Cosmology Vacuum Uniformity Vacuum Energy Uniform

Uniformity of the vacuum

The vacuum is not empty. In modern physics it is a seething backdrop of quantum fields, and a central question is whether its energy density is the same everywhere. The classical answer, built into the standard cosmological model, is yes: the cosmological constant is, by definition, uniform across space. This uniformity is what lets cosmologists treat the vacuum as a smooth background that accelerates the expansion of the universe.

The Recognition Science framework approaches the question from a different starting point. It models reality as a discrete record of recognition events, where each event carries a forced cost. The framework's library of machine-checked theorems contains a declaration, vacuum_energy_uniform, that proves a specific statement: the energy density of the vacuum is the same at every location. The proof rests on a symmetry axiom: the framework's carrier space, a three-dimensional grid of integer points, has no distinguished location. A shift of the whole grid by any amount leaves the relevant functions unchanged.

Within the framework, the vacuum's energy comes from a fixed fraction of its modes being "phase-locked," meaning they are committed entries in the ledger with a maintenance cost. That fraction is 11/16, a combinatorial property of the framework's mode structure, independent of position. The theorem then states that the energy per voxel, the framework's unit cell of space, is a constant positive value. The proof is a simple reflexivity check: the function that assigns this constant to every point is, by construction, symmetric under all translations.

What the theorem does not do is identify this framework-internal energy with the physical vacuum energy density measured by cosmologists. The bridge from the framework's phase-locked modes to the stress-energy tensor of general relativity is a hypothesis, not a proved result. The theorem establishes structural uniformity within the framework's own model; the physical identification is a separate, open step. This is the honest scope: a proved statement about a model, not a measurement of the cosmos.

/-- The vacuum energy density function is spatially uniform. -/
theorem vacuum_energy_uniform :
    VoxelSymmetric (fun _ => phaseLockEnergy) :=
  ⟨fun _ _ => rfl⟩

/-- Voxel symmetry: no distinguished location on the carrier.
    This is the ℤ³ translation invariance from VoxelSymmetry. -/
structure VoxelSymmetric (f : ℤ × ℤ × ℤ → ℝ) : Prop where
  shift_invariant : ∀ (v d : ℤ × ℤ × ℤ), f (v.1 + d.1, v.2.1 + d.2.1, v.2.2 + d.2.2) = f v

theorem passive_fraction_lt_one : passiveFraction < 1 := by
  unfold passiveFraction; norm_num

/-- The vacuum J-cost is non-negative (since passiveFraction > 0 and E_coh > 0). -/
theorem vacuum_energy_pos : phaseLockEnergy > 0 := by
  unfold phaseLockEnergy
  exact mul_pos passive_fraction_pos E_coh_pos

What this page does not claim

The theorem does not prove that the framework's vacuum energy equals the physically measured vacuum energy density. The theorem does not derive the value of the cosmological constant from first principles. The theorem does not apply to non-uniform vacuum states, such as those near a black hole.

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

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