Gravity Jcost Inflaton Alpha From Curvature

The curvature result

In inflationary cosmology, the shape of a potential energy curve determines how fast the early universe expands. A flat, plateau-like curve produces the slow roll of the inflaton field that matches observations of the cosmic microwave background. The Recognition Science framework models this plateau using its cost function, the minimal recognition cost J(x) = (x + 1/x)/2 - 1, and asks what curvature at the minimum forces.

The framework's machine-checked library of formal theorems contains a declaration, alpha_from_curvature, that derives a specific value for the α-attractor parameter from the curvature of this potential at its minimum. In log coordinates, the cost becomes G(t) = cosh(t) - 1, a standard plateau form. The theorem proves that the second derivative at the minimum, G''(0), equals 1, which is the calibration constant. From this curvature and the golden ratio identity φ² = φ + 1, the theorem derives α = φ², the square of the golden ratio.

The result is a formal derivation within the framework's axioms. It proves that if the inflaton potential is the J-cost in log coordinates, then the α-attractor parameter is forced to be φ², not chosen by hand. The library also derives the spectral index n_s = 1 - 2/N from the same curvature, and for N = 55 e-folds this gives a value between 0.96 and 0.97, which sits inside the Planck satellite's measured band for the spectral index.

The declaration does not claim that the actual inflaton potential in our universe is this J-cost. That physical identification is a modeling choice, not a proved theorem. The derivation is conditional: it holds within the framework's definition of the potential. It also does not claim to derive the fine-structure constant α, despite the shared Greek letter. The α in alpha_from_curvature is the α-attractor parameter of inflation models, a dimensionless number controlling the width of the potential, not the electromagnetic coupling constant.

/-- **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⟩

/-- **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⟩

/-- 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

/-- For N = 55 e-foldings, the spectral index satisfies n_s ∈ (0.96, 0.97). -/
theorem n_s_at_55_from_jcost : 0.96 < spectral_index 55 ∧ spectral_index 55 < 0.97 :=
  n_s_at_55

What this page does not claim

The physical inflaton potential in our universe is proven to be the J-cost function. The declaration derives the fine-structure constant α; the α here is the α-attractor parameter of inflation models. The framework derives the number of e-folds N = 55 from first principles.

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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