Cosmology Baryogenesis3 From Jcost

What the module proves

In cosmology, baryogenesis is the process that would explain why the universe contains more matter than antimatter. The measured imbalance is about six baryons for every billion photons. A module in the Recognition Science framework, Baryogenesis3_FromJCost, was written to connect that imbalance to the framework's cost function, a forced measure of recognition effort. The file's docstring records that plan, but the formal content stops far short of it.

The module defines a cost for two real numbers m and e as J(m/e), where J is the framework's cost function. The machine-checked library then establishes three facts. First, the cost is zero when m equals e. Second, the cost is nonnegative for positive inputs. Third, the number phi minus 3/2 is positive, where phi is the golden ratio. These are general properties of the cost function, not results about baryogenesis. The definition of domainCost never refers to baryons, photons, or any physical quantity; it is a template applied to arbitrary numbers.

The docstring itself is explicit about this gap. It says the Lean code establishes nothing specific to the subject, because the cost is defined without reference to one. The file's research note sketches a hoped-for connection: the observed baryon asymmetry of about 6e-10 might be expressed as J(phi) divided by phi to some rung power, giving a number like 1.74e-5, which is not close to the measured value without additional suppression. That note is a plan, not a result.

What the module does establish, in plain language, is a small certificate: the cost function behaves sensibly at equality and for positive inputs, and the golden ratio exceeds 1.5. These facts are checked in the machine-checked library, but they say nothing about why the universe has more matter than antimatter. The physical claim remains open.

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]

What this page does not claim

This module proves any physical baryon asymmetry. This module derives the measured value 6e-10. The research note in the docstring is a proved result.

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)

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Derived articles

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