Cosmology Cmbacoustic Peak Ratios Cmb Acoustic Peak Ratios One Statement

The peak ratio theorem

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 is a series of acoustic peaks: ripples in the density of matter and radiation from sound waves that traveled through the early universe. Cosmologists measure the angular positions of these peaks on the sky, labeled ℓ₁, ℓ₂, ℓ₃, to learn about the universe's geometry and composition. The first peak sits at ℓ₁ = 220.0, the second at ℓ₂ = 540.3, and the third at ℓ₃ = 814.6, according to the Planck 2018 data.

The Recognition Science framework contributes a structural claim about the underlying wavenumber (a measure of spatial frequency, the inverse of wavelength) peaks, not the observed angular positions. Its machine-checked library of formal theorems proves that the ratios of the first three wavenumber peaks are exactly the golden ratio φ ≈ 1.618 and its square: k₂/k₁ = φ, k₃/k₂ = φ, and k₃/k₁ = φ². This is a THEOREM, derived from the framework's earlier structure-formation work with no new axioms. The proof also establishes numerical bands: k₂/k₁ lies between 1.61 and 1.62, and k₃/k₁ lies between 2.59 and 2.63.

Here is the distinction the framework draws. The observed angular ratios from Planck are ℓ₂/ℓ₁ ≈ 2.456 and ℓ₃/ℓ₁ ≈ 3.703. These are not φ or φ², and the framework's own theorem explicitly proves that ℓ₂/ℓ₁ ≠ φ. The gap arises because the mapping from wavenumber space to angular multipole space involves the angular diameter distance, a projection factor that varies with cosmology. The φ-rational prediction applies to the bare wavenumbers, which are not directly measured in the CMB but could in principle be recovered from galaxy surveys like BOSS or DESI that measure baryon acoustic oscillations (BAO) in three dimensions.

In Recognition Science, the wavenumber ratios are a proved structural fact. The numerical match to the observed angular ratios, after accounting for projection geometry, is explicitly labeled a HYPOTHESIS, deferred to a full transfer-function calculation with Boltzmann-hierarchy codes like CAMB or CLASS. The falsifier is concrete: any direct k-space measurement of the second-to-first or third-to-first BAO peak ratios falling outside the predicted φ-band by more than 5% would refute the claim. The theorem itself, however, stands independent of any observation, as a statement about the framework's internal structure-formation model.

/-- 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

/-- 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 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 angular-multipole ratios ℓ₂/ℓ₁ and ℓ₃/ℓ₁ equal φ or φ². The framework has derived the full CMB power spectrum or the transfer function from first principles. Any claim about the physical reality of the wavenumber peaks beyond the framework's internal model.

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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