Cosmology Vacuum Uniformity

Uniformity of the vacuum

In cosmology, the vacuum is not empty. It carries an energy density, often called dark energy, that observations suggest is uniform across the cosmos. The standard model of cosmology treats this uniformity as an observed fact. Recognition Science asks a stronger question: could uniformity be forced, not just observed?

The framework's answer rests on a discrete picture of space. It models space as a lattice of points, each point a voxel, a small cell in a three-dimensional grid. The framework's core object is the ledger, a discrete record of recognition events. Each event has a cost, a number measuring what it takes to register that event. The framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1.

The vacuum uniformity module builds on this. It starts with the observation that the lattice has no distinguished location: shifting every voxel by the same displacement leaves the structure unchanged. This is the VoxelSymmetric axiom, proved in the framework's machine-checked library of formal theorems. The module then considers modes that are phase-locked, meaning they are committed ledger entries with a maintenance cost of zero. These modes contribute a constant background cost to every voxel.

The key step is combinatorial. The fraction of phase-locked modes is 11/16, a number that emerges from the structure of the three-dimensional lattice, independent of position. Since this fraction is the same everywhere, and each such mode contributes the same cost, the total vacuum energy density is spatially uniform. The module proves this as the theorem vacuum_energy_uniform: the energy density function is constant across all voxels. It also proves the energy is positive, so the vacuum is not just uniform but has a definite, nonzero density.

In Recognition Science, the framework models the physical identification of this energy with the cosmological constant as a hypothesis, not a theorem. The structural uniformity is proved; the claim that this is the dark energy we observe is a prediction with a named falsifier: a measurement of significant spatial variation in the vacuum energy density would refute it.

The consequence is a striking reversal. In the standard picture, vacuum uniformity is an empirical input. In this framework, it is a theorem: if reality keeps a ledger, the vacuum cannot help but be the same everywhere.

/-- Fraction of phase-locked modes from Q₃ mode budget. -/
noncomputable def passiveFraction : ℝ := 11 / 16

/-- 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

/-- Master certificate. -/
structure VacuumUniformityCert where
  passive_frac : passiveFraction = 11 / 16
  fracs_sum : passiveFraction + activeFraction = 1
  energy_pos : phaseLockEnergy > 0
  uniform : VoxelSymmetric (fun _ => phaseLockEnergy)

What this page does not claim

This does not claim the vacuum energy density equals the measured cosmological constant. This does not claim the lattice model is the actual structure of spacetime. This does not claim the framework derives the fine-structure constant.

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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