Cosmology Dark Energy Eos Phase Locked

Phase-Locked Vacuum

In cosmology, the equation of state parameter w is the ratio of pressure to energy density. For dark energy, observations are consistent with w = -1, the value that makes the energy density constant as the universe expands. The Recognition Science framework contains a formal structure called PhaseLocked that aims to derive, not assume, this value.

In the framework's ledger picture, a discrete record of recognition events, a phase-locked mode is one whose recognition state does not update. The structure PhaseLocked defines a ratio that is fixed at 1, and a cost function J that evaluates to zero at that ratio. The theorem phase_locked_energy_constant states that this cost is the same at every tick, meaning the energy of such a mode is constant over time.

From this tick-independence, the framework derives the equation of state. A constant energy density, one that does not change with volume, satisfies the relativistic condition ρ + p = 0 for a Lorentz-invariant vacuum. The theorem w_eq_neg_one proves that for any ConstantEnergyContribution, the equation of state parameter equals -1. The certificate darkEnergyEOSCert packages both the existence of a vacuum mode and the derivation of w = -1.

The derivation is a formal theorem in the machine-checked library, with zero axioms beyond the standard three. It replaces a previous definitional choice, w := -1, with a proved statement. The framework's claim is that the constant energy density, which is the defining feature of a cosmological constant, forces the equation of state to be exactly -1.

What PhaseLocked does not claim is that the universe's dark energy is actually phase-locked. The structure establishes a conditional statement: if a mode is phase-locked, then its energy is constant, and if a contribution has constant energy density, then w = -1. It does not prove that such a mode exists in the physical universe, nor does it predict the measured value of the dark energy density itself. The framework provides a formal mechanism, not an observational claim.

/-- 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 framework does not prove that the universe's dark energy is phase-locked. The framework does not derive the measured value of the dark energy density. The framework does not claim that w is exactly -1 for all possible dark energy models, only for those with constant energy density.

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.

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