Cosmology Cmb Power Spectrum Peaks V3 Cmbpeak Pos V3 Cert

What the certificate proves

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. Its temperature map shows ripples, and those ripples have a characteristic angular size. When astronomers plot the ripple strength against angular scale, they see a series of peaks: the first at multipole moment l_1 around 220, the second near 540, the third near 800. These peaks are a standard probe of cosmology, used to measure the universe's geometry and contents.

The Recognition Science declaration CMBPeakPos_v3Cert is a machine-checked certificate: a small package of three formal facts about a function called domainCost, defined as J(m/e), where J is the framework's cost function. The three facts are elementary: the cost is zero when the two inputs are equal, the cost is never negative for positive inputs, and the number phi minus 3/2 is positive. The certificate proves these three statements hold for its definition of domainCost, nothing more.

The subject line of the file records an intended connection: the peak ratios 540/220 and 800/220 are close to powers of the golden ratio phi, suggesting a phi-power spacing. But the formal certificate does not define m or e in terms of the CMB. The declaration uses the same generic template as 2383 sibling modules, all stating the same three facts about J(m/e). Without a subject-specific definition of m and e, the certificate proves no property of acoustic peaks, no spacing law, and no connection between the CMB and the golden ratio.

In plain terms: the certificate is a valid but empty shell. It guarantees that a certain abstract cost function has three basic properties, and it is honest about its own limits. The research note above the code says what would make it a theorem about the CMB: a definition of m and e in the subject's own terms. Until then, the peak ratios remain an observation, not a derived result.

theorem cert_inhabited : Nonempty CMBPeakPos_v3Cert := ⟨cert⟩

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

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

What this page does not claim

The certificate does not prove that CMB peak ratios follow a phi-power law. The certificate does not establish any connection between the golden ratio and the cosmic microwave background. The certificate does not validate the research note's observational claim about peak positions.

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)

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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