Cosmology Dark Energy Eos Phase Locked Energy Constant

The phase-locked energy

Dark energy is the name cosmologists give to the unknown something that is accelerating the expansion of the universe. Its defining feature is an equation of state parameter, written w, that relates pressure to energy density. For the simplest form of dark energy, a cosmological constant, w equals exactly -1. The Recognition Science framework contains a machine-checked library of formal theorems, and in that library a declaration called phase_locked_energy_constant establishes a precise statement about one kind of vacuum state.

The framework models reality as maintaining a ledger, a discrete record of recognition events with a forced cost. The cost function J(x) = (x + 1/x)/2 - 1 measures how far a state is from unity, with J(1) = 0. A phase-locked mode is a state that has committed its ledger entry at x = 1, so its cost is zero and stays zero at every tick of the framework's internal counter. The theorem phase_locked_energy_constant states that for any such mode, the cost at one tick equals the cost at any other tick. That is a tautology in the formal system, because the cost is a function of the ratio alone and the ratio is fixed at 1, but it is the hinge on which the derivation turns.

The physical step comes next. The framework argues that a contribution whose energy density does not change with volume or expansion time must, in general relativity, have pressure p = -ρ, which gives w = p/ρ = -1. The library proves this as the theorem w_eq_neg_one: for any constant energy contribution with positive density, the equation of state parameter is exactly -1. The framework then assembles these pieces into a certificate, a structure that packages the existence of a phase-locked vacuum mode and the universal w = -1 result.

What the declaration does not claim is just as important as what it proves. The theorem phase_locked_energy_constant is a statement about a formal object in the framework's ledger model; it does not by itself prove that the physical universe contains such a mode. The derivation from constant energy density to w = -1 relies on the relativistic stress-energy tensor being proportional to the metric, which is a standard physics assumption, not something the framework derives from scratch. The framework does not claim that dark energy is a cosmological constant in the observable universe; it claims that within its model, a phase-locked vacuum mode has exactly that equation of state.

/-- 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)]

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 prove that the physical universe contains a phase-locked vacuum mode. The derivation of w = -1 assumes the standard relativistic form of the stress-energy tensor, not a framework-derived result. The framework does not claim that dark energy is a cosmological constant in the observable universe, only that within its model a phase-locked mode has that equation of state.

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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