Qft Higgs Self Coupling Rs

The Higgs self-coupling in RS

In the Standard Model of particle physics, the Higgs boson interacts with itself. The strength of that three-Higgs interaction is the Higgs self-coupling, usually written λ_H. It is not a free parameter in the way the electron mass is; the theory predicts it from the Higgs mass m_H and the vacuum expectation value v, through λ_H = m_H²/(2v²). With the measured values m_H = 125.25 GeV and v = 246 GeV, that formula gives λ_H ≈ 0.130. The number matters because it is one of the few quantities in the Standard Model that the theory does not simply absorb; measuring it tests whether the Higgs field really is the simple quartic field the model assumes.

Recognition Science (RS) approaches such numbers from a different direction. Its central object is the cost, a measure of how expensive it is for reality to recognize one scale in terms of another. The framework proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. The framework's library of machine-checked formal theorems then derives constants like the golden ratio from this J. The Higgs self-coupling enters as a candidate application: RS identifies the ratio m_H/v with the cost input, so that J(m_H/v) should approximate λ_H. Plugging in the numbers gives J(125.25/246) ≈ 0.118, which lands about 9% below the Standard Model value of 0.130.

The module in the RS library, Higgs_Self_Coupling_RS.lean, does not prove that 0.118 equals 0.130. Its docstring is explicit: the Lean code proves three general facts about J, namely that it vanishes when its two inputs are equal, that it is nonnegative for positive inputs, and that φ - 3/2 is positive. These are true of J for any positive inputs, not specific to the Higgs. The module defines a cost for the Higgs ratio but does not define what m and e mean in Higgs terms; that missing definition is what would turn the general facts into a statement about the Higgs. The 9% gap is therefore a research note, not a result.

What the module does establish is a template. It shows how a physical coupling could be expressed as a cost, and it records the exact point where the derivation stops. The three proved facts are the skeleton any future Higgs-specific theorem would build on. The gap between 0.118 and 0.130 is not a failure of the framework; it is a precise, honest measure of what remains to be understood. A reader can see exactly what is proved, what is conjectured, and where the missing definition sits.

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

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

The RS value 0.118 is not derived from first principles; it is an identification of the cost input with m_H/v. The module does not prove that the Higgs self-coupling equals the cost function value. The three proved facts about J are general properties, not statements specific to the Higgs boson.

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

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