Cosmology Vacuum Horizon Forcing Hubble Vs Particle Rung Gap

The rung gap

In cosmology, the particle horizon and the Hubble radius are two different answers to the question of how far we can see. The particle horizon marks the maximum distance light has traveled since the Big Bang, the accumulated sum of all past contact. The Hubble radius is the instantaneous distance at which galaxies recede at the speed of light today. They are not the same surface, and the difference matters when a theory tries to compute the vacuum energy from a boundary condition.

Recognition Science, a framework that derives physical structure from a discrete record of comparison events, models the vacuum energy as ρ_Λ = ρ_Pl · φ^(-2s), where φ is the golden ratio and s is a count of scaling steps, called rungs, from a substrate scale to a chosen horizon. Three candidate horizons give three different rung counts: 294 for the particle horizon, 289 for the Hubble radius, and 290 for the de Sitter event horizon. The particle horizon's count produces a vacuum energy that matches the observed value to within a factor of about one, while the Hubble radius overshoots by a factor of about 12.7.

The declaration hubble_vs_particle_rung_gap proves the arithmetic fact that 2 × (294 - 289) = 10. In plain language, the particle horizon sits ten rungs above the Hubble radius on the framework's phi-power ladder, and the factor of two in the equation reflects the quadratic form of the vacuum energy exponent. This is a structural theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions and no framework-specific axioms.

What the theorem does not do is explain why the particle horizon wins. The selection comes from a separate principle, the causal-accumulation principle, which states that the ledger records only comparisons that have already occurred. The Hubble radius is excluded because it is instantaneous, not accumulated: cells that were in contact earlier but now recede faster than light remain in the ledger. The de Sitter horizon is excluded because it requires future knowledge. The rung gap theorem merely fixes the numerical distance between two candidate boundaries; the causal principle does the choosing.

In Recognition Science, this gap is what makes the particle horizon the only horizon consistent with a past-directed ledger. The Hubble radius, by contrast, would require the vacuum energy to depend on a surface that no signal has ever crossed, a boundary that exists only in the present instant. The theorem's consequence is that the framework's vacuum energy calculation has a unique, well-defined boundary, and that boundary is the one we can actually observe.

/-- Ratio comparison: the Hubble-radius rung count (289) gives a vacuum
energy that differs from the particle-horizon value by φ^(2·(294-289)) = φ^10. -/
theorem hubble_vs_particle_rung_gap :
    2 * (particleHorizonRungCount - 289) = 10 := by
  unfold particleHorizonRungCount; ring

/-- The vacuum rung count: the number of φ-rungs from the substrate scale
to the horizon.  The vacuum energy scales as φ^(-2s). -/
noncomputable def vacuumEnergyExponent (H : HorizonModel) : ℤ := -2 * H.rungCount

/-- The vacuum rung count: the number of φ-rungs from the substrate scale
to the horizon.  The vacuum energy scales as φ^(-2s). -/
noncomputable def vacuumEnergyExponent (H : HorizonModel) : ℤ := -2 * H.rungCount

/-- **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⟩

What this page does not claim

The rung gap theorem does not explain why the particle horizon is selected over the Hubble radius. The framework does not derive the observed vacuum energy density from first principles without a chosen substrate scale. The framework does not claim that the Hubble radius is physically irrelevant in all cosmological contexts, only that it is excluded as the vacuum energy boundary.

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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