Gravity Rsbaryogenesis Alpha Inflaton Alt

The inflaton parameter

In cosmology, the inflaton is the field thought to have driven the universe's rapid early expansion. Its potential energy curve is often written with a parameter α that controls the width of the potential. The Recognition Science framework's declaration alpha_inflaton_alt establishes, as a proved result in its machine-checked library of formal results, that this parameter equals φ⁻², where φ is the golden ratio, approximately 1.618. The result states that the definition of α_inflaton is definitionally equal to φ⁻², meaning the identification is not an approximation but an exact identity within the framework's formalism.

The declaration itself is a single line: theorem alpha_inflaton_alt : alpha_inflaton = phi ^ (-(2 : ℕ)) := rfl. The proof is by reflexivity, which in a proof assistant means the two sides are syntactically the same after unfolding definitions. This is a MODEL-level claim about how the framework defines the inflaton parameter, not a derivation from deeper principles. The framework's broader baryogenesis work, which this declaration supports, links this parameter to a mechanism for the matter-antimatter asymmetry of the universe, but the result itself only pins down the value of α in terms of φ.

What the declaration does not claim is more important than what it does. It does not prove that φ⁻² is the physically correct value of the inflaton parameter in our universe. It does not derive the parameter from the framework's foundational forcing chain that produces other constants. It does not make any prediction about observable quantities. The result is a formal statement about the internal consistency of a definition, not an empirical claim. The framework's own documentation lists α_infl = φ⁻² as an 'alpha-attractor parameter' for the inflaton potential, but the result alone provides no physical justification for why this value should be chosen over any other.

The value φ⁻² is approximately 0.382. In the context of the framework's baryogenesis mechanism, this parameter appears alongside other φ-derived quantities like the CP-odd couplings λ_CP = φ⁻⁷ and κ_CP = φ⁻⁹. The framework's library proves these couplings are positive and less than one, and that the baryon asymmetry prediction η_B ≈ 5.1 × 10⁻¹⁰ falls within 20 percent of the observed value 6.1 × 10⁻¹⁰. These are separate results. The alpha_inflaton_alt result stands alone as a definitional identity, and its role is to fix a number that appears in the framework's model of inflation, not to establish that number's physical necessity.

/-- α_inflaton = 1/φ² = (φ-1)² (using φ² = φ+1). -/
theorem alpha_inflaton_alt : alpha_inflaton = phi ^ (-(2 : ℝ)) := rfl

/-- The α-attractor parameter for the inflaton potential.
    V(χ) = V₀ tanh²(χ / (√6 φ)) with α = φ⁻².

    Note: The Universe-Origin paper uses α = φ², while the Baryogenesis
    paper uses α = φ⁻². These correspond to different parameterization
    conventions:
    - α = φ² in the "natural" convention (large-field inflation)
    - α = φ⁻² in the "inverse" convention (small-field with Planck suppression)

    Both give the same spectral predictions when N is adjusted. -/
noncomputable def alpha_inflaton : ℝ := phi ^ (-(2 : ℝ))

theorem eta_B_within_20_percent :
    eta_B_fractional_offset < 0.20 := by
  unfold eta_B_fractional_offset eta_B_prediction eta_B_observed
  norm_num

What this page does not claim

The result does not prove that φ⁻² is the physically correct value of the inflaton parameter in our universe. The result does not derive the inflaton parameter from the framework's foundational forcing chain. The result makes no prediction about observable cosmological quantities.

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

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