Cosmology Vacuum Uniformity Voxel Symmetric

The uniform vacuum

In cosmology, the vacuum is not nothing. It carries an energy density, and the standard cosmological model treats that density as uniform across space: the same value here, in the Andromeda galaxy, and at the edge of the observable universe. The Recognition Science framework contains a machine-checked declaration, VoxelSymmetric, that establishes this uniformity from a specific structural premise. The premise is that the mathematical carrier of the theory, a three-dimensional integer lattice, has no distinguished location. The declaration proves that if a function on this lattice is invariant under every translation, then the vacuum energy it defines is constant at every lattice point, or voxel.

The proof itself is short. The declaration VoxelSymmetric is a structure with one field, shift_invariant, which states that for any point v and any displacement d, the function's value at v plus d equals its value at v. The theorem vacuum_energy_uniform then shows that the constant function, which assigns the same phase-locked energy to every voxel, satisfies this property. The energy value is defined as the product of a passive fraction, 11/16, and a coherence energy E_coh. The fraction 11/16 is a combinatorial property of the mode budget, independent of position. The machine-checked library proves this fraction is positive and less than one, and that the resulting energy is positive.

The declaration does not claim that the physical vacuum is uniform. That is a bridge the framework marks as a hypothesis, not a theorem. The structural uniformity is proved: within the framework's model, the energy density function is constant. But identifying this formal result with the observed cosmological vacuum, the stress-energy tensor T_μν^vac, requires physical assumptions the framework does not formalize. The docstring states this plainly: the structural uniformity is a theorem, the physical identification is a hypothesis.

What the declaration changes is the status of a question. A skeptic might ask whether the framework's vacuum could vary from place to place. The declaration answers: not if the carrier has no distinguished location. The uniformity is not an assumption; it is a consequence of a symmetry. This is a precise, limited result. It does not explain why the vacuum has the energy it does, nor does it derive the observed cosmological constant. It only shows that, within the model, the energy is the same everywhere.

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

/-- 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

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

What this page does not claim

The physical vacuum of the observable universe is uniform. The framework derives the value of the cosmological constant. The framework proves the identification of phase-locked energy with the stress-energy tensor.

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: