Cosmology Dark Energy Equation Of State Bit Correction Bound

The correction bound

In cosmology, dark energy is often modeled by a single number, w, the ratio of its pressure to its energy density. The cosmological constant, the simplest model, has w = -1 exactly. Observations allow w to drift slightly from that value, and the question is how far. The Recognition Science framework's ledger, a discrete record of recognition events, produces a bound on that drift.

The bound is stated as a definition, not a proved theorem. The declaration bitCorrectionBound sets the maximum deviation to phi - 3/2, where phi is the golden ratio, approximately 1.618. This evaluates to about 0.118. The definition is exact in closed form: because 1/phi equals phi - 1, the expression simplifies precisely to phi - 3/2. The framework's library, a machine-checked collection of formal theorems, records this as a definition with no axioms and no unfinished proofs.

The bound applies to the equation of state's present-day value, w_0. The framework predicts w_0 lies in the interval (-1 - J(phi), -1), approximately (-1.13, -1), where J(phi) is the same 0.118. This interval is the phantom-Carnot ceiling, a name reflecting that the bound comes from a thermodynamic-style limit within the framework's cost structure. The baseline w = -1 remains the cosmological constant case, and the framework's library also certifies that it distinguishes five dark energy models: cosmological constant, quintessence, phantom, quintom, and holographic.

What the bound does not do is prove that dark energy's equation of state actually deviates from -1. It establishes a ceiling on the deviation, not the deviation itself. It also does not derive the value of w_0 from first principles; it sets a limit. The bound is a definitional choice within the framework, not an empirical measurement or a derived theorem about the universe's actual expansion history.

/-- 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

/-- 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

/-- 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

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

What this page does not claim

The bound does not prove that w_0 actually differs from -1. The bound does not derive the measured value of w_0 from first principles. The bound is a definitional ceiling, not a theorem about the universe's expansion history.

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)

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