Cosmology Cmbpolarization Ratio3 From Jcost Cmbpolar Ratio3 Cert

A certificate and its limits

In cosmology, the cosmic microwave background (CMB) carries two types of polarization patterns: E-modes, which are curl-free, and B-modes, which have a curl. B-modes are of special interest because primordial gravitational waves are one of the few sources that can generate them. The ratio of B-mode to E-mode power is often expressed through the tensor-to-scalar ratio r, a parameter that quantifies the amplitude of primordial gravitational waves relative to density fluctuations. A common target for next-generation experiments is r around 0.01 to 0.001, with current upper limits near 0.03.

The Recognition Science framework models a ledger, a discrete record of recognition events, and assigns a cost to each event. The framework's central result is a unique cost function J(x) = (x + 1/x)/2 - 1, which is forced by five plain conditions and proved in a machine-checked library of formal theorems. The declaration CMBPolarRatio3Cert is a small structure inside this library. It packages three general facts about the cost function: it vanishes when the two inputs are equal, it is nonnegative for positive inputs, and a certain threshold value phi - 3/2 is positive. These facts are proved for any positive real numbers m and e, with no reference to cosmology.

The intended application was to relate the B/E ratio to the tensor-to-scalar ratio r through the formula B/E = r / (8 * J(phi)), which at r = 0.017 would give about 1.8 percent. This appears in a research note in the source file, not as a theorem. The note explicitly states that the module proves nothing specific to this subject, because the cost function is defined as J(m/e) without any definition of m and e in cosmological terms. The certificate establishes the abstract properties of the cost function, not the polarization ratio.

In Recognition Science, this certificate is a building block: it shows the cost function behaves as expected on positive inputs, which is a necessary condition for any physical application. But the bridge from the abstract cost to the CMB polarization is not built. The certificate is a general-purpose tool, not a cosmological result. What it does not claim is equally clear: it does not derive the tensor-to-scalar ratio, it does not predict a specific B-mode signal, and it does not connect the cost function to any measured polarization data.

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)

structure CMBPolarRatio3Cert 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 derive the tensor-to-scalar ratio r. The certificate does not predict a specific B-mode polarization signal. The certificate does not connect the cost function to any measured CMB data.

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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