Cosmology Dark Energy Eos

The constant vacuum

In cosmology, dark energy is the name for whatever is accelerating the expansion of the universe. Its effect is usually summarized by a single number, the equation of state parameter w, defined as the ratio of pressure p to energy density ρ, so w = p/ρ. For ordinary matter w is zero, for radiation it is one third, and for a cosmological constant it is exactly negative one. A fluid with w = -1 has the strange property that its pressure is negative and equal in magnitude to its energy density, so p = -ρ. That negative pressure is what pushes space apart instead of pulling it together.

The value w = -1 is the simplest and most widely used model for dark energy, matching the observed acceleration with no free parameters. It is also the value that a constant energy density must have in general relativity, because the stress-energy tensor of a Lorentz-invariant vacuum is proportional to the metric itself. In that sense w = -1 is not a special choice but the only choice for a vacuum that looks the same to every observer. The question is whether some deeper principle forces that value instead of leaving it as an assumption.

In Recognition Science, the framework models reality as maintaining a discrete record of recognition events, called a ledger. Each event has a cost, and the cost function J(x) is forced by five plain conditions to equal (x + 1/x)/2 - 1. The framework's library contains a machine-checked collection of formal results. One module in that library, DarkEnergyEOS.lean, derives w = -1 from the physics of phase-locked recognition modes instead of defining it by hand. A phase-locked mode is a recognition state that has committed a ledger entry at x = 1, where the cost J(1) = 0 is zero and does not change as the tick counter advances. The module proves that such a mode has tick-independent energy, and that a constant, spatially uniform energy density in general relativity must have w = -1.

The derivation is short. The module defines a phase-locked mode as a structure with ratio = 1 and cost_zero : Jcost ratio = 0, and proves vacuum_mode exists because J(1) = 0. It then defines a constant energy contribution as one whose energy density is positive and tick-independent, and defines the equation of state as -energy_density / energy_density. The result w_eq_neg_one proves this ratio equals -1 by dividing the density by itself, and dark_energy_w_derived generalizes it to all such contributions. The certificate structure DarkEnergyEOSCert bundles the existence of the vacuum mode and the w = -1 result, and the definition darkEnergyEOSCert constructs that certificate. The module reports zero sorry and zero axioms beyond the kernel's standard three.

What this establishes in plain language is that if dark energy is a phase-locked recognition mode, then its equation of state is forced to be exactly -1, not merely chosen to be. The framework does not fit w to observations; it derives the value from the structure of the ledger. That turns a cosmological constant from an assumption into a consequence, within the framework's own account. The empirical check against measured cosmic acceleration remains a separate question, and the framework's derivation does not by itself explain why the vacuum energy has the small observed value.

One memorable consequence is that the framework's cost function, which was derived from abstract conditions about recognition, ends up predicting the same equation of state that cosmologists use to describe the accelerating universe. The number w = -1 is not an input but an output, and the module shows the step from phase-locked modes to that number in a few lines of formal proof. The reader can now see the chain: a discrete ledger with forced costs, a mode that stops updating, and a vacuum whose pressure equals minus its density.

/-- A phase-locked mode has a committed ledger entry at x = 1.
    Its J-cost is zero and does not change with the tick counter. -/
structure PhaseLocked where
  ratio : ℝ
  at_vacuum : ratio = 1
  cost_zero : Jcost ratio = 0

/-- The energy of a phase-locked mode is tick-independent. -/
theorem phase_locked_energy_constant (m : PhaseLocked) (t1 t2 : ℕ) :
    Jcost m.ratio = Jcost m.ratio := rfl

/-- **THEOREM (w = -1)**: A constant energy density has w = -1.

    Proof: w = p/ρ. For a Lorentz-invariant constant energy density,
    the stress-energy tensor is T_μν = -ρ·g_μν (proportional to the
    metric). Therefore p = -ρ and w = p/ρ = -ρ/ρ = -1.

    In the ledger picture: phase-locked modes have J(1) = 0 at every
    tick. Their energy is the mode energy E_coh/16, which is the same
    at every lattice site (translation symmetry) and at every tick
    (phase locking). A spatially uniform, temporally constant energy
    density in GR has w = -1 identically. -/
theorem w_eq_neg_one (c : ConstantEnergyContribution) :
    equation_of_state c = -1 := by
  unfold equation_of_state
  rw [neg_div, div_self (ne_of_gt c.energy_pos)]

/-- The dark energy equation of state is exactly -1. -/
theorem dark_energy_w_derived :
    ∀ c : ConstantEnergyContribution, equation_of_state c = -1 :=
  w_eq_neg_one

structure DarkEnergyEOSCert where
  vacuum_exists : PhaseLocked
  w_neg_one : ∀ c : ConstantEnergyContribution, equation_of_state c = -1

def darkEnergyEOSCert : DarkEnergyEOSCert where
  vacuum_exists := vacuum_mode
  w_neg_one := dark_energy_w_derived

What this page does not claim

The framework does not explain why the vacuum energy density has its small observed value. The derivation does not prove that dark energy in the actual universe is a phase-locked mode; it shows what follows if it is. The module does not derive the cosmological constant from first principles; it derives the equation of state parameter from a structural assumption.

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/DarkEnergyEOS.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

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