Cosmology Cosmic Microwave Background From Rs Second Peak Ratio

A ratio with a band

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. Imprinted on that glow are acoustic peaks: ripples of slightly hotter and colder plasma that froze into the sky. Cosmologists label them by multipole moment ℓ, a measure of angular scale. The first peak sits at ℓ₁ = 220, and the second at roughly ℓ₂ ≈ 507. Their ratio, ℓ₂/ℓ₁, is about 2.305, a number that falls inside the observationally favored band (2.3, 2.4).

In Recognition Science (RS), a framework that builds physical structure from a forced cost function, the library of formal theorems records a definition: secondPeakRatio is the rational number 507/220. A machine-checked theorem, secondPeakRatio_band, proves that this rational number lies strictly between 2.3 and 2.4. That is the entire content of the declaration: it certifies an arithmetic fact about a chosen rational, not a derivation of the ratio from first principles. The number 507/220 is a definitional choice, an identification, not a consequence of the framework's axioms.

What the declaration does not claim is as precise as what it does. It does not derive the value 507/220 from the framework's forcing chain; no theorem in the pack connects secondPeakRatio to the golden ratio, the cost function, or any deeper RS structure. The band (2.3, 2.4) is a statement about the rational 507/220, not a proof that the true cosmological ratio must lie there. The library also contains a separate definition for the first peak, firstPeak = 44 × 5 = 220, with a theorem that this equals the Planck measured value, but that is a distinct claim, not part of secondPeakRatio.

For a reader, the practical takeaway is the shape of the claim. The declaration is a certificate of arithmetic: 507 divided by 220 is a number in the observed band. It is a small, honest piece of bookkeeping, not a physical law. The framework's larger ambitions, such as deriving the peak positions from the cost function, remain targets, not results. This declaration is the kind of groundwork that a larger derivation would build on, but it does not itself perform that derivation.

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

/-- Second peak ratio ∈ (2.3, 2.4). -/
def secondPeakRatio : ℚ := 507 / 220  -- approximate ℓ₂/ℓ₁ ≈ 2.305

theorem firstPeak_matches_planck : firstPeak = firstPeakPlanck := by decide

What this page does not claim

No derivation of the second peak ratio from the framework's cost function or forcing chain. No proof that the true cosmological ratio must equal 507/220. No claim that the band (2.3, 2.4) is a derived physical prediction rather than a chosen interval.

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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