Gravity Jcost Inflaton Calibration Forces Alpha

The calibration result

In inflationary cosmology, the inflaton is the hypothetical field whose slow roll down a potential energy hill drove the exponential expansion of the early universe. The shape of that potential determines observable quantities like the spectral index, which measures how density fluctuations vary with scale. The Recognition Science framework's library of formal theorems contains a result, calibration_forces_alpha, that connects this potential's curvature to the golden ratio.

The framework models the inflaton potential using its cost function J(x) = (x + 1/x)/2 - 1. In logarithmic coordinates t = ln(x), this becomes G(t) = cosh(t) - 1, a plateau-shaped potential. The theorem states that the curvature at the minimum, G''(0), equals 1, and that this calibration sets the α-attractor parameter to α = φ², where φ is the golden ratio (1 + √5)/2. This follows from the identity φ² = φ + 1, which the framework's library proves.

What this means physically is that the framework's forced cost function, when used as an inflaton potential, produces a specific prediction for the α-attractor family of inflation models. The spectral index derived from this potential is n_s = 1 - 2/N, where N is the number of e-folds of expansion. For N = 55, this gives n_s ≈ 0.964, which falls within the range measured by the Planck satellite (0.96 to 0.97). This is a derived prediction, not a fitted parameter.

The theorem does not claim to derive the fine-structure constant α (the electromagnetic coupling constant). The name calibration_forces_alpha refers to the α-attractor parameter, not the fine-structure constant. The framework's expression for the fine-structure constant, which lands within 5.6 parts per million of the measured value, uses a seed of 44π that is an identification, not a derived coupling. That derivation remains open.

/-- The α-attractor curvature equals the calibration of J-cost.
    Both are forced to 1 (or equivalently, to φ² = φ+1 at the next level)
    by the uniqueness theorem for J. -/
theorem calibration_forces_alpha :
    Real.cosh 0 = 1 ∧ alpha_attractor = phi ^ 2 := by
  exact ⟨Real.cosh_zero, phi_sq_eq.symm ▸ alpha_attractor_eq_phi_plus_one⟩

/-- 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 fine-structure constant is not derived by this theorem; its seed 44π is an identification, not a derived coupling. The theorem does not prove that the framework's inflaton model is the actual physical mechanism of cosmic inflation. The α-attractor parameter here is not the fine-structure constant, despite the similar symbol.

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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