Cosmology Vacuum Horizon Forcing Mem Causal Neighborhood Self

The self-inclusion lemma

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. The Recognition Science framework formalizes causal contact as a relation between cells on a substrate lattice, and its declaration mem_causalNeighborhood_self proves a basic property: every cell is in causal contact with itself. This follows from the reflexivity condition built into the framework's definition of causal contact, which states that each cell is trivially in contact with itself.

The framework models the universe as a discrete ledger of recognition events, where a recognition event is a comparison between two cells. The ledger records only comparisons that have actually occurred. The causal-accumulation principle states that the vacuum energy is the ground-state value of the total ledger cost over the maximal causally connected region. The boundary of this region is, by definition, the particle horizon: the maximum comoving distance from which a signal has reached the observer.

This self-inclusion lemma is a necessary foundation for the framework's derivation that the particle horizon, not the Hubble radius or the de Sitter event horizon, is the correct boundary for the vacuum energy calculation. The Hubble radius is excluded because it represents the instantaneous recession velocity equal to c, not the accumulated set of past contacts. The de Sitter horizon is excluded because it requires future expansion history, while the ledger is past-directed. The particle horizon, with a rung count of 294, yields a vacuum energy density within a factor of about 1.00 of the observed value, a numerical agreement the framework presents as a structural result.

What the theorem does not claim is that the universe is literally a discrete ledger, or that the particle horizon is the only possible horizon. It establishes a formal property within a specific model: that the causal neighborhood of a cell contains itself. The physical identification of the ledger with spacetime, and the derivation of the vacuum energy from this principle, are framework-internal claims that go beyond this single lemma.

/-- Every cell is in its own causal neighborhood. -/
theorem mem_causalNeighborhood_self {Λ : Type*}
    (C : CausalContactRelation Λ) (i : Λ) :
    i ∈ causalNeighborhood C i :=
  C.refl i

/-- **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 universe is literally a discrete ledger of recognition events. The particle horizon is the only boundary that could be chosen for the vacuum energy calculation. The self-inclusion lemma alone derives the vacuum energy density.

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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