Cosmology Cmbacoustic Peak Ratios Phi Sq Band

The phi-squared band

The golden ratio φ, approximately 1.618, is the number that solves r² = r + 1. Its square is therefore φ + 1, approximately 2.618. The declaration phi_sq_band is a machine-checked theorem stating that φ² lies strictly between 2.59 and 2.63. This is a narrow numerical window: it excludes 2.6 and 2.62 while including 2.618, the true value.

This band matters in cosmology because of how sound waves moved through the hot, dense early universe. Pressure waves in the primordial plasma left imprints on the cosmic microwave background, appearing as a series of acoustic peaks in the temperature fluctuation spectrum. The peaks occur at characteristic angular scales, labeled by multipole numbers ℓ. The framework's structural claim is that the ratios of the wavenumbers of these peaks follow the golden ratio: the second-to-first ratio equals φ, the third-to-first equals φ². The phi_sq_band theorem supplies the numerical content of that claim for the third-to-first ratio.

In Recognition Science, the framework models the universe's structure as arising from a discrete recognition process, and the golden ratio emerges as the unique self-similar scaling from its forcing chain. The library's theorems prove these wavenumber ratios at the bare wavenumber level, with no new axioms. The phi_sq_band theorem is one step in that proof, establishing the interval that the third-to-first ratio must occupy.

The declaration does not claim that the observed angular multipole ratios from Planck 2018 data match this band. The measured ℓ₂/ℓ₁ ratio is about 2.456, and ℓ₃/ℓ₁ about 3.703, neither close to φ or φ². The framework's own library proves that the Planck angular ratio is not equal to φ. The mismatch arises because the observed angular scales involve a projection from wavenumber space to angular multipole space, a geometric factor that depends on the angular diameter distance. The φ-rational prediction applies to the bare wavenumbers, which would require direct measurements of the acoustic peak wavenumbers, such as from baryon acoustic oscillation surveys like BOSS or DESI.

What the theorem establishes, then, is a precise, checkable consequence of the framework's structural assumptions. It pins down a numerical band for a ratio that the framework predicts at the wavenumber level, and it separates that prediction from the angular observations that do not directly test it. The falsifier is named: any direct k-space measurement of the second-to-first or third-to-first BAO peak ratios outside the predicted φ-band by more than 5 percent would contradict the framework's claim.

/-- φ² lies in the band `(2.59, 2.63)`. (Tight: `(1.61)² = 2.5921`,
`(1.62)² = 2.6244`.) -/
theorem phi_sq_band : (2.59 : ℝ) < phi ^ 2 ∧ phi ^ 2 < 2.63 := by
  refine ⟨?_, ?_⟩
  · have h := phi_gt_onePointSixOne
    have h_pos := phi_pos
    have : (1.61 : ℝ) ^ 2 ≤ phi ^ 2 := by
      have h_le : (1.61 : ℝ) ≤ phi := le_of_lt h
      apply pow_le_pow_left₀ (by norm_num : (0 : ℝ) ≤ 1.61) h_le
    have h_15_sq : (1.61 : ℝ) ^ 2 = 2.5921 := by norm_num
    linarith
  · have h := phi_lt_onePointSixTwo
    have h_pos := phi_pos
    have : phi ^ 2 ≤ (1.62 : ℝ) ^ 2 := by
      have h_le : phi ≤ (1.62 : ℝ) := le_of_lt h
      apply pow_le_pow_left₀ (le_of_lt h_pos) h_le
    have h_162_sq : (1.62 : ℝ) ^ 2 = 2.6244 := by norm_num
    linarith

/-- 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-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

What this page does not claim

The observed Planck angular multipole ratios match the φ-rational band. The phi_sq_band theorem alone predicts the absolute positions of the acoustic peaks. The framework derives the full transfer function of cosmological perturbations.

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.

Derived articles

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