Gravity Jcost Inflaton Slow Roll Epsilon Vanishes

The vanishing slow-roll parameter

In cosmology, the slow-roll approximation describes a period of inflation where a scalar field, the inflaton, rolls slowly down its potential energy hill. The approximation's validity is measured by two parameters, ε and η, which must be small. The theorem slow_roll_epsilon_vanishes establishes a precise fact about a specific potential: at the point t = 0, the value of ε is exactly zero. This is a statement about the model's mathematical structure, not a direct observation of the early universe.

The potential in question is derived from a framework called Recognition Science, which models the cost of recognizing an event. In this framework, the cost function J(x) = (x + 1/x)/2 - 1, when expressed in logarithmic coordinates, becomes G(t) = cosh(t) - 1. This is a plateau-shaped potential, a common shape in inflation models. The theorem proves that at t = 0, which corresponds to the vacuum state, the slow-roll parameter ε is zero. This means the field is exactly at a critical point, a local minimum, where it is not rolling at all.

The proof is a simple calculation. The definition of ε is (sinh(t))² / (2 * (cosh(t))²). At t = 0, sinh(0) = 0, so the entire expression is 0. The theorem is part of a larger set of results in the framework's library, a machine-checked collection of formal theorems. This particular result is a lemma, a stepping stone. It is used to build a certificate that the J-cost potential satisfies the conditions for inflation, including the bound ε ≤ 1/2 and a spectral index in the Planck band.

What the theorem does not claim is that this potential is the one realized in our universe. It does not prove that inflation happened, nor that the J-cost potential is the correct model of the early universe. It is a mathematical result about a specific function. The connection between this abstract cost function and the physical inflaton field is a modeling choice, not a proven fact. The framework itself is a mathematical structure, and its application to physics is a hypothesis that must be tested against observation.

/-- **THEOREM**: ε vanishes at the vacuum (t = 0, where inflation ends).
    This confirms J-cost generates a slow-roll inflationary potential. -/
theorem slow_roll_epsilon_vanishes : slow_roll_epsilon 0 = 0 := by
  unfold slow_roll_epsilon
  simp [Real.sinh_zero]

/-- G is the J-cost in log coordinates. -/
theorem G_is_Jcost_log (t : ℝ) : G t = Real.cosh t - 1 := rfl

/-- The first slow-roll parameter ε.
    Standard form for V = G: ε = (V')² / (2(V+1)²)
    For G = cosh(t) − 1: V+1 = cosh(t), V' = sinh(t).
    So ε = sinh²(t) / (2 cosh²(t)) = (tanh(t))² / 2. -/
def slow_roll_epsilon (t : ℝ) : ℝ :=
  Real.sinh t ^ 2 / (2 * Real.cosh t ^ 2)

What this page does not claim

The theorem does not prove that the J-cost potential is realized in the physical universe. The theorem does not claim that inflation actually occurred. The theorem does not derive the fine-structure constant or any other physical constant.

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

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