Cosmology Cosmic Microwave Background From Rs Second Peak Ratio Band

The second peak ratio

The cosmic microwave background (CMB) is the oldest light in the universe, a faint glow left over from 380,000 years after the Big Bang. Imprinted on that glow are acoustic peaks: ripples in the plasma of the early universe that became slightly hotter and cooler spots in the radiation. Cosmologists label these peaks by a multipole number, ℓ, where higher ℓ means smaller angular size on the sky. The first peak, ℓ₁, is the largest ripple; the second peak, ℓ₂, is the next one out. Planck, the European Space Agency's space observatory that mapped the CMB from 2009 to 2013, measured ℓ₁ at 220 with an uncertainty of about 0.5, and the ratio ℓ₂/ℓ₁ falls between 2.3 and 2.4.

That ratio band is what the Recognition Science declaration secondPeakRatio_band certifies. In plain terms, the declaration proves a simple arithmetic fact: the number 507/220, which is approximately 2.305, lies strictly between 2.3 and 2.4. The proof is machine-checked, meaning a computer program verified every step of the reasoning, with no gaps and no unproven assumptions. The declaration also packages this with related facts: the first peak equals 44 times 5, which is exactly 220, and that value matches the Planck measurement. The whole package, called CMBCert, bundles these pieces into a single certified statement about the CMB peaks.

In Recognition Science, the framework models reality as a discrete record of recognition events, and from that record it derives constants and structures. Within this framework, the number 44 is called the baryon rung and 5 is the configuration dimension; their product gives the first peak. The framework's library, a machine-checked collection of formal theorems, proves that 44 × 5 = 220 and that 220 lies in the measured window of 215 to 225. For the second peak, the framework chooses the ratio 507/220 as its representative value and proves that this value falls inside the observed band.

What the declaration does not claim is just as important. It does not prove that the ratio 507/220 is the actual measured value of ℓ₂/ℓ₁; it only proves that this chosen number lies within the observed range. It does not derive the numbers 44 and 5 from first principles in this file; those are definitional choices, not derived results. And it does not claim that the framework's model explains why the CMB has these peaks, only that a specific arithmetic relationship holds. The value 507/220 is a definition, not a measurement or a prediction.

The practical consequence is a certified consistency check. Anyone can now verify that the framework's chosen numbers land inside the window that Planck's data established, with the arithmetic fully checked by machine. That is a small but solid anchor: a formal proof that a particular ratio, chosen within the framework, is compatible with what telescopes actually see.

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

theorem firstPeak_eq : firstPeak = 220 := by decide

theorem firstPeak_matches_planck : firstPeak = firstPeakPlanck := by decide

What this page does not claim

The declaration does not prove that 507/220 is the measured value of the second peak ratio; it only proves this chosen number lies within the observed band. The declaration does not derive the numbers 44 and 5 from first principles; they are definitional choices in the framework. The declaration does not claim the framework explains the existence of acoustic peaks in the CMB.

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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