Cosmology Vacuum Horizon Forcing Causal Accumulation Selects Particle Horizon

Horizon selection

In cosmology, a horizon is a boundary in space beyond which an observer cannot see. The three standard candidates are the particle horizon, the Hubble radius, and the de Sitter event horizon. The particle horizon marks the greatest distance from which light has had time to reach us since the Big Bang, roughly 46 billion light-years. The Hubble radius, about 14 billion light-years, is where galaxies currently recede at the speed of light. The de Sitter event horizon, near 17 billion light-years, is the limit of what we will ever see if cosmic expansion continues as it does now.

These three horizons lead to three different predictions for the energy of empty space, the vacuum energy. In the Recognition Science framework, that energy is computed from a ledger, a discrete record of which regions of space have ever exchanged a signal. The record only includes pairs that have been in causal contact, meaning a signal could have traveled between them since the start. The framework's principle is that the vacuum energy comes from the largest region that has fully accumulated such a record. That region's boundary is, by definition, the particle horizon. The Hubble radius fails because it is an instantaneous measure, not an accumulated one: two points that once exchanged a signal remain in the ledger even if they now recede faster than light. The de Sitter horizon fails because it depends on the future expansion history, while the ledger is strictly past-directed, recording only comparisons that have already happened.

The formal statement, causal_accumulation_selects_particle_horizon, is a theorem in the framework's machine-checked library of formal theorems. It proves, from the definitions, that the particle horizon is the only one of the three that is causally accumulated and does not require future information. The theorem is a structural result about the framework's own model of horizons; it is not a measurement of the sky. The numerical agreement that motivates it is separate: the framework's vacuum energy formula, using a rung count of 294 for the particle horizon, lands within about 1 percent of the observed value, while the Hubble radius overshoots by a factor of about 12.7 and the de Sitter horizon by about 5.8. That comparison is an empirical check, not part of the theorem.

What the theorem does not claim is just as important. It does not prove that the particle horizon is the correct horizon for all cosmological purposes; it only shows that, within this framework's causal-accumulation principle, the particle horizon is the selected one. It does not derive the observed vacuum energy from first principles; the rung count of 294 is a definitional input, not a proved number. And it does not say that the Hubble radius or de Sitter horizon are physically meaningless, only that they do not satisfy the framework's specific condition of being a past-directed, accumulated causal record. The theorem is a precise statement about a formal model, and its value is in making that selection rule explicit and checkable.

/-- **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]

/-- 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 theorem does not prove that the particle horizon is the correct horizon for all cosmological purposes. The theorem does not derive the observed vacuum energy from first principles; the rung count is a definitional input. The theorem does not say the Hubble radius or de Sitter horizon are physically meaningless.

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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