Cosmology Cosmic Microwave Background From Rs Config Dim

The CMB peak decomposition

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. Its temperature is not perfectly uniform; it carries small ripples that astronomers plot as a power spectrum against a quantity called the multipole moment ℓ. The first and largest of these ripples, the first acoustic peak, appears at ℓ₁ = 220. The Planck satellite, which mapped the CMB from 2009 to 2013, measured this peak at 220 ± 0.5, and that value is a standard result in modern cosmology.

Within the Recognition Science framework, a machine-checked library of formal theorems, the number 220 is not left as a bare measurement. The framework defines two integers: baryonRung, set to 44, and configDim, set to 5. Their product is 220, and a theorem in the library proves that this product equals the first peak value. The framework's claim is arithmetic: the measured peak can be decomposed as 44 × 5. It also defines a second peak ratio as 507/220, about 2.305, and proves this number lies between 2.3 and 2.4, matching the observed ratio of the second to the first peak.

What configDim does not claim is just as important as what it does. It does not claim that the number 5 is derived from the framework's deeper principles; it is a definition, a chosen integer, not a proved consequence. The framework does not claim that this decomposition predicts the CMB peak from first principles, nor that it explains why the peak is at 220 rather than some other value. The match with Planck's measurement is a numerical agreement, not a derivation: the framework notes the coincidence that 44 × 5 equals 220, and the library certifies the arithmetic, but it does not claim a physical mechanism that forces the peak to that location.

The honest summary is that configDim is a label for the number 5 in one particular decomposition of a measured cosmological quantity. The framework's contribution is to make the arithmetic explicit and machine-checked, and to record that the decomposition sits within the observed band. What remains open is whether the factors 44 and 5 have any deeper origin in the framework's own structure, and whether the second peak ratio's band is a genuine prediction or a post-hoc fit.

def baryonRung : ℕ := 44

def configDim : ℕ := 5

theorem firstPeak_eq : firstPeak = 220 := by decide

/-- Planck measured value 220 ± 0.5. -/
def firstPeakPlanck : ℕ := 220

theorem secondPeakRatio_band : (2.3 : ℝ) < (secondPeakRatio : ℝ) ∧ (secondPeakRatio : ℝ) < 2.4 := by
  unfold secondPeakRatio
  constructor <;> norm_num

What this page does not claim

configDim is not derived from the framework's foundational principles; it is a definitional choice. The framework does not claim a physical mechanism that forces the CMB peak to ℓ = 220. The match with Planck's value is a numerical agreement, not a prediction from first principles.

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/CosmicMicrowaveBackgroundFromRS.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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