Cosmology Vacuum Horizon Forcing

Choosing the horizon

In cosmology, a horizon is a boundary beyond which an observer cannot see or reach. The standard model of the universe offers three candidates: the particle horizon, the farthest distance light has traveled since the Big Bang; the Hubble radius, where galaxies recede at the speed of light; and the de Sitter event horizon, the limit of what we will ever see given cosmic expansion. Each one implies a different value for the vacuum energy, the energy density of empty space, and only one matches observations.

Recognition Science approaches this with a discrete ledger. The framework models reality as a lattice of cells that exchange comparisons, and the cost of those comparisons is recorded in a ledger, a discrete record of events. The causal-accumulation principle states that the vacuum energy is the ground state of the total ledger cost over the maximal region where cells have actually been in contact. This selects the particle horizon, because it is the only boundary defined by past contact alone. The Hubble radius fails because it is an instantaneous measure, and the de Sitter horizon fails because it requires knowing the future expansion of the universe.

The module VacuumHorizonForcing formalizes this selection in a machine-checked library of formal theorems. It defines a causal contact relation that is reflexive, symmetric, and permanent: once two cells have compared, they stay in contact. It then proves that the particle horizon satisfies the causal-accumulation property while the other two do not. The central theorem, causal_accumulation_selects_particle_horizon, states this distinction explicitly. The library also records the rung count of 294 from the substrate scale to the particle horizon, and the resulting vacuum energy exponent of -588, which reproduces the observed vacuum energy density to within a factor of about one.

What this establishes is a structural reason for a cosmological coincidence. The vacuum energy is not a free parameter but a consequence of which horizon you choose, and the choice is forced by the principle that only past events count. The framework does not measure the vacuum energy; it derives the selection rule that makes the measured value the only consistent one.

/-- **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⟩

/-- **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

This does not claim that the vacuum energy density is derived from first principles without any input. This does not claim that the particle horizon is the only horizon with any causal significance. This does not claim that the framework's ledger model is the standard cosmological model.

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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