Cosmology Cmbacoustic Peak Ratios Planck Ratio Not Directly Phi

The angular ratio check

The cosmic microwave background (CMB) is the oldest light in the universe, and its temperature fluctuations show a series of acoustic peaks: regular bumps in a plot of fluctuation strength against angular scale. Cosmologists use the positions of these peaks, labeled ℓ₁, ℓ₂, ℓ₃, as a standard ruler to measure the geometry and contents of the universe. The Planck satellite's 2018 release gives the first three peak positions as ℓ₁ = 220.0, ℓ₂ = 540.3, and ℓ₃ = 814.6, so the observed ratio of the second peak to the first is 540.3 / 220.0 ≈ 2.456.

Within the Recognition Science framework, a machine-checked library of formal theorems derives a structural prediction for the peak positions at the level of wavenumber, the spatial frequency of the oscillations. The framework's library proves that the ratio of the second peak's wavenumber to the first is exactly the golden ratio φ, and the third to the first is exactly φ². The golden ratio is the number satisfying φ² = φ + 1, approximately 1.618. These are unconditional theorems about the framework's model of structure formation, stated for any positive starting wavenumber.

The declaration planck_ratio_not_directly_phi is a formal honesty check. It proves that the observed angular-multipole ratio ℓ₂/ℓ₁, which is about 2.456, is not equal to φ. The proof is a direct numerical contradiction: the observed ratio lies between 2.45 and 2.46, while φ lies between 1.61 and 1.62. The theorem does not say the framework's prediction is wrong. It says the bare angular ratio is the wrong test. The structural prediction lives at the wavenumber level, and the observed angular positions include a projection factor from wavenumber space to angular-multipole space that depends on the angular diameter distance. Only at a fixed distance would the two ratios coincide.

In Recognition Science, this distinction matters. The framework's claim is about the underlying wavenumber peaks, not about the directly observed angular peaks. The declaration therefore names a falsifier: a direct measurement of the wavenumber peak ratios from baryon acoustic oscillation surveys such as BOSS or DESI, if it fell outside the predicted band of 1.61 to 1.62 for the second-to-first ratio by more than five percent, would count against the structural prediction. The angular-multipole comparison is a separate, deferred hypothesis about the projection geometry, not a theorem.

/-- Planck 2018 first acoustic peak: `ℓ_1 = 220.0`. -/
def planck_l_1 : ℝ := 220.0

/-- Planck 2018 second acoustic peak: `ℓ_2 = 540.3`. -/
def planck_l_2 : ℝ := 540.3

/-- Planck 2018 third acoustic peak: `ℓ_3 = 814.6`. -/
def planck_l_3 : ℝ := 814.6

/-- The second-to-first peak ratio is exactly `φ`. -/
theorem ratio_2_1 (k_0 : ℝ) (h : 0 < k_0) :
    k_peak k_0 2 / k_peak k_0 1 = phi :=
  peak_2_1_ratio k_0 h

/-- The observed angular-multipole ratio is **not** the bare φ-rational
ratio: the projection geometry from k-space to ℓ-space introduces a
factor that depends on the angular diameter distance. The *bare
wavenumber ratio* is the φ-rational prediction, recoverable from
direct k-space BAO measurements. -/
theorem planck_ratio_not_directly_phi :
    planck_ratio_2_1 ≠ phi := by
  intro h_eq
  have h_planck := planck_ratio_2_1_value
  rw [h_eq] at h_planck
  have h_phi_lt := phi_lt_onePointSixTwo
  linarith

theorem planck_ratio_2_1_value : 2.45 < planck_ratio_2_1 ∧ planck_ratio_2_1 < 2.46 := by
  unfold planck_ratio_2_1 planck_l_2 planck_l_1
  refine ⟨?_, ?_⟩ <;> norm_num

/-- The 2-1 ratio band: `(1.61, 1.62)`. -/
theorem ratio_2_1_band (k_0 : ℝ) (h : 0 < k_0) :
    1.61 < k_peak k_0 2 / k_peak k_0 1 ∧
    k_peak k_0 2 / k_peak k_0 1 < 1.62 := by
  rw [ratio_2_1 k_0 h]
  exact ⟨phi_gt_onePointSixOne, phi_lt_onePointSixTwo⟩

What this page does not claim

The framework's structural prediction for the wavenumber peaks is confirmed by any direct measurement. The observed angular-multipole ratios match the golden ratio or any simple multiple of it. The Planck angular peak positions themselves are derived from the framework.

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/CMBAcousticPeakRatios.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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