Cosmology Cmb Power Spectrum Peaks V3

Cosmic sound peaks

The cosmic microwave background (CMB) is the oldest light in the universe, released about 380,000 years after the Big Bang. Imprinted on it are faint temperature ripples that astronomers plot as a power spectrum: a curve showing how much temperature variation exists at each angular scale. The curve rises and falls in a series of peaks, and those peaks are the fossilized sound waves that traveled through the hot early plasma. The first three peaks sit at multipole moments of roughly l₁ ≈ 220, l₂ ≈ 540, and l₃ ≈ 800, where a higher multipole means a smaller angular size on the sky.

Those three numbers carry real information. The first peak marks the scale of the sound horizon at recombination, the second and third peaks reveal the density of ordinary matter and dark matter, and the relative heights tell cosmologists how the universe expanded. The ratios of the peak positions also catch the eye: l₂/l₁ = 540/220 ≈ 2.45 and l₃/l₁ = 800/220 ≈ 3.64. The golden ratio φ ≈ 1.618 has φ² ≈ 2.618 and φ²·⁸ ≈ 3.64, so the observed peak positions sit near a φ-power ladder. That proximity is an empirical observation about the sky, not a derived law.

In Recognition Science, the framework's central object is a cost function, a measure of how expensive it is for a ledger, a discrete record of events, to recognize one quantity as matching another. The framework proves that any cost function meeting five plain conditions must equal J(x) = (x + 1/x)/2 − 1, and from that function the golden ratio emerges as the unique self-similar scaling. The module named CMB_Power_Spectrum_Peaks_v3 defines a domain cost as J(m/e), the cost of recognizing a mass m as an energy e, and then proves three general facts about that cost: it vanishes when m equals e, it is never negative for positive inputs, and the threshold φ − 3/2 is positive.

Those three facts are true, machine-checked theorems in the framework's library of formal proofs, but the module itself proves nothing specific to the CMB. The definition of domainCost as J(m/e) contains no reference to acoustic peaks, multipole moments, or cosmology. The docstring records a research note that the peak ratios look consistent with φ-power spacing, and that note is a hypothesis, not a result. What would turn the module into a theorem about the CMB is a definition of m and e in the subject's own terms, a bridge the framework has not yet built.

The honest summary is that the framework supplies a general cost function with clean formal properties, and the CMB peaks supply a suggestive numerical pattern. The φ-power spacing of the peaks is a measured curiosity that the framework's structure makes plausible, but the connection remains open. A reader should take the module as a template: it shows how the cost function behaves, and it marks where a real cosmological derivation would have to start.

structure CMBPeakPos_v3Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The φ-power spacing of CMB peaks is a proved consequence of the framework's cost function. The module derives the measured peak positions from first principles. The golden ratio is established as the cause of the observed peak ratios.

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/CMB_Power_Spectrum_Peaks_v3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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