Cosmology Cosmic Microwave Background From Rs First Peak Matches Planck

The first peak match

The cosmic microwave background (CMB) is the oldest light in the universe, released about 380,000 years after the Big Bang. Its temperature map shows faint ripples, and those ripples have a characteristic scale. Cosmologists describe that scale with the multipole moment ℓ, a number that measures angular size on the sky: a smaller ℓ means a larger patch of sky. The first acoustic peak, the strongest ripple, appears at ℓ₁ = 220. The Planck satellite, which mapped the CMB from 2009 to 2013, measured that peak at 220 ± 0.5.

The Recognition Science framework, which derives physical constants from a theory of recognition events, produces its own number for this peak. Its calculation starts with two internal quantities: baryonRung = 44 and configDim = 5. The framework's firstPeak is defined as their product, 44 × 5 = 220. The declaration firstPeak_matches_planck is a machine-checked theorem stating that this product equals the Planck value. The proof is a direct computation; the library checks it with zero unproved assumptions. The framework also places the ratio of the second peak to the first peak in the interval (2.3, 2.4), consistent with the observed ratio of about 2.3.

What the theorem does not do is explain why the CMB has an acoustic peak at all, or why baryonRung and configDim take the values 44 and 5. Those numbers are definitions in the framework, not derived results. The match is a numerical coincidence between a defined product and a measured value, verified by arithmetic. It is not a physical derivation of the peak's origin. The framework's library proves the equality 44 × 5 = 220 and the inequality for the second peak ratio; it does not prove that the CMB must have these peaks, nor that the framework's constants are the reason the universe has this structure.

The practical value of the declaration is modest but real: it documents, in a machine-checked form, that the framework's arithmetic agrees with a precise cosmological measurement. The agreement is exact at the integer level, though the Planck measurement carries an uncertainty of ±0.5. The theorem does not claim the framework predicted the peak before Planck measured it, nor that the framework's model of baryons or configuration space is physically correct. It claims only that a specific product of two defined integers equals the measured peak position. The broader question, whether the framework's recognition-based derivation of these constants is physically meaningful, remains outside the scope of this theorem.

theorem firstPeak_matches_planck : firstPeak = firstPeakPlanck := by decide

/-- ℓ₁ = baryonRung × configDim = 220. -/
def firstPeak : ℕ := baryonRung * configDim

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 theorem does not derive the origin of the CMB acoustic peaks. The theorem does not prove the framework's recognition model is physically correct. The theorem does not claim the framework predicted the peak before Planck measured it.

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)

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