Cosmology Tensor To Scalar Ratio From Rs R Lt One

The ratio bound

The tensor-to-scalar ratio r is a number cosmologists use to describe gravitational waves left over from the universe's earliest moments. A value of zero means no such waves; larger values mean stronger waves. The standard picture expects r to be small, and the Recognition Science framework derives a specific prediction: r = 2 / (45φ²), where φ is the golden ratio, giving a number between 0.015 and 0.020.

The declaration tensor-to-scalar ratio r_lt_one is a formal theorem in the framework's machine-checked library of formal theorems. It proves that the derived value is less than 1. The proof works by substituting the golden ratio identity φ² = φ + 1, then using the known fact that φ is greater than 1.61 to show the denominator 45(φ + 1) is large enough. The result is a basic consistency check: the framework's prediction stays in the physically plausible range rather than blowing up to an absurd value.

The theorem does not claim the ratio is measured, observed, or confirmed by any telescope. It does not say the value 0.015 to 0.020 is correct physics. It only establishes a mathematical property of a definition: the number defined by that formula is less than one. The framework separately proves the ratio sits in the band (0.015, 0.020), but that band is a derived statement about the formula, not a comparison against experimental data.

In Recognition Science, the framework models the early universe's gravitational wave production through its forcing chain, and this ratio is one output. The r_lt_one theorem is a guardrail: it ensures the model's prediction is not pathological. A reader should take away that the framework produces a finite, bounded, positive number for r, and that the bound is machine-checked. What remains open is whether any actual observation will match the band.

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

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

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]

What this page does not claim

The theorem does not claim any measured value for r matches the prediction. It does not claim the band (0.015, 0.020) is confirmed by cosmological data. It does not claim the formula r = 2 / (45φ²) is derived from first principles without any model choice.

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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