Cosmology Dark Energy Eos Equation Of State

The equation of state in Recognition Science

In standard cosmology, the simplest way to describe dark energy is with a single number, w, defined as the ratio of its pressure to its energy density. A value of w = -1 corresponds to a cosmological constant: a form of energy whose density remains constant even as the universe expands. This is the leading model for the observed accelerated expansion, but it is usually put in by hand. The Recognition Science framework derives this value from its own internal logic rather than assuming it.

The derivation starts with a ledger, a discrete record of recognition events. The framework models a phase-locked mode as a state whose recognition cost is zero and does not change with time. A committed ledger entry sits at the value x = 1, where the cost function J(1) = 0. The framework then defines a constant energy contribution as one whose energy density is positive and independent of the tick counter, the discrete time step of the ledger. The equation_of_state declaration computes w as the negative of the energy density divided by itself, which is -1.

The theorem w_eq_neg_one proves that for any such constant energy contribution, the equation of state parameter is exactly -1. The proof uses the thermodynamic relation dE = -p dV: if energy density does not change with volume, the pressure must be negative. For a Lorentz-invariant vacuum, the stress-energy tensor is proportional to the metric, giving p = -ρ and hence w = -1. The framework's library, a machine-checked collection of formal theorems, records this as a theorem with no unproved assumptions.

What this does not claim is that dark energy in the actual universe has been measured to be exactly -1. The theorem applies to a mathematical structure defined within the framework, a constant energy contribution. Observational cosmology constrains w to be close to -1, but the framework's result is a derivation from its own axioms, not a measurement of the cosmos. It also does not claim that the framework's model of dark energy is the only possible one; it shows that within its own definitions, w = -1 follows necessarily.

/-- **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)]

What this page does not claim

The theorem does not measure the actual value of w in the universe. The framework does not claim its model of dark energy is the only possible one. The result does not prove that the universe's expansion is accelerating.

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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