Cosmology Inflaton Potential Structural Slow Roll Epsilon Pos

The positivity proof

In cosmology, slow-roll inflation is described by two small parameters, ε and η, that measure how gradually the inflaton field's potential flattens. The parameter ε is defined as ε = 1/(2φ⁵), where φ is the golden ratio, approximately 1.618. The declaration slowRollEpsilon_pos is a formal proof that this ε is strictly greater than zero: 0 < 1/(2φ⁵).

The proof is a short chain of arithmetic facts. It uses the positivity of 1, the positivity of 2, and the positivity of φ⁵, which follows because φ itself is positive. The result is checked by the machine-checked library of formal theorems, with no unproved assumptions or axioms. It is a theorem about a definition, not a claim about the physical universe.

What the declaration does not claim is broader. It does not say that ε is small, that inflation occurs, or that the potential V(χ) matches any observation. It only establishes the sign of one defined quantity. The value of ε, about 0.0557, and its role in slow-roll dynamics are separate facts, not part of this proof.

In Recognition Science, this positivity fact is one entry in a certificate that bundles five regime counts, the e-fold count of 44, a Fibonacci identity for φ⁵, positivity of both slow-roll parameters, and a spectral index band. The certificate is a collection of formal statements, each individually checked. The positivity proof is the smallest of them, and its scope is exactly that smallness.

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

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

What this page does not claim

The declaration does not claim that ε is small or that slow-roll inflation occurs. The declaration does not claim that the potential V(χ) matches any observational data. The declaration does not claim that the golden ratio φ is physically fundamental.

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.

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