Cosmology Dark Energy Eos Constant Energy Contribution

A constant energy contribution

In standard cosmology, dark energy is the name for whatever is causing the accelerated expansion of the universe. The simplest model treats it as a constant energy density, a fixed amount of energy in every cubic centimeter of space that does not dilute as the universe expands. The key question is how the pressure of this energy relates to its density, a ratio written as w. For an ordinary gas, w is positive. For dark energy, observations point to w being very close to -1, meaning the energy exerts a kind of negative pressure, a suction that drives the expansion faster.

The mathematical object called a constant energy contribution in the Recognition Science framework is a precise way of stating this idea. It is defined as a structure with three parts: a positive real number for the energy density, a proof that this number is greater than zero, and a condition that the energy does not depend on time. The framework then defines the equation of state parameter as the negative of the energy density divided by itself, which simplifies to -1. A theorem in the framework's machine-checked library of formal theorems proves this result: for any such constant energy contribution, the equation of state parameter is exactly -1.

The physical reasoning behind the theorem is the relativistic condition for a vacuum. In general relativity, a Lorentz-invariant vacuum has a stress-energy tensor proportional to the metric, which forces the pressure to be the negative of the energy density. The framework's contribution is to derive this condition from its own ledger picture, where a phase-locked mode has a committed record at a ratio of 1, and its recognition cost is zero at every tick. The energy of such a mode is constant across space and time, so it behaves as a constant energy density and therefore has w = -1.

What the declaration does not claim is that this constant energy density is the actual dark energy in our universe. It proves a conditional statement: if you have a constant energy contribution, then its equation of state is -1. It does not prove that such a contribution exists in nature, nor does it derive the measured value of the dark energy density. The framework also does not claim that w is exactly -1 for all possible dark energy models; it only applies to the specific case of a constant energy density, not to models where the energy changes with time or space.

/-- **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)]

/-- Energy density of a constant (tick-independent) contribution.
    A contribution whose energy doesn't change with volume satisfies
    the thermodynamic relation dE = -p dV.
    If E is constant: dE = 0 = -p dV, so either p = 0 (trivial) or
    we need the relativistic form: ρ + p = 0 for a Lorentz-invariant
    vacuum, giving p = -ρ and w = p/ρ = -1. -/
structure ConstantEnergyContribution where
  energy_density : ℝ
  energy_pos : 0 < energy_density
  tick_independent : True

What this page does not claim

The declaration does not prove that a constant energy contribution exists in nature. The declaration does not derive the measured value of the dark energy density. The declaration does not apply to dark energy models with a time-varying equation of state.

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expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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