Cosmology Cmbacoustic Peak Ratios Ratio 2 1 Band

The 2-1 peak band

The cosmic microwave background (CMB) is the oldest light in the universe, released about 380,000 years after the Big Bang. Imprinted on that light is a series of acoustic peaks: regular bumps in temperature fluctuations that trace sound waves in the primordial plasma. The first three peaks have angular positions measured by the Planck satellite in 2018 at multipole moments ℓ₁ = 220.0, ℓ₂ = 540.3, and ℓ₃ = 814.6. The ratio of the second peak to the first, ℓ₂/ℓ₁, is about 2.456.

In Recognition Science, the framework models these peaks as standing waves on a discrete lattice of recognition events. Its machine-checked library of formal theorems derives, from the structure of that lattice, a prediction for the bare wavenumber ratio of the peaks. The declaration ratio_2_1_band proves that the second-to-first wavenumber ratio lies strictly between 1.61 and 1.62. That band contains the golden ratio φ, which satisfies φ² = φ + 1 and equals (1 + √5)/2 ≈ 1.618. The declaration is unconditional: it holds for any positive starting wavenumber k₀.

The band is a numerical consequence of a deeper structural result. The framework proves that the bare wavenumber ratios are exactly φ-rational: k₂/k₁ = φ, k₃/k₂ = φ, and k₃/k₁ = φ². The band declaration then follows from two proved bounds on φ itself, 1.61 < φ < 1.62. These are declarations in the framework's library, derived without adding new axioms.

The framework does not claim that the observed angular ratio 2.456 matches the band. It explicitly proves the opposite: planck_ratio_not_directly_phi shows that ℓ₂/ℓ₁ ≠ φ. The reason is projection geometry. The observed multipole ratio ℓ₂/ℓ₁ equals the wavenumber ratio only if the angular diameter distance is the same for both peaks, which it is not. The φ-rational prediction applies at the wavenumber level, recoverable from direct k-space measurements of baryon acoustic oscillations, such as those from BOSS or DESI. The comparison to Planck data is therefore a hypothesis, not a declaration, and its falsifier is a direct k-space measurement outside the band by more than 5%.

What the band changes is the testability of the framework. It turns a structural claim about a lattice into a concrete numerical window, (1.61, 1.62), that future BAO surveys can confirm or refute. The declaration does not say the CMB peaks themselves are golden; it says the underlying wavenumber spectrum is, and it names the measurement that would decide.

/-- 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 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

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 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 MASTER CERTIFICATE.** Five clauses.

1. `r_21`: bare wavenumber `k_2/k_1 = φ`.
2. `r_32`: bare wavenumber `k_3/k_2 = φ`.
3. `r_31`: bare wavenumber `k_3/k_1 = φ²`.
4. `r_21_band`: `k_2/k_1 ∈ (1.61, 1.62)`.
5. `r_31_band`: `k_3/k_1 ∈ (2.59, 2.63)`.

The Planck angular-multipole ratios differ from the bare wavenumber
ratios by the projection-geometry factor; the φ-rational prediction
is at the *wavenumber* level, recoverable from direct k-space BAO
measurements (BOSS, DESI). -/
structure CMBAcousticPeakRatiosCert where
  r_21 : ∀ k_0 : ℝ, 0 < k_0 → k_peak k_0 2 / k_peak k_0 1 = phi
  r_32 : ∀ k_0 : ℝ, 0 < k_0 → k_peak k_0 3 / k_peak k_0 2 = phi
  r_31 : ∀ k_0 : ℝ, 0 < k_0 → k_peak k_0 3 / k_peak k_0 1 = phi ^ 2
  r_21_band : ∀ k_0 : ℝ, 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
  r_31_band : ∀ k_0 : ℝ, 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

What this page does not claim

The observed angular multipole ratio ℓ₂/ℓ₁ equals the golden ratio. The framework derives the fine-structure constant or any other coupling constant. The Planck 2018 data themselves confirm the φ-rational prediction.

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)

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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