Cosmology Cosmic Microwave Background From Rs First Peak Planck

The first acoustic peak

The cosmic microwave background (CMB) is the oldest light in the universe, released about 380,000 years after the Big Bang. Its temperature is not perfectly uniform: it carries tiny ripples, and those ripples have a characteristic angular scale. Cosmologists describe that scale with the multipole number ℓ, where larger ℓ means smaller angles on the sky. The first acoustic peak, the strongest of the ripples, appears at ℓ₁ ≈ 220. The Planck satellite measured this peak with high precision: 220 ± 0.5.

In Recognition Science (RS), the framework models the CMB peak not as a free parameter but as a product of two framework-internal numbers. The first is the baryon rung, a discrete step on the framework's ladder of particle masses, set to 44. The second is the configuration dimension, a count of spatial degrees of freedom in the framework's geometry, set to 5. The framework's declaration firstPeakPlanck defines the measured value as 220, and a companion theorem proves that the framework's own firstPeak, defined as 44 × 5, equals that value exactly. The same file also defines a second peak ratio, 507/220 ≈ 2.305, and proves it lies in the observed band (2.3, 2.4).

What the declaration does not claim is just as important. The equality firstPeak = 220 is a theorem about arithmetic: 44 times 5 is 220. The match to the Planck value is a definitional identification, not a derivation from deeper physics. The framework does not explain why the baryon rung is 44 or why the configuration dimension is 5; those are chosen constants, not forced outcomes. The second peak ratio is an approximation, not an exact prediction, and the framework does not derive the full acoustic peak structure of the CMB. The declaration establishes a numerical coincidence, checked against a measurement, and leaves the physical mechanism open.

theorem firstPeak_eq : firstPeak = 220 := by decide

theorem firstPeak_matches_planck : firstPeak = firstPeakPlanck := by decide

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

What this page does not claim

The framework does not derive the value 44 or 5 from first principles. The framework does not predict the full CMB power spectrum, only the first peak location and a band for the second peak ratio. The match to Planck is an empirical check, not a derivation of the CMB from RS axioms.

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.

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