Physics Cpviolation3 From Jcost

A placeholder for kaon decay

In particle physics, CP violation is the small asymmetry between matter and antimatter behavior in certain decays. The neutral kaon system shows it through the parameter ε_K, measured at about 2.23 × 10⁻³. The module CPViolation3_FromJCost in the Recognition Science library aims at this quantity, but its formal content stops well short of the physics.

The file defines domainCost, a cost function that measures the price of recognition, as J(m/e), where J is the forced cost function J(x) = (x + 1/x)/2 − 1. The Lean code proves three general facts about this cost: it equals zero when the two inputs are equal, it is never negative for positive inputs, and the number φ − 3/2 is positive. These are true for any positive real numbers m and e; they say nothing about kaons, mesons, or weak decays.

The docstring records a research note, not a result. It tries to match ε_K by computing J(φ)^(3/2) = 0.0405, which is too large, then J(φ)^3 = 0.0016, which is close to 0.002 but not to the measured 2.23 × 10⁻³. The note itself admits the first attempt fails and the second is only "consistent" in a loose sense. The formal theorems prove none of this.

What the module does establish, cleanly, is a template. The three proved facts appear in a structure called CPVKaon3Cert, and the library shows this structure is inhabited. That is a certificate that the cost function has the three basic properties, not that it describes kaon physics. To make this a theorem about CP violation, someone would need to define m and e in the kaon system's own terms, and that definition does not exist here.

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 statement about kaon decay or CP violation. The numerical match between J(φ)^3 and 0.002 is a theorem; it is a research note with no formal backing.

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/Physics/CPViolation3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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