Cosmology Structure Formation From Bit Peak 2 1 Ratio

Peak spacing in the cosmic microwave background

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 are a series of acoustic peaks: ripples in the density of matter that show up as hotter and cooler spots. Astronomers measure the wavenumber of each peak, essentially how tightly packed the ripples are on the sky. The first peak is the largest and broadest; the second and third are smaller and sharper. What matters for this page is the spacing between those peaks, the ratio of one wavenumber to the next.

In the Recognition Science framework, the ledger, a discrete record of recognition events, forces a preferred scaling factor: the golden ratio φ, about 1.618. The framework's machine-checked library of formal theorems defines a sequence of wavenumbers k_n = k_0 · φ^n, where k_0 is any positive base scale and n counts the peak. The declaration peak_2_1_ratio proves that, for this defined sequence, the ratio of the second peak's wavenumber to the first is exactly φ. A companion theorem shows the third-to-second ratio is also φ, so the third-to-first ratio is φ². These ratio statements hold for any positive base scale k_0; they are parameter-free in the ratio sector.

The library marks the ratio structure as a theorem: it is a proved consequence of the definition of k_peak. The numerical match to actual Planck and DESI data is a separate hypothesis, with a named falsifier: any of the first three CMB acoustic peaks observed at a wavenumber ratio more than 5% off the predicted φ or φ² values. The library does not claim that the actual universe follows this ladder; it claims that if the framework's model of structure formation is right, the ratios should be φ and φ².

What the declaration does not do is also important. It does not predict the absolute positions of the peaks, only the ratios. It does not prove that the real CMB peaks are spaced by φ; that is an empirical question. It does not derive the base scale k_0 from first principles. The theorem is about a defined sequence, not about the cosmos directly. The step from the formal ladder to the observed sky is a hypothesis, and it is one that observations could in principle falsify.

/-- 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-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 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 real CMB peaks are observed to be spaced by the golden ratio. The base scale k_0 is derived from first principles. The theorem applies to any sequence other than the defined k_n = k_0 · φ^n.

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.

Derived articles

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