Cosmology Baryogenesis3 From Jcost Baryogen3 Cert

The certificate's scope

In the Recognition Science framework, a cost function measures the price of a recognition event, a discrete record of something happening. The framework's central theorem forces this cost to take one specific form, J(x) = (x + 1/x)/2 - 1. The declaration Baryogen3Cert is a small bundle of three proved facts about that cost function when it is applied to a ratio of two numbers, m divided by e.

The three facts are general and simple. First, when the two numbers are equal, the cost is zero. Second, for positive inputs, the cost is never negative. Third, the golden ratio phi, about 1.618, is greater than 1.5. Each is proved in the machine-checked library of formal theorems, and together they form a certificate, a named package of results that other proofs can rely on.

Despite the name, Baryogen3Cert does not establish anything about baryogenesis, the physical process that generated the excess of matter over antimatter in the early universe. The certificate's cost function is defined as J(m/e) without any connection to particle masses or photon counts. The research note attached to the module records an intended direction, where the matter-antimatter asymmetry eta_B might be derived from this cost, but the note itself states the Lean code proves nothing specific to the subject. The numerical estimate in that note, roughly 1.74e-5, is far from the measured value of about 6e-10, and the note calls for an additional suppression factor that is not defined.

What the certificate does provide is a clean, reusable foundation. Any future work that needs a cost function vanishing at equality, staying nonnegative, and having a positive threshold at phi minus 1.5 can cite Baryogen3Cert directly. The physical bridge, defining m and e in terms of real particles and fields, remains open.

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]

What this page does not claim

The certificate proves no statement about the matter-antimatter asymmetry eta_B. The numerical value 1.74e-5 in the research note is not a derived prediction, only a recorded estimate. The certificate does not define m or e in terms of any physical particle or field.

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/Baryogenesis3_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.

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