Physics Higgs Coupling3 From Jcost

The module and its scope

The Higgs boson's self-coupling is the strength at which the Higgs field interacts with itself, a number the Standard Model predicts from the Higgs mass and the vacuum expectation value. In the usual formula, lambda = m_H^2 / (2 v^2), plugging in 125 GeV for the mass and 246 GeV for the vacuum value gives about 0.129. That number matters because measuring it tests whether the Standard Model's simplest mechanism holds.

In Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1 is a proved measure of mismatch between two quantities. The module Higgs_Coupling3_FromJCost defines a cost for a ratio m/e, where m and e are real numbers, and then proves three facts: the cost is zero when m equals e, it is never negative for positive inputs, and the golden-ratio constant phi minus 3/2 is positive. These are all general properties of the cost function itself.

What the module does not do is connect those facts to the Higgs boson. The definition of domainCost uses the ratio m/e without any specification of what m and e mean in particle physics. The docstring notes the intended identification: Higgs self-coupling equals J(phi), with the empirical value 0.129 against the framework's 0.118, a 9 percent deviation. But that identification is a research note, not a theorem.

In Recognition Science, the framework models the Higgs self-coupling as J(phi) = 0.118, which is the cost at the golden ratio. The empirical value 0.129 differs by about 9 percent. The module itself proves none of this; it only proves the three generic cost properties. The physical bridge, defining m and e in terms of Higgs parameters, remains open.

The distinction matters for anyone reading the library: a module named after a physical quantity may contain only generic mathematics. The honest takeaway is that the framework has a candidate formula for the Higgs self-coupling, but the machine-checked part stops at the cost function's basic behavior. The physics is a hypothesis awaiting a definition.

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 module does not prove that the Higgs self-coupling equals J(phi). The module does not define the Higgs mass or vacuum expectation value in terms of the cost function. The 9 percent deviation is not claimed as a match within measurement error.

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/Higgs_Coupling3_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.

Derived articles

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