Gravity Jcost Inflaton

The inflaton potential

Inflation is the leading theory for the first instants of the universe: a period of extraordinarily rapid expansion, driven by a field called the inflaton. The inflaton's potential energy, a function of the field's value, determines how the expansion proceeds. The standard slow-roll approximation describes inflation when the potential is flat enough that the field rolls slowly. Two parameters, ε and η, measure that flatness; inflation proceeds while both stay small.

A well-known example is the Starobinsky potential, which produces a plateau at large field values. In log coordinates, where t = ln(x), this potential takes the form cosh(t) − 1. This is also exactly the shape of the J-cost function from Recognition Science when written in those coordinates. The J-cost is the unique function forced by a recognition composition law, and its log-coordinate form is G(t) = cosh(t) − 1.

In Recognition Science, the framework models the inflaton potential as this same J-cost. The framework's library proves that this potential has a minimum at t = 0, where the cost is zero and the first derivative vanishes, making it a critical point. The second derivative at that point equals 1, which is the calibration constant. From this curvature, the framework derives the α-attractor parameter α = φ², where φ is the golden ratio, and the spectral index n_s = 1 − 2/N for a number of e-folds N.

For N = 55 e-folds, the spectral index is 1 − 2/55 ≈ 0.9636, which lies within the 0.96 to 0.97 band reported by the Planck satellite. The framework also derives the tensor-to-scalar ratio r = 12φ²/N². These are the predictions of the Starobinsky model, recovered here from first principles rather than assumed.

What changes is that the potential is not chosen. It is the unique function satisfying the recognition composition law, and its log-coordinate form is the Starobinsky plateau. The framework's library proves that the slow-roll parameter ε vanishes at the vacuum and is bounded by 1/2 everywhere, and that η equals 1. The master certificate `InflationFromJCostCert` packages these results with zero axioms.

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

/-- G(0) = 0: the vacuum (x = 1, t = 0) has zero cost. -/
theorem G_at_zero : G 0 = 0 := by
  unfold G; simp [Real.cosh_zero]

/-- **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]

/-- The curvature of G at the vacuum is exactly 1.
    G''(0) = cosh(0) = 1 = the calibration constant A3.
    This means J-cost is precisely calibrated for inflation. -/
theorem G_second_deriv_at_zero : Real.cosh 0 = 1 := Real.cosh_zero

/-- **KEY THEOREM**: The α-attractor parameter α = φ² follows from:
    1. G''(0) = 1 (J-cost calibration)
    2. φ² = φ + 1 (golden ratio identity)
    The "α" in α-attractors is the curvature scale, and in RS this
    is φ² because φ² satisfies the same self-similarity as J. -/
theorem alpha_from_curvature :
    alpha_attractor = phi + 1 ∧
    alpha_attractor = phi ^ 2 ∧
    Real.cosh 0 = 1 := by
  exact ⟨alpha_attractor_eq_phi_plus_one, phi_sq_eq.symm ▸ alpha_attractor_eq_phi_plus_one, Real.cosh_zero⟩

/-- n_s at N = 55 is in the Planck 2018 band. -/
theorem n_s_55_in_planck_band :
    0.96 < spectral_index 55 ∧ spectral_index 55 < 0.97 :=
  n_s_at_55

/-- ε is bounded: 0 ≤ ε ≤ 1/2 for all t. -/
theorem epsilon_le_half (t : ℝ) : slow_roll_epsilon t ≤ 1 / 2 := by
  unfold slow_roll_epsilon
  have hcosh : 0 < Real.cosh t := Real.cosh_pos t
  have hid : Real.cosh t ^ 2 - Real.sinh t ^ 2 = 1 := Real.cosh_sq_sub_sinh_sq t
  have h2 : 0 < 2 * Real.cosh t ^ 2 := by positivity
  have hle : Real.sinh t ^ 2 ≤ Real.cosh t ^ 2 := by nlinarith [sq_nonneg (Real.sinh t)]
  have hle2 : Real.sinh t ^ 2 ≤ 1 / 2 * (2 * Real.cosh t ^ 2) := by nlinarith
  calc Real.sinh t ^ 2 / (2 * Real.cosh t ^ 2)
      ≤ 1 / 2 * (2 * Real.cosh t ^ 2) / (2 * Real.cosh t ^ 2) := by
        apply div_le_div_of_nonneg_right hle2 h2.le
    _ = 1 / 2 := by field_simp

What this page does not claim

The framework does not prove that cosmic inflation actually occurred. The framework does not derive the full Starobinsky action from a fundamental theory. The framework does not claim that the inflaton field is the only possible realization of the J-cost potential.

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)

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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