Gravity Jcost Inflaton Inflation From Jcost Cert

The certificate

Inflation is the idea that the universe expanded at an enormous rate in its first instant. Cosmologists describe that expansion with a potential, a curve that tells how much energy a field has at each value. In the Recognition Science framework, a specific potential emerges from a proved result rather than being chosen by hand.

The framework's central theorem forces a particular cost function, J(x) = (x + 1/x)/2 − 1, from five plain conditions. The declaration inflation_from_jcost_cert packages the consequences of applying that cost to inflation. In log coordinates, the cost becomes G(t) = cosh(t) − 1, a plateau-shaped curve that rises steeply then flattens. This is the same family as the Starobinsky potential, a well-studied model in cosmology.

The certificate proves several properties of this potential. It has a minimum at zero cost, a critical point at that minimum, and a curvature there equal to the framework's calibration constant. The slow-roll parameter ε, which measures how gently the field rolls, vanishes at the vacuum. The spectral index, a number that describes the distribution of density fluctuations, follows the formula 1 − 2/N, where N is the number of e-folds of expansion. At N = 55, that index lands between 0.96 and 0.97, inside the band favored by Planck satellite data.

What the certificate does not do is prove that inflation happened. It does not derive the measured value of the spectral index from first principles; it shows that a particular potential, forced by the framework's axioms, produces a value within an accepted range. The number 55 for e-folds is an input, not an output. The certificate also does not prove that the framework's cost function is the actual inflaton potential of our universe. It establishes a mathematical consistency, not an observational confirmation.

/-- **THE INFLATION FROM J-COST THEOREM**: Every ingredient in the
    inflationary prediction chain is forced by the J-cost uniqueness.
    Zero free parameters. -/
theorem inflation_from_jcost_cert : InflationFromJCostCert where
  G_is_jcost := fun _ => rfl
  vacuum_zero_cost := G_at_zero
  epsilon_zero := slow_roll_epsilon_vanishes
  epsilon_bounded := epsilon_le_half
  calibration := Real.cosh_zero
  alpha_from_phi := phi_sq_eq.symm ▸ alpha_attractor_eq_phi_plus_one
  spectral_formula := fun N => rfl
  n_s_planck := n_s_at_55

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

/-- **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 spectral index formula 1 − 2/N follows from the α-attractor
    with α = φ²: n_s = 1 − 2/N is the standard slow-roll result
    when ε ≪ 1 (plateau regime). -/
theorem n_s_from_jcost (N : ℝ) (hN : 0 < N) :
    spectral_index N = 1 - 2 / N := rfl

/-- 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

What this page does not claim

The certificate does not prove that inflation occurred in the actual universe. The certificate does not derive the measured spectral index from first principles; it shows a value within an accepted range. The certificate does not prove the framework's cost function is the true inflaton 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)

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