Cosmology Cmbacoustic Peak Ratios Ratio 3 1

The third peak ratio

The cosmic microwave background (CMB) is the oldest light in the universe, and its sky map shows a series of acoustic peaks: ripples in the hot plasma of the early universe that froze into the pattern we see. The first peak marks the largest such ripple, the second a smaller one, the third smaller still. Cosmologists routinely measure the angular positions of these peaks on the sky, labeled l1, l2, l3. In the standard picture, the ratio of the third peak's position to the first is a number around 3.7, a value that depends on the detailed physics of the plasma and the geometry of the universe.

The Recognition Science framework approaches the same peaks from a different direction. Its library of machine-checked formal theorems derives, from the framework's core assumptions about recognition cost, a prediction about the underlying wavenumbers k1, k2, k3, the spatial frequencies of the density fluctuations that later became the peaks. The declaration ratio_3_1 states a precise structural fact: for any positive starting wavenumber k0, the ratio of the third peak wavenumber to the first is exactly phi squared, where phi is the golden ratio, approximately 1.618. Since phi squared equals phi plus one, this means k3 is about 2.618 times k1, a ratio that falls in the numerical band between 2.59 and 2.63.

This is a theorem in the framework's sense: it is proved from the framework's axioms, with no additional assumptions, and it holds for every positive k0. The proof is short, a direct consequence of the framework's earlier result that adjacent peak wavenumbers are separated by the golden ratio. The theorem does not, by itself, say anything about what we observe on the sky. The observed angular positions l1, l2, l3 are not the same as the wavenumbers k1, k2, k3. The projection from wavenumber space to the angular sky involves the angular diameter distance, a geometric factor that shifts the ratios. The framework's own library records that the observed angular ratio l2/l1 is about 2.456, which is not the golden ratio, and proves this difference explicitly.

What the framework claims, then, is a statement about the bare wavenumbers, not about the directly observed angular peaks. The test of the prediction is a direct measurement of wavenumbers, for example through baryon acoustic oscillations in galaxy surveys like BOSS or DESI. The framework's falsifier is named: any direct k-space measurement of the second-to-first or third-to-first peak ratio that falls outside the predicted golden-ratio band by more than five percent would refute the structural claim. Until such a measurement is made, the wavenumber prediction remains a theorem within the framework and an open empirical question for cosmology.

/-- 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

/-- The 3-1 ratio band: `(2.59, 2.63)`. -/
theorem ratio_3_1_band (k_0 : ℝ) (h : 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 := by
  rw [ratio_3_1 k_0 h]
  exact phi_sq_band

/-- 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 CMB angular peak ratios match the golden ratio prediction. The framework derives the full CMB power spectrum or the angular diameter distance. The wavenumber prediction has been confirmed by any existing cosmological measurement.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: