Cosmology Cmbpolarization Ratio3 From Jcost

Cosmic polarization ratio

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 polarization, the direction in which its light waves vibrate, splits into two patterns: E-modes, which are curl-free, and B-modes, which have a twist. Primordial gravitational waves, ripples in spacetime from the universe's first instant, are expected to generate a tiny B-mode signal, while ordinary density fluctuations produce mostly E-modes. The ratio of B-mode to E-mode power is therefore a direct probe of those early gravitational waves.

In Recognition Science, a framework that derives physical structure from a forced cost of recognition, the B/E ratio is expressed as B/E = r / (8 * J(phi)), where r is the tensor-scalar ratio and J(phi) is the framework's central cost function evaluated at the golden ratio phi. The tensor-scalar ratio r measures the amplitude of primordial gravitational waves relative to density fluctuations. With r = 0.017, a value consistent with current upper limits, this formula gives B/E = 0.017 / 0.944 = 1.8 percent. This is a structural prediction, not a fitted parameter, within the framework.

The module named CMBPolarizationRatio3FromJCost in the framework's machine-checked library of formal theorems proves three general facts about the cost function Jcost(m/e): it vanishes when m equals e, it is nonnegative for positive inputs, and the quantity phi - 3/2 is positive. These are universal properties of the cost function, not specific to cosmology. The module does not define what m and e mean for the CMB, so it proves nothing about the actual polarization ratio. The research note attached to the module records the intended connection, but the formal content stops at the general cost function facts.

The framework's contribution here is a candidate structural formula for a measurable cosmological quantity. The formula's ingredients, the golden ratio and the forced cost function, come from the framework's derivation of physical constants. Whether the 1.8 percent prediction matches future observations of the B-mode signal remains to be seen. The framework treats this as an empirical check, not a theorem. The module itself is honest about its scope: it establishes properties of the cost function, not the cosmology.

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)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

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

What this page does not claim

The module proves the B/E ratio equals 1.8 percent; it only records the formula as a research note. The framework derives the tensor-scalar ratio r; the value 0.017 is an input, not an output. The B-mode signal has been measured; it remains a target for current and future CMB experiments.

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/CMBPolarizationRatio3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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