Cosmology Dark Energy Eos Dark Energy W Derived

The derived equation of state

In cosmology, the equation of state parameter w is the ratio of pressure to energy density, p/ρ, for a given component of the universe. For the cosmological constant, the energy density associated with empty space itself, this ratio is exactly -1. The Recognition Science framework contains a formal theorem, dark_energy_w_derived, which derives this value from its own foundational principles rather than assuming it as a definition. The theorem states that for any constant energy contribution, the equation of state parameter equals -1.

The derivation begins with the framework's central concept of a ledger, a discrete record of recognition events. A phase-locked mode is one that has committed a ledger entry at the value x = 1, where the framework's cost function J(x) equals zero. This cost is tick-independent, meaning it does not change as the framework's internal counter advances. The energy of such a mode is therefore constant over time. The theorem then applies the standard thermodynamic relation for a Lorentz-invariant vacuum: a constant energy density ρ implies pressure p = -ρ, hence w = p/ρ = -1.

The proof is carried out in the machine-checked library, a collection of formal theorems verified by a proof assistant. The library shows that a constant energy density has w = -1, and the theorem dark_energy_w_derived generalizes this to all such contributions. This replaces a previous definitional statement, where w was simply set to -1, with a derived result. The framework's library contains no unproven assumptions (0 sorry, 0 axiom) for this result.

What the theorem does not claim is equally important. It does not claim that dark energy in the observed universe actually has w = -1; that remains an empirical question for observational cosmology. It does not claim that the framework's model of dark energy is the correct physical description. The theorem establishes a conditional statement: if a contribution has constant energy density, then its equation of state parameter is -1. The physical identification of dark energy with such a contribution is a separate modeling step, not part of the formal proof.

/-- 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

/-- **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)]

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

What this page does not claim

The theorem does not assert that observed dark energy has w = -1. The theorem does not establish that the framework's model of dark energy is physically correct. The theorem does not derive the value of the energy density itself, only the ratio w.

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)

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Derived articles

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