Cosmology Primordial Spectrum R Prediction

The tensor-to-scalar ratio

The tensor-to-scalar ratio r compares the amplitude of primordial gravitational waves (tensor perturbations) to the amplitude of density fluctuations (scalar perturbations) in the early universe. Cosmologists measure it through the polarization patterns they leave in the cosmic microwave background. A detection of r above roughly 0.01 would be a major clue about the energy scale of inflation, the rapid expansion thought to have seeded cosmic structure.

Within the Recognition Science framework, the declaration r_prediction states a numerical prediction for this ratio. The framework models primordial fluctuations as arising from fluctuations in a quantity it calls cost, a measure of the price of recognition events in its discrete ledger of reality. The prediction is that r equals (φ - 1)^4, where φ is the golden ratio, approximately 1.618. Since φ - 1 is about 0.618, the fourth power is about 0.146. The theorem r_prediction, proved in the framework's machine-checked library of formal theorems, establishes only a weaker bound: it proves that (φ - 1)^4 lies strictly between 0.1 and 0.2.

The framework's own comments acknowledge a discrepancy. The intended prediction of r ≈ 0.02 would require (2 - φ)^4, which equals 0.382^4 ≈ 0.021. The declaration r_prediction as written uses (φ - 1)^4, which is about 0.146, not 0.02. The theorem therefore does not establish the intended value. It proves a range that contains both the intended 0.02 and the computed 0.146, but it does not single out either number.

What r_prediction does not claim is as important as what it proves. It does not claim that the tensor-to-scalar ratio is measured to be 0.146. The observed upper bound in the framework's own data is 0.06. It does not claim that the golden ratio derivation is complete; the connection between the spectral tilt and φ is stated as an approximation within 15 percent, not as an exact equality. And it does not claim that the prediction has been confirmed by observation. The framework's falsifier structure explicitly allows for the possibility that r contradicts the prediction.

theorem r_prediction :
    -- r ≈ 0.02 is a testable RS prediction
    -- NOTE: The comment "(φ-1)⁴ = 0.382⁴" is incorrect.
    -- φ - 1 ≈ 0.618 (the golden ratio conjugate), so (φ-1)⁴ ≈ 0.146.
    -- The correct value 0.021 would be (2-φ)⁴ = 0.382⁴.
    -- For now, we prove a weaker bound: 0.1 < (φ-1)⁴ < 0.2
    0.1 < rs_prediction_r ∧ rs_prediction_r < 0.2 := by
  unfold rs_prediction_r
  -- φ - 1 ≈ 0.618, so (φ-1)⁴ ≈ 0.146
  -- Using bounds: 1.61 < φ < 1.62, so 0.61 < φ-1 < 0.62
  have h_phi_gt : phi - 1 > 0.61 := by
    have h := phi_gt_onePointSixOne
    linarith
  have h_phi_lt : phi - 1 < 0.62 := by
    have h := phi_lt_onePointSixTwo
    linarith
  -- 0.61^4 ≈ 0.138 > 0.1, 0.62^4 ≈ 0.148 < 0.2
  have h_low : (0.61 : ℝ)^4 > 0.1 := by norm_num
  have h_high : (0.62 : ℝ)^4 < 0.2 := by norm_num
  have h_phi_pos : phi - 1 > 0 := by linarith [one_lt_phi]
  constructor
  · calc 0.1 < (0.61 : ℝ)^4 := h_low
       _ < (phi - 1)^4 := by
           apply pow_lt_pow_left₀ h_phi_gt (by norm_num) (by norm_num)
  · calc (phi - 1)^4 < (0.62 : ℝ)^4 := by
           apply pow_lt_pow_left₀ h_phi_lt (le_of_lt h_phi_pos) (by norm_num)
       _ < 0.2 := h_high

theorem r_prediction :
    -- r ≈ 0.02 is a testable RS prediction
    -- NOTE: The comment "(φ-1)⁴ = 0.382⁴" is incorrect.
    -- φ - 1 ≈ 0.618 (the golden ratio conjugate), so (φ-1)⁴ ≈ 0.146.
    -- The correct value 0.021 would be (2-φ)⁴ = 0.382⁴.
    -- For now, we prove a weaker bound: 0.1 < (φ-1)⁴ < 0.2
    0.1 < rs_prediction_r ∧ rs_prediction_r < 0.2 := by
  unfold rs_prediction_r
  -- φ - 1 ≈ 0.618, so (φ-1)⁴ ≈ 0.146
  -- Using bounds: 1.61 < φ < 1.62, so 0.61 < φ-1 < 0.62
  have h_phi_gt : phi - 1 > 0.61 := by
    have h := phi_gt_onePointSixOne
    linarith
  have h_phi_lt : phi - 1 < 0.62 := by
    have h := phi_lt_onePointSixTwo
    linarith
  -- 0.61^4 ≈ 0.138 > 0.1, 0.62^4 ≈ 0.148 < 0.2
  have h_low : (0.61 : ℝ)^4 > 0.1 := by norm_num
  have h_high : (0.62 : ℝ)^4 < 0.2 := by norm_num
  have h_phi_pos : phi - 1 > 0 := by linarith [one_lt_phi]
  constructor
  · calc 0.1 < (0.61 : ℝ)^4 := h_low
       _ < (phi - 1)^4 := by
           apply pow_lt_pow_left₀ h_phi_gt (by norm_num) (by norm_num)
  · calc (phi - 1)^4 < (0.62 : ℝ)^4 := by
           apply pow_lt_pow_left₀ h_phi_lt (le_of_lt h_phi_pos) (by norm_num)
       _ < 0.2 := h_high

/-- The tensor-to-scalar ratio r < 0.06 (Planck + BICEP/Keck). -/
noncomputable def tensor_to_scalar_upper_bound : ℝ := 0.06

What this page does not claim

This answer does not claim that the tensor-to-scalar ratio has been measured to be 0.146. This answer does not claim that the golden ratio derivation of the spectral tilt is exact. This answer does not claim that the framework's prediction has been confirmed by observation.

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/PrimordialSpectrum.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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