Cosmology Cmbacoustic Peak Ratios Planck Ratio 2 1 Value

The measured ratio

The cosmic microwave background (CMB) is the oldest light in the universe, a fossil glow released about 380,000 years after the Big Bang. Imprinted on that glow is a pattern of hot and cold spots, and within that pattern are acoustic peaks: regular bumps in the power spectrum that record sound waves traveling through the dense early plasma. The first peak marks the largest scale of those waves, the second peak marks the next harmonic, and so on. Cosmologists measure the positions of these peaks by an angular multipole number, denoted ℓ. Planck 2018, the European Space Agency mission that mapped the CMB with unprecedented precision, reported the first peak at ℓ₁ = 220.0 and the second at ℓ₂ = 540.3.

Dividing those two numbers gives the ratio ℓ₂/ℓ₁ = 540.3 / 220.0 ≈ 2.456. The Recognition Science declaration planck_ratio_2_1_value is a machine-checked theorem stating that this ratio lies strictly between 2.45 and 2.46. That is a narrow band: it confirms the arithmetic of the two measured peak positions and encodes them as a single, exact interval. The declaration does not invent new data. It takes the two Planck values as definitions and proves, by direct computation, that their quotient falls in that range. The result is a formal certificate that the arithmetic is correct, not a new measurement of the sky.

The context matters. The Recognition Science framework models the universe as storing a discrete record of events, which it calls a ledger, a record whose structure forces certain ratios to appear. Within that framework, a separate theorem derives that the bare wavenumber ratios between acoustic peaks should equal the golden ratio φ ≈ 1.618, so that k₂/k₁ = φ and k₃/k₁ = φ². The observed angular ratio 2.456 is not φ, and the library proves this explicitly: the theorem planck_ratio_not_directly_phi shows ℓ₂/ℓ₁ ≠ φ. The gap exists because the angular multipole ℓ is not the same as the wavenumber k; the projection from k-space to ℓ-space depends on the angular diameter distance to the last scattering surface.

What the declaration establishes, then, is a precise statement about the measured angular ratio, not a confirmation of the golden-ratio prediction. The prediction lives at the wavenumber level, where direct measurements of baryon acoustic oscillations from surveys like BOSS or DESI could test it. The falsifier is explicit: any direct k-space measurement of the second-to-first or third-to-first BAO peak ratios outside the predicted φ-band by more than 5 percent would refute the structural claim. The Planck angular ratio, by contrast, is a measured fact, and the declaration simply certifies its value within a tight interval.

theorem planck_ratio_2_1_value : 2.45 < planck_ratio_2_1 ∧ planck_ratio_2_1 < 2.46 := by
  unfold planck_ratio_2_1 planck_l_2 planck_l_1
  refine ⟨?_, ?_⟩ <;> norm_num

/-- Planck 2018 first acoustic peak: `ℓ_1 = 220.0`. -/
def planck_l_1 : ℝ := 220.0

/-- Planck 2018 second acoustic peak: `ℓ_2 = 540.3`. -/
def planck_l_2 : ℝ := 540.3

/-- 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 declaration does not confirm the golden-ratio prediction for the angular multipole ratio. The declaration is not a new measurement of the cosmic microwave background. The declaration does not establish that the bare wavenumber ratios equal φ; that is a separate theorem.

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.

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