Cosmology Dark Energy Phi Dilution Derivation Carrier Dimension Eq Spacetime

The carrier dimension

In the Recognition Science framework, the dark-energy occupancy fraction θ is the share of recognition events that survive as the universe's dark energy. An earlier law read this fraction off the dimension of the Einstein field equations, calling that number the dark-energy carrier dimension. The theorem carrier_dimension_eq_spacetime establishes that this carrier dimension equals the framework's forced spacetime dimension, and both are 4.

The result is a theorem in the framework's machine-checked library of formal theorems. It proves that the number used in the earlier dark-energy law, darkEnergyCarrierDimension, is exactly the same as the spacetime dimension spacetime_dim that emerges from the framework's forcing chain. The proof is short: both quantities are already proved equal to 4, so the theorem simply identifies them. This is not a new physical assumption; it is a bookkeeping identification that closes a gap in the earlier derivation.

What the theorem does not do is derive the value 4 from scratch. The equality carrier_dimension_eq_spacetime relies on two prior results: darkEnergyCarrierDimension_eq_four and spacetime_dim_eq_four. The first comes from the Einstein field equation structure; the second comes from the framework's Alexander duality argument for three spatial dimensions plus one temporal octave. The theorem itself only says these two numbers agree, not why either is 4.

The practical consequence is that the dark-energy dilution law now has a single, consistent dimension to use. The occupancy fraction θ = φ⁻⁴ is derived from two premises: multiplicative composition across independent dimensions and single-dimension self-similar attenuation. The exponent 4 is the forced spacetime dimension. The carrier dimension theorem ensures the earlier law's dimension and the forced spacetime dimension are the same object, so the derivation is coherent rather than accidental.

/-- The EFE carrier dimension that the prior law read off is exactly the forced spacetime
dimension. -/
theorem carrier_dimension_eq_spacetime :
    DarkEnergyPhiDilutionLaw.darkEnergyCarrierDimension = SpacetimeEmergence.spacetime_dim := by
  rw [DarkEnergyPhiDilutionLaw.darkEnergyCarrierDimension_eq_four,
      SpacetimeEmergence.spacetime_dim_eq_four]

/-- The EFE carrier dimension that the prior law read off is exactly the forced spacetime
dimension. -/
theorem carrier_dimension_eq_spacetime :
    DarkEnergyPhiDilutionLaw.darkEnergyCarrierDimension = SpacetimeEmergence.spacetime_dim := by
  rw [DarkEnergyPhiDilutionLaw.darkEnergyCarrierDimension_eq_four,
      SpacetimeEmergence.spacetime_dim_eq_four]

/-- The EFE carrier dimension that the prior law read off is exactly the forced spacetime
dimension. -/
theorem carrier_dimension_eq_spacetime :
    DarkEnergyPhiDilutionLaw.darkEnergyCarrierDimension = SpacetimeEmergence.spacetime_dim := by
  rw [DarkEnergyPhiDilutionLaw.darkEnergyCarrierDimension_eq_four,
      SpacetimeEmergence.spacetime_dim_eq_four]

What this page does not claim

The theorem does not derive the value 4 from first principles; it identifies two previously proved values. The theorem does not establish that dark energy exists or that the framework's cosmology matches observation. The theorem does not claim the Einstein field equations are derived within the framework.

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/DarkEnergyPhiDilutionDerivation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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