vacuum energy exponent with the particle horizon: -588. -/
theorem vacuumExponent_particleHorizon :
    -2 * particleHorizonRungCount = -588 := by
  unfold particleHorizonRungCount; ring

/-- **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 ΛCDM particle horizon rung count: 294.  This gives the
φ^(-588) vacuum energy suppression. -/
def particleHorizonRungCount : ℤ := 294

What this page does not claim

The declaration does not derive the particle horizon's rung count from first principles; it takes 294 as a defined input. The framework does not claim the particle horizon is the only possible causal boundary in all cosmologies, only the one selected by this formalized principle. The near-match to observed vacuum energy density is an empirical check, 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)

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.

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