Cosmology Inflaton Potential Structural Inflaton Cert

The certification record

In cosmology, inflation is the brief exponential expansion of the early universe, and its physics is usually summarized by a potential, a mathematical function that describes how the energy of a hypothetical scalar field, the inflaton, changes as the field rolls toward a minimum. The Recognition Science declaration InflatonCert is a machine-checked certificate: a single data structure that packages six structural claims about one such potential, V(χ), into one object. It does not derive the potential from first principles; it records that six specific properties, each proved separately, hold together.

The certificate's first claim is that the potential has exactly five canonical phase regimes, a finite list of distinct behaviors: a slow-roll plateau, a slow-roll slope, a hilltop decline, a reheating phase, and a post-reheating radiation era. The second claim fixes the number of e-folds, the natural logarithm of the total expansion factor, at 44, a count the framework associates with a gap in its structural ladder. The third claim is a Fibonacci identity: the fifth power of the golden ratio φ equals 5φ + 3, a relation that connects the potential's parameters to the framework's preferred scaling constant.

The remaining three claims concern the slow-roll approximation, the standard way to describe a field rolling slowly enough that its kinetic energy is negligible. The certificate asserts that the two slow-roll parameters, ε and η, are both positive, and that the spectral index n_s, which measures how the density fluctuations left over from inflation vary with scale, lies in the narrow band between 0.955 and 0.957. The certificate's own values are ε = 1/(2φ⁵), η = 1/φ⁵, and n_s - 1 = -2/45, which places n_s at about 0.9556.

What the certificate does not claim is just as important. It does not claim that this potential is the one realized in nature, nor that the framework derived it from its foundational axioms. The five regimes are a definitional choice, a model, not a theorem. The e-fold count of 44 is a definition. The positivity of ε and η is proved, but only for the specific definitions chosen. The spectral index band is a proved numerical inequality, but the comparison of that band to any measured cosmological value, such as the Planck satellite's constraint on n_s, is not part of the certificate; it is an empirical check left to the reader. The certificate is a compact, mechanically verified summary of what the framework's library has established about this particular structural potential, nothing more.

theorem inflatonRegime_count : Fintype.card InflatonRegime = 5 := by decide

/-- e-fold count N_e = 44 (gap-45 ladder). -/
def efoldCount : ℕ := 44

/-- φ⁵ = 5φ + 3 (Fibonacci identity). -/
theorem phi5_eq : phi ^ 5 = 5 * phi + 3 := by
  have h2 := phi_sq_eq
  have h3 : phi ^ 3 = 2 * phi + 1 := by nlinarith
  have h4 : phi ^ 4 = 3 * phi + 2 := by nlinarith
  nlinarith

theorem slowRollEpsilon_pos : 0 < slowRollEpsilon := by
  unfold slowRollEpsilon
  apply div_pos one_pos
  exact mul_pos (by norm_num) (pow_pos phi_pos 5)

theorem slowRollEta_pos : 0 < slowRollEta := by
  unfold slowRollEta
  exact div_pos one_pos (pow_pos phi_pos 5)

/-- n_s - 1 = -2/45 gives n_s ∈ (0.955, 0.957). -/
theorem spectralIndex_band :
    ((0.955 : ℝ) < 1 - 2/45) ∧ (1 - 2/45 < (0.957 : ℝ)) := by
  refine ⟨?_, ?_⟩ <;> norm_num

What this page does not claim

The certificate does not claim that this potential is the one realized in nature. The certificate does not claim that the potential is derived from the framework's foundational axioms. The certificate does not claim that its spectral index band matches any measured cosmological value.

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/Cosmology/InflatonPotentialStructural.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.

Derived articles

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