Cosmology Dark Energy Equation Of State Dark Energy Eo Scert

The certification record

In cosmology, the equation of state of dark energy is a number, usually written w, that describes how the pressure of dark energy relates to its density. A value of w = -1 is the baseline for a cosmological constant, the simplest explanation of the accelerating expansion of the universe, where the energy density stays constant in time. The Recognition Science framework's library, a machine-checked collection of formal theorems, contains a small structure named DarkEnergyEoSCert that records two facts about this baseline.

The first fact is that the framework's baseline value for w is exactly -1. The second is that the framework distinguishes exactly five models of dark energy: the cosmological constant, quintessence, phantom, quintom, and holographic models. The certificate packages these two statements together, and its construction is verified with no unproved assumptions. This is a formal bookkeeping device, not a new physical claim about the universe.

In Recognition Science, the framework also derives a bound on how far the true value of w might deviate from -1. That bound is |δw| ≤ J(φ), where J(φ) = φ - 3/2 ≈ 0.118, and φ is the golden ratio. This means the framework's prediction for w lies in the interval (-1.118, -1). The bound is stated in closed form, not as an approximation, and it is described as an exact phantom-Carnot ceiling.

What the certificate does not claim is important. It does not assert that the framework has measured the value of w, nor does it claim that the bound is a confirmed observational result. The certificate establishes the formal baseline and the model count, and the bound is a separate derivation within the framework. The certificate itself is a definitional structure, not a theorem about the physical universe.

/-- Dark energy EoS w_0 = -1 (cosmological constant baseline). -/
def wLambda : ℝ := -1

theorem darkEnergyModelCount : Fintype.card DarkEnergyModel = 5 := by decide

def darkEnergyEoSCert : DarkEnergyEoSCert where
  five_models := darkEnergyModelCount
  baseline_w := rfl

/-- BIT correction bound: `|δw| ≤ J(φ)`. This is the exact phantom-Carnot ceiling
`J(φ) = φ − 3/2 ≈ 0.118` in closed form (not an approximation): since `1/φ = φ − 1`,
the earlier obfuscated form `1/φ − 3/2 + 1` equals `φ − 3/2` exactly. -/
noncomputable def bitCorrectionBound : ℝ := phi - 3 / 2

What this page does not claim

The certificate does not assert a measured value for the dark energy equation of state. The bound is a derivation within the framework, not an observational confirmation. The certificate does not claim that the phantom model is physically realized.

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/DarkEnergyEquationOfState.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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