Cosmology Dark Energy Equation Of State Depth Delta Bound Pos

The delta bound

In cosmology, the dark energy equation of state, w(z), describes how the pressure of dark energy relates to its density as the universe expands. The standard model, ΛCDM, assumes w = -1, a constant. More flexible models allow w to vary with redshift, z, and cosmologists have proposed five canonical forms: ΛCDM, wCDM, the w0wa CPL parametrization, quintessence, and phantom models.

Within the Recognition Science framework, these five models are treated as the five configurations of a discrete depth dimension. The framework's central object is the ledger, a discrete record of recognition events. The framework models the deviation from ΛCDM, written as δ, where w = -1 + δ, as being bounded by a specific constant. This constant, deltaBound, is defined as 1 divided by the golden ratio raised to the fifth power, φ⁵. The golden ratio, φ, is approximately 1.618, so φ⁵ is about 11.09, making deltaBound approximately 0.09017.

The declaration deltaBound_pos is a formally verified theorem in the framework's machine-checked library of formal theorems. It proves that this bound is a positive number, meaning it is greater than zero. This is a foundational property: the bound is not zero and not negative, so the deviation from the cosmological constant is always a strictly positive quantity. A second theorem, deltaBound_small, proves that this bound is less than 0.1, confirming that the framework's allowed deviation is small.

What this does not claim is that dark energy must deviate from a cosmological constant, nor does it claim that the measured value of w(z) is exactly -1 + δ. The theorem only establishes the mathematical positivity of a specific bound. It does not assert that this bound is physically correct or that it matches any observational data. The framework's claim is structural: within its model, the deviation is a positive, small number. Whether that structure describes the actual universe remains a question for observation.

theorem deltaBound_pos : 0 < deltaBound := by
  unfold deltaBound
  exact div_pos one_pos (pow_pos phi_pos 5)

theorem deltaBound_pos : 0 < deltaBound := by
  unfold deltaBound
  exact div_pos one_pos (pow_pos phi_pos 5)

What this page does not claim

The theorem does not claim that the deviation δ is physically realized or that its value is measured. It does not claim that the framework's bound matches any specific observational data. It does not claim that the five canonical models are the only possible ones.

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

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