Cosmology Structure Formation From Bit Peak 3 2 Ratio

The acoustic peak ratio

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: slight hot and cold spots that appear as a ripple pattern when astronomers plot the brightness of the CMB against angular scale. The peaks are not evenly spaced. Their positions, set by the physics of sound waves in the early universe, are one of the strongest observational tests of cosmological models. The ratio of the wavenumber of the third peak to the second peak, written k₃/k₂, is a number that any theory of structure formation must predict.

In the Recognition Science framework, that ratio is not fitted to data. The framework defines a sequence of characteristic wavenumbers kₙ = k₀ · φⁿ, where φ is the golden ratio, the number (1 + √5)/2 ≈ 1.618, and k₀ is a base scale. The declaration peak_3_2_ratio is a proved theorem in the framework's machine-checked library of formal theorems: for any positive base scale k₀, the ratio of the third peak to the second peak is exactly φ. The same theorem also gives k₂/k₁ = φ and k₃/k₁ = φ². Because the ratio is a quotient, the base scale k₀ cancels out: the prediction is scale-invariant, meaning it does not depend on the overall size of the pattern, only on its shape.

The framework's claim is precise. The theorem proves a mathematical identity about the sequence kₙ; it does not, by itself, assert that the real CMB obeys that sequence. The match to actual observations, such as data from the Planck satellite or the Dark Energy Spectroscopic Instrument (DESI), is a separate hypothesis. The framework names a falsifier: if any of the first three CMB acoustic peaks is observed at a wavenumber ratio more than 5% off the predicted φ or φ² values, the hypothesis fails. As of the library's certificate, the theorem is proved with no gaps and no extra axioms, while the numerical agreement with data is an empirical check, not a proven fact.

What the declaration does not claim is just as important. It does not claim that the CMB peaks are the only place the golden ratio appears, nor that the framework's base scale k₀ is derived from first principles. It does not claim that the ratio φ is a fundamental constant of nature in the same sense as the speed of light. The theorem is a statement about a defined sequence; the physical interpretation, that the early universe's sound waves follow this sequence, is a hypothesis the framework offers for testing. The distinction keeps the mathematics clean and the physics honest: a proved identity about a ladder of numbers, and a falsifiable prediction about the sky.

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

/-- The second-to-first peak ratio is `φ`. -/
theorem peak_2_1_ratio (k_0 : ℝ) (h : 0 < k_0) :
    k_peak k_0 2 / k_peak k_0 1 = phi :=
  k_peak_adjacent_ratio k_0 1 h

/-- The third-to-first peak ratio is `φ²`. -/
theorem peak_3_1_ratio (k_0 : ℝ) (h : 0 < k_0) :
    k_peak k_0 3 / k_peak k_0 1 = phi ^ 2 := by
  unfold k_peak
  have h_phi_ne : phi ≠ 0 := ne_of_gt phi_pos
  have h_k0_ne : k_0 ≠ 0 := ne_of_gt h
  field_simp

/-- The peak ratios are independent of the base scale `k_0`. -/
theorem peak_ratios_scale_invariant
    (k_0 k_0' : ℝ) (n m : ℕ) (h : 0 < k_0) (h' : 0 < k_0') :
    k_peak k_0 (n + m) / k_peak k_0 n = k_peak k_0' (n + m) / k_peak k_0' n := by
  unfold k_peak
  have h_phi_ne : phi ≠ 0 := ne_of_gt phi_pos
  have h_k0_ne : k_0 ≠ 0 := ne_of_gt h
  have h_k0'_ne : k_0' ≠ 0 := ne_of_gt h'
  have h_pow_n_ne : phi ^ n ≠ 0 := pow_ne_zero n h_phi_ne
  -- Both sides simplify to phi^m.
  have h_lhs : k_0 * phi ^ (n + m) / (k_0 * phi ^ n) = phi ^ m := by
    rw [pow_add]; field_simp
  have h_rhs : k_0' * phi ^ (n + m) / (k_0' * phi ^ n) = phi ^ m := by
    rw [pow_add]; field_simp
  rw [h_lhs, h_rhs]

/-- **STRUCTURE FORMATION FROM BIT MASTER CERTIFICATE (Track F4).** -/
structure StructureFormationFromBITCert where
  k_pos : ∀ k_0 n, 0 < k_0 → 0 < k_peak k_0 n
  adjacent_ratio : ∀ k_0 n, 0 < k_0 →
    k_peak k_0 (n + 1) / k_peak k_0 n = phi
  peak_3_1_eq_phi_sq : ∀ k_0, 0 < k_0 →
    k_peak k_0 3 / k_peak k_0 1 = phi ^ 2
  scale_invariant : ∀ k_0 k_0' n m, 0 < k_0 → 0 < k_0' →
    k_peak k_0 (n + m) / k_peak k_0 n = k_peak k_0' (n + m) / k_peak k_0' n

What this page does not claim

The theorem does not assert that the real CMB obeys the golden-ratio sequence; that is a hypothesis. The framework does not derive the base scale k₀ from first principles. The golden ratio is not claimed to be a fundamental constant of nature in the same sense as the speed of light.

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/StructureFormationFromBIT.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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