Cosmology Structure Formation From Bit Peak 3 1 Ratio

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: bright and dark rings that correspond to sound waves that were frozen in when the universe became transparent. Cosmologists measure the angular scale of these peaks to learn the geometry and composition of the cosmos. The first peak sets the overall scale; the ratios between the first, second, and third peaks are sensitive to the density of ordinary matter and other parameters.

In the Recognition Science framework, the matter power spectrum P(k), which describes how much structure exists at different scales, inherits a specific structure from the framework's core kernel. The framework defines a ladder of characteristic wavenumbers k_n = k_0 · φ^n, where φ is the golden ratio (approximately 1.618) and k_0 is a base scale. A machine-checked library of formal theorems proves that the ratio of adjacent peak wavenumbers is exactly φ, and that the ratio of the third peak to the first is exactly φ². These are theorems about the defined ladder, not about the observed sky.

The declaration peak_3_1_ratio is one of those theorems. It states that for any positive base scale k_0, the ratio k_peak k_0 3 / k_peak k_0 1 equals φ². The proof is a short algebraic manipulation: it unfolds the definition of k_peak, uses the fact that φ is positive and nonzero, and simplifies the resulting fraction. The theorem also holds for any positive k_0, which means the ratio is independent of the base scale. This scale invariance is a separate theorem in the same file.

In Recognition Science, the framework models the CMB acoustic peaks as following this φ-ladder. The framework's library shows that the peak ratios are positive constants of φ, independent of the base scale k_0. This is a structural claim about the framework's own model, not a measurement of the actual sky. The framework's status line marks the φ-rational ratio structure as a theorem, but the numerical match to Planck or DESI data is explicitly a hypothesis, not a theorem.

The framework's prediction is falsifiable: 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 pack's date, no such observation is cited. The theorem itself does not claim that the real CMB peaks follow this ladder; it claims only that the defined ladder has these ratios. The empirical check against Planck or DESI data remains an open hypothesis.

/-- 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

/-- Adjacent peak ratio is exactly `φ`. -/
theorem k_peak_adjacent_ratio (k_0 : ℝ) (n : ℕ) (h : 0 < k_0) :
    k_peak k_0 (n + 1) / k_peak k_0 n = phi := 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_pow_n_ne : phi ^ n ≠ 0 := pow_ne_zero n h_phi_ne
  rw [pow_succ]
  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]

What this page does not claim

The theorem does not claim that the observed CMB peaks follow the φ-ladder. The theorem does not claim that the framework's model matches Planck or DESI data. The theorem does not claim that the base scale k_0 has a specific physical value.

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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