Qft Top Quark Yukawa Rs

Top quark Yukawa in RS

In particle physics, the Yukawa coupling of a quark measures how strongly it interacts with the Higgs field, which gives particles mass. For the top quark, the heaviest known particle, this coupling is approximately 0.9927, close to 1. The standard model does not predict this number; it is an input measured from experiment. Recognition Science (RS) starts from a different question: what if this near-unity value is not an accident but a consequence of a deeper cost principle?

The classical story begins with the measured top quark mass, about 172.69 GeV, and the Higgs vacuum expectation value, about 246.22 GeV. The Yukawa coupling is defined as y_t = √2 * m_t / v, which gives 0.9927. In RS, the framework models a ledger, a discrete record of recognition events, where each event carries a cost. The central cost function J(x) = (x + 1/x)/2 - 1 measures how far a ratio x is from unity. When x equals 1, J(1) = 0; the cost vanishes. The top Yukawa, being almost exactly 1, sits at the unique point where this cost is zero.

In Recognition Science, the framework's library proves three general facts about this cost function applied to a ratio m/e. First, the cost vanishes when m equals e. Second, the cost is nonnegative for any positive m and e. Third, the constant φ - 3/2, called the canonical threshold, is positive. These are theorems in the machine-checked library of formal theorems. They are fully general statements about the cost function, not specific to the top quark.

The module named qft-top-quark-yukawa-rs defines a cost function for the top Yukawa as J(m/e), where m and e are placeholders. It proves the three general facts above and packages them into a certificate structure. But the module does not define what m and e are in physical terms; it does not assert that m is the top mass or e is the Higgs vacuum expectation value. The research note in the module records the intended idea, that the top Yukawa is the unique J-cost-unity coupling, but that remains a hypothesis, not a proved result.

What this means in plain language: the framework shows that if you accept its cost function, then a coupling of exactly 1 is special, because it is the only value with zero cost. The measured top Yukawa is close enough to 1 to invite the question. But the framework has not yet derived the top mass from first principles, and it has not closed the gap between the measured 0.9927 and the ideal 1. The module is a template, not a finished physical theory.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The top Yukawa is proved to equal 1 in RS. The module derives the top mass from the cost function. The near-unity value of the top Yukawa is explained by the framework.

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

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