Cosmology Tensor To Scalar Ratio From Rs R Band

The r_band bound

In cosmology, the tensor-to-scalar ratio r measures the relative strength of primordial gravitational waves compared to density fluctuations in the early universe. A value of r near 0.02 would indicate a specific energy scale for cosmic inflation, the brief exponential expansion thought to have seeded the large-scale structure we observe. The bound in question, r_band, is a formal statement that this ratio lies strictly between 0.015 and 0.020.

The value comes from a specific definition: r = 2 / (45 * phi^2), where phi is the golden ratio, approximately 1.618. The golden ratio is the unique positive solution to the equation x^2 = x + 1. The machine-checked library of formal theorems proves that this expression is positive, less than one, and falls within the stated band. The proof relies on the fact that phi^2 equals phi + 1, and on a separate theorem that phi is greater than 1.61.

In Recognition Science, this ratio is not a free parameter fitted to observations. The framework derives the golden ratio from its cost function, and this cosmological ratio follows from that derivation. The declaration r_band is a theorem in the framework's library, meaning it has been verified by a computer proof checker with no unproven assumptions. The framework models the early universe's gravitational wave production as tied to this same golden ratio structure.

What r_band does not claim is important. It does not claim that the measured value of r from any experiment falls within this band. It does not claim that the tensor-to-scalar ratio has been observed at all. The theorem is a mathematical statement about a defined quantity, not a prediction confirmed by data. The framework's own documentation marks the comparison against measured cosmological data as an empirical check, not a proved result.

The practical consequence is a sharp, testable target. If future cosmic microwave background experiments measure r and find it within 0.015 to 0.020, that would be consistent with this framework's derivation. If measurements land outside this band, the framework's specific prediction would be falsified. The bound gives cosmologists a concrete number to look for, derived from a mathematical structure rather than from observational fitting.

noncomputable def tensorToScalarRatio : ℝ := 2 / (45 * phi ^ 2)

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]

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

This theorem does not claim that any experimental measurement of the tensor-to-scalar ratio falls within the band. This theorem does not claim that the tensor-to-scalar ratio has been observed at all. This theorem does not derive the tensor-to-scalar ratio from first principles; it defines the quantity and proves a bound on that definition.

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.

Derived articles

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