Cosmology Vacuum Uniformity Vacuum Energy Pos

The uniformity theorem

In cosmology, the vacuum energy is the energy that remains in a volume of space when all matter and radiation have been removed. One of the deepest questions about it is whether this energy is the same everywhere, or whether it might vary from place to place. The Recognition Science framework's ledger, a discrete record of recognition events, offers a structural answer: its theorem vacuum_energy_pos establishes that the vacuum energy density is spatially uniform, meaning it takes the same positive value at every location.

The argument rests on a symmetry. The framework models space as a three-dimensional grid of cells, called voxels. A stated axiom says that this grid has no distinguished location: shifting the entire grid by any amount leaves the description unchanged. From this, the framework derives that the energy contributed by phase-locked modes, which are committed entries in the ledger, is constant per voxel. The theorem vacuum_energy_uniform states this constancy directly, and vacuum_energy_pos adds that the constant is positive, not zero.

The positivity follows from two stated facts. First, the fraction of phase-locked modes is 11/16, a number that lies strictly between zero and one. Second, the coherence energy E_coh is positive. The product of a positive fraction and a positive energy is positive, so the phase-locked energy per voxel is positive. The uniformity, meanwhile, is a direct consequence of the grid's translation invariance: a function that ignores its argument and returns the same value everywhere is trivially shift-invariant.

It is important to separate what the theorem establishes from what it does not. The theorem establishes a structural property: within the framework's model, the vacuum energy density is uniform and positive. It does not establish that this model's vacuum energy equals the observed cosmological constant, nor does it derive the numerical value of the vacuum energy density. The identification of the framework's phase-locked energy with the physical stress-energy tensor is a hypothesis, not a result of the framework. The framework's own documentation marks this distinction explicitly: the uniformity is a theorem, while the physical identification is a hypothesis.

What the theorem changes is the shape of the vacuum energy question. Instead of asking whether the vacuum energy might vary, the framework says that within its axioms, uniformity is forced by the structure of the grid itself. The remaining question is whether the framework's model corresponds to the real universe, a question that no formal proof can settle by itself.

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

/-- 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

theorem passive_fraction_pos : passiveFraction > 0 := by
  unfold passiveFraction; norm_num

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

/-- 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 derive the numerical value of the vacuum energy density. The theorem does not prove that the framework's vacuum energy equals the observed cosmological constant. The theorem does not establish that space is actually a discrete grid; that is a modeling choice.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: