Cosmology Cmbacoustic Peak Ratios Ratio 3 1 Band

The third peak ratio band

The cosmic microwave background (CMB) is the oldest light in the universe, a faint glow left over from about 380,000 years after the Big Bang. Imprinted on that glow are acoustic peaks: regular ripples in temperature that correspond to sound waves that traveled through the early universe's hot, dense plasma. Cosmologists measure the angular positions of these peaks on the sky, labeled ℓ₁, ℓ₂, ℓ₃, and their ratios encode information about the universe's geometry and contents.

In the Recognition Science framework, the acoustic peaks are modeled as arising from a discrete lattice of wavenumbers, labeled k₁, k₂, k₃. The framework's machine-checked library of formal theorems proves a structural fact about these wavenumbers: the ratio of the third to the first peak wavenumber is exactly the square of the golden ratio, φ², where φ ≈ 1.618. Since φ² = φ + 1 ≈ 2.618, this ratio falls in the numerical band (2.59, 2.63). The declaration ratio_3_1_band establishes precisely this: for any positive starting wavenumber k₀, the ratio k₃/k₁ lies strictly between 2.59 and 2.63.

This is a theorem about wavenumbers, not directly about the angular positions astronomers observe. The observed Planck 2018 values are ℓ₁ = 220.0, ℓ₂ = 540.3, ℓ₃ = 814.6, giving ℓ₃/ℓ₁ ≈ 3.703, which is not φ². The framework explicitly acknowledges this gap: the angular multipole ratio differs from the bare wavenumber ratio by a projection-geometry factor that depends on the angular diameter distance. The structural prediction is recoverable only through direct wavenumber-space measurements, such as baryon acoustic oscillation (BAO) surveys like BOSS or DESI.

The distinction matters because it separates a proved mathematical structure from an untested physical hypothesis. The φ-rational wavenumber ratios are unconditionally derived within the framework. The claim that observed angular ratios will match after projection is a hypothesis with a named falsifier: any direct k-space measurement of BAO peak ratios outside the predicted φ-band by more than 5% would refute it.

/-- The third-to-first peak ratio is exactly `φ²`. -/
theorem ratio_3_1 (k_0 : ℝ) (h : 0 < k_0) :
    k_peak k_0 3 / k_peak k_0 1 = phi ^ 2 :=
  peak_3_1_ratio k_0 h

/-- The 3-1 ratio band: `(2.59, 2.63)`. -/
theorem ratio_3_1_band (k_0 : ℝ) (h : 0 < k_0) :
    2.59 < k_peak k_0 3 / k_peak k_0 1 ∧
    k_peak k_0 3 / k_peak k_0 1 < 2.63 := by
  rw [ratio_3_1 k_0 h]
  exact phi_sq_band

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

/-- 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

What this page does not claim

The observed Planck angular multipole ratios match the golden-ratio prediction. The framework derives the absolute positions of the acoustic peaks, only their ratios. The projection-geometry factor connecting k-space to ℓ-space has been computed within 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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