Physics Zboson Width3 From Jcost

The Z boson width

The Z boson is a heavy particle that carries the weak nuclear force. Its decay width, written Γ_Z, measures how quickly it falls apart into lighter particles; the measured value is 2.4952 GeV. A GeV is a unit of energy, and here it tells you the particle's lifetime: a larger width means a shorter life. The standard model of particle physics predicts this width from the Z boson's mass, its coupling to other particles, and the weak mixing angle.

In Recognition Science, the framework models the Z boson's width using its own cost function. The cost function, written J(x), is a mathematical measure of how far a ratio x is from 1; it equals (x + 1/x)/2 - 1, and it is zero when x equals 1. The module defines a domain cost as J(m/e), where m and e are two masses or energies. The framework's calculation gives Γ_Z = M_Z * alpha / (phi * sin^2(theta_W)) = 91.2 * 0.0073 / (1.618 * 0.231) = 1.78 GeV. This is close to the measured 2.4952 GeV, but not equal.

What the module actually proves is much more modest. The machine-checked library of formal theorems proves three general facts about the cost function: it vanishes when the two inputs are equal, it is never negative for positive inputs, and the constant phi - 3/2 is positive. These facts are true for any positive numbers m and e. The module does not prove that the Z boson width equals 1.78 GeV, because the definition of m and e in terms of Z boson physics is missing. The calculation in the research note is a target, not a theorem.

The gap matters. The framework's cost function has a unique shape, forced by five plain conditions, and it appears in many contexts. But applying it to a specific particle requires a separate step: defining what m and e mean for that particle. Until that definition exists, the Z boson width remains an open problem in the framework. The module is a template, shared verbatim with 2383 sibling modules, showing how a proof would look once the physics is added.

noncomputable def cert : ZWidth3Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

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 framework proves the Z boson width equals 1.78 GeV. The module contains a definition of m and e in terms of Z boson physics. The close numerical match is a theorem rather than a research note.

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/ZBoson_Width3_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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