peak_3_2_ratio
The theorem shows that the wavenumber ratio between the third and second CMB acoustic peaks equals the golden ratio phi. Modelers of structure formation in the BIT framework cite this when confirming the phi-ladder in the power spectrum. The proof consists of a direct one-line application of the adjacent-peak ratio result at the second peak.
claimLet $k_n = k_0 phi^n$ be the wavenumber of the nth acoustic peak for positive base scale $k_0$. Then $k_3 / k_2 = phi$.
background
The module derives the matter power spectrum from the BIT kernel, imposing a phi-ladder on the characteristic wavenumbers. The definition k_peak sets $k_n = k_0 phi^n$. The upstream adjacent ratio theorem establishes that consecutive peaks obey $k_{n+1} / k_n = phi$ for any positive $k_0$ (unfolded via the explicit power form and cancellation of nonzero terms). This places the third-to-second ratio inside the sequence of phi-rational peak positions that the module proves for the first three CMB peaks.
proof idea
The proof invokes the adjacent peak ratio theorem at n=2, passing the positivity hypothesis on k_0. No further reduction is required.
why it matters in Recognition Science
The result is called directly by the ratio_3_2 theorem in the CMBAcousticPeakRatios module to obtain the same statement. It fills the phi-rational ratio structure asserted in the module documentation for the first three acoustic peaks. The construction remains independent of the base scale k_0 in the ratio sector.
scope and limits
- Does not fix the absolute scale of the peaks or the value of k_0.
- Does not extend the ratio claim to peaks beyond the third.
- Does not compare the predicted ratios against specific observational datasets.
- Does not derive the phi value from the underlying Recognition Science forcing chain.
Lean usage
theorem ratio_3_2 (k_0 : ℝ) (h : 0 < k_0) : k_peak k_0 3 / k_peak k_0 2 = phi := peak_3_2_ratio k_0 hformal statement (Lean)
63theorem peak_3_2_ratio (k_0 : ℝ) (h : 0 < k_0) :
64 k_peak k_0 3 / k_peak k_0 2 = phi :=
proof body
Term-mode proof.
65 k_peak_adjacent_ratio k_0 2 h
66
67/-- The third-to-first peak ratio is `φ²`. -/