Cosmology Cmbacoustic Peak Ratios

Acoustic peak ratios

The cosmic microwave background (CMB) is the oldest light in the universe, a snapshot of sound waves that traveled through the hot plasma of the early cosmos. These waves left their imprint as a series of peaks in the temperature fluctuations we observe. The first three peaks, at angular multipoles ℓ₁ = 220.0, ℓ₂ = 540.3, and ℓ₃ = 814.6 from Planck 2018, are the classic signature of baryon acoustic oscillations. In standard cosmology, the peak positions depend on the sound horizon and the angular diameter distance to the surface of last scattering.

In Recognition Science, the framework models the universe's structure formation through a discrete lattice of recognition events, a ledger of interactions with forced costs. Within this account, the wavenumbers of the acoustic peaks, the spatial frequencies of the sound waves, are not free parameters. The framework proves that the ratios of the bare wavenumbers follow the golden ratio φ, where φ satisfies φ² = φ + 1 and equals (1 + √5)/2 ≈ 1.618. Specifically, the second peak's wavenumber divided by the first is exactly φ, the third divided by the second is exactly φ, and the third divided by the first is exactly φ². These are stated as theorems in the framework's machine-checked library of formal theorems.

The framework also provides numerical bands for these ratios. The second-to-first ratio lies strictly between 1.61 and 1.62, and the third-to-first ratio lies between 2.59 and 2.63. These bands follow from the golden ratio's own bounds. The key caveat is that these predictions apply to the wavenumbers, not directly to the angular multipoles we observe. The observed ℓ₂/ℓ₁ ≈ 2.456 differs from φ because the projection from wavenumber space to angular space introduces a factor depending on the angular diameter distance. The framework explicitly proves that the observed angular ratio is not equal to φ.

The honest test of this structural prediction, therefore, is not the Planck angular peaks but direct measurements of wavenumbers, such as those from baryon acoustic oscillation surveys like BOSS or DESI. The framework predicts that the second-to-first and third-to-first BAO peak wavenumber ratios will fall within the φ-bands. This is a falsifiable hypothesis: any direct k-space measurement outside the predicted bands by more than 5 percent would refute it. The structural claim at the wavenumber level is proved; the numerical match to observation remains an open empirical check.

/-- 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 third-to-second peak ratio is exactly `φ`. -/
theorem ratio_3_2 (k_0 : ℝ) (h : 0 < k_0) :
    k_peak k_0 3 / k_peak k_0 2 = phi :=
  peak_3_2_ratio k_0 h

/-- 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 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⟩

/-- 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

/-- 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

/-- **CMB ACOUSTIC PEAK RATIOS ONE-STATEMENT.** Three structural facts:

(1) The bare wavenumber peak ratio `k_2/k_1 = φ`, in the band `(1.61, 1.62)`.
(2) The bare wavenumber peak ratio `k_3/k_1 = φ² = φ + 1`, in the band
    `(2.59, 2.63)`.
(3) The Planck angular-multipole ratio `ℓ_2/ℓ_1 ≈ 2.456` differs from
    the bare φ-rational prediction by the projection-geometry factor;
    the test of the structural prediction is at the bare wavenumber
    level, via direct k-space BAO measurements (BOSS, DESI). -/
theorem cmb_acoustic_peak_ratios_one_statement (k_0 : ℝ) (h : 0 < k_0) :
    k_peak k_0 2 / k_peak k_0 1 = phi ∧
    k_peak k_0 3 / k_peak k_0 1 = phi ^ 2 ∧
    planck_ratio_2_1 ≠ phi :=
  ⟨ratio_2_1 k_0 h, ratio_3_1 k_0 h, planck_ratio_not_directly_phi⟩

What this page does not claim

The observed Planck angular multipole ratios themselves are not equal to the golden ratio or its square. The numerical match to the Planck data is not a theorem; it is a falsifiable hypothesis pending a full transfer-function calculation. The framework does not derive the absolute values of the peak wavenumbers, only their ratios.

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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