Cosmology Tensor To Scalar Ratio From Rs Tensor Ratio Cert

A certified band

In cosmology, the tensor-to-scalar ratio r measures the relative strength of primordial gravitational waves against density fluctuations in the early universe. A smaller r means gravitational waves were weaker; a larger r means they were stronger. Observational campaigns such as BICEP and the Planck satellite have spent years trying to measure r, because its value would discriminate between competing theories of cosmic inflation.

The Recognition Science framework's machine-checked library of formal theorems contains a certificate named TensorRatioCert. The certificate establishes, with a formal proof, that the framework's own expression for r, namely 2/(45φ²) where φ is the golden ratio, lies strictly between 0.015 and 0.020. The proof also shows r is positive and less than one. This is a narrow, precise claim: a band, not a point value.

The band itself is notable because it sits in a region cosmologists care about. Many simple inflation models predict r values in the range of 0.01 to 0.1, and current observational upper limits hover near 0.03. A prediction in the 0.015 to 0.020 window is therefore testable by next-generation experiments. The framework's certificate does not, however, claim that this band matches any particular measurement; it only certifies the arithmetic of its own formula.

What the certificate does not claim is just as important as what it proves. It does not say the formula 2/(45φ²) was derived from first principles within the framework; the derivation, if any, is not part of this certificate. It does not claim the band is a prediction that has been confirmed by observation. And it does not assert that the tensor-to-scalar ratio in the real universe actually falls in this range. The certificate is a formal statement about a number defined inside the framework, nothing more.

theorem r_band : (0.015 : ℝ) < tensorToScalarRatio ∧ tensorToScalarRatio < 0.020 := by
  constructor
  · unfold tensorToScalarRatio
    rw [phi2_eq]
    have h1 := phi_gt_onePointSixOne
    have hpos : (0:ℝ) < 45 * (phi + 1) := by nlinarith
    have hlt : 45 * (phi + 1) < 45 * 2.63 := by nlinarith [phi_lt_onePointSixTwo]
    have hup : 2 / (45 * 2.63) ≤ 2 / (45 * (phi + 1)) := by
      apply div_le_div_of_nonneg_left (by norm_num) hpos (by nlinarith)
    linarith [show (0.015:ℝ) < 2 / (45 * 2.63) from by norm_num]
  · unfold tensorToScalarRatio
    rw [phi2_eq]
    have h1 := phi_gt_onePointSixOne
    have hpos : (0:ℝ) < 45 * (phi + 1) := by nlinarith
    have hgt : 45 * (phi + 1) > 45 * 2.59 := by nlinarith
    have hlo : 2 / (45 * (phi + 1)) ≤ 2 / (45 * 2.59) := by
      apply div_le_div_of_nonneg_left (by norm_num) (by nlinarith) (by nlinarith)
    linarith [show (2 : ℝ) / (45 * 2.59) < 0.020 from by norm_num]

structure TensorRatioCert where
  r_pos : 0 < tensorToScalarRatio
  r_band : (0.015 : ℝ) < tensorToScalarRatio ∧ tensorToScalarRatio < 0.020

theorem r_pos : 0 < tensorToScalarRatio :=
  div_pos (by norm_num) (mul_pos (by norm_num) (pow_pos phi_pos 2))

theorem r_lt_one : tensorToScalarRatio < 1 := by
  unfold tensorToScalarRatio
  rw [phi2_eq]
  have h1 := phi_gt_onePointSixOne
  have hpos : (0:ℝ) < 45 * (phi + 1) := by nlinarith
  rw [div_lt_iff₀ hpos]
  nlinarith

What this page does not claim

The certificate does not claim the formula 2/(45φ²) was derived from first principles within the framework. It does not claim the band has been confirmed by any observation. It does not assert that the real universe's tensor-to-scalar ratio falls in this range.

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/TensorToScalarRatioFromRS.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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