Cosmology Cmbpolarization3 From Jcost Cmbpolar3 Cert

The certificate's scope

The ledger (a discrete record of events) in Recognition Science assigns a cost to any ratio of two positive quantities. The certificate named CMBPolar3Cert establishes three plain facts about that cost. First, when the two quantities are equal, the cost is exactly zero. Second, for any two positive quantities, the cost is never negative. Third, a specific threshold value, the golden ratio minus 1.5, is greater than zero. These are the certificate's entire mathematical content.

The certificate's name suggests a connection to the cosmic microwave background's E-mode polarization. However, the surrounding documentation states directly that the formal proof contains nothing specific to that subject. The cost function is defined as J(x) = (x + 1/x)/2 - 1, applied to a ratio m/e, without any definition of what m and e mean for polarization. The research note above the code records where the idea was meant to go, not a result. A real theorem about E-mode polarization would need to define m and e in terms of the actual physics.

What the certificate does provide is a reusable template. The same three facts, universally quantified, appear in over two thousand sibling modules. The library proves them once in a shared location, and each module re-exports them. This is a structural convenience, not a cosmological discovery. The numerical speculation in the note, suggesting an E/T ratio near 55.7 percent or 11.8 percent, is an unverified research comment, not a proved claim.

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 certificate does not claim any relationship between the cost function and actual E-mode polarization data. The certificate does not claim that the golden ratio appears in the cosmic microwave background. The numerical ratios in the research note are not proved results.

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/CMBPolarization3_FromJCost.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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