Cosmology Cosmic Microwave Background From Rs

Cosmic Microwave Background

The cosmic microwave background (CMB) is the faint glow left over from the hot early universe, first measured by Penzias and Wilson in 1965. Its temperature map is not perfectly smooth; it shows small ripples, and those ripples have a characteristic pattern of peaks when plotted against angular scale. The first and largest of these, the first acoustic peak, appears at a multipole moment of ℓ₁ = 220, as measured by the Planck satellite with an uncertainty of about 0.5. This peak corresponds to the scale of the largest sound wave that could travel through the plasma before recombination.

The multipole moment ℓ is a standard way to measure angular size on the sky: higher ℓ means smaller angles. The first peak's position is set by the sound horizon at recombination, the distance sound could travel in the early plasma, projected onto today's sky. In the standard cosmological model this depends on the density of matter and the expansion rate. The measured value 220 is a precise number that any successful theory of the early universe must reproduce.

In Recognition Science, the framework models the CMB peak positions as consequences of its underlying structure. Its library of machine-checked formal theorems defines a quantity called baryonRung, set to 44, and configDim, set to 5. The first peak is then defined as their product: firstPeak = 44 × 5 = 220. A theorem in the library proves this equality, and another proves that this matches the Planck value exactly. The library also defines the ratio of the second peak to the first as 507/220, approximately 2.305, and proves that this ratio lies between 2.3 and 2.4, matching the observed range.

These are not free parameters fitted to the data; the framework derives 44 and 5 from its forcing chain, which starts from a single cost function and leads to constants like the golden ratio and the number of spatial dimensions. The claim is that the same structure that forces three dimensions also forces the CMB's acoustic peak positions. The agreement with Planck is exact for the first peak, within the stated band for the second peak ratio, and the library certifies these arithmetic facts with no unproved assumptions.

What this means for a reader is that the framework offers a single derivation that connects the largest observable structure in the universe to a small set of integers. The CMB peaks are not an isolated coincidence; they are one more place where the framework's forced constants reappear. The honest caveat is that the identification of 44 and 5 with physical quantities is a modeling choice, not a theorem, and the full physical bridge from recognition events to plasma acoustics remains 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 fine-structure constant alpha. The assignment of 44 and 5 to baryon and configuration dimensions is a modeling choice, not a proved physical derivation. The framework does not provide a full physical model of the CMB plasma or its acoustic oscillations.

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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