Physics Wboson Width3 From Jcost

A placeholder module

The W boson is a heavy elementary particle that carries the weak nuclear force. Its decay width, the spread of energies it can decay into, is a measured quantity: 2.085 GeV. A module named WBoson_Width3_FromJCost in the Recognition Science framework's machine-checked library of formal theorems was intended to derive this width from the framework's cost function, but the file itself states plainly that it proves nothing specific to the W boson.

The module defines cost, a measure of recognition effort, as Jcost(m/e), where m and e are real numbers. The three theorems it proves are generic: the cost is zero when the two inputs are equal, the cost is nonnegative for positive inputs, and the number phi minus 3/2 is positive. These facts follow from the definition of Jcost alone and would hold for any pair of positive numbers, not just for quantities related to the W boson.

The file's own documentation explains the gap. To make the module a theorem about the W boson, one would need to define m and e in terms of the particle's mass and other physical constants. The research note inside the file records an attempt: Gamma_W = J(phi)^(1/2) * M_W/pi = 8.79 GeV, which does not match the measured value. A second attempt using the standard formula Gamma_W = M_W * alpha/sin^2(theta_W) gives 2.54 GeV, closer to 2.085 GeV but still not exact. Neither calculation is proved in the Lean code.

What the module does establish, in plain language, is a small set of formal facts about the cost function itself. It shows that the cost of recognizing a quantity as equal to itself is zero, that costs never go negative, and that a particular threshold constant is positive. These are real theorems, checked by the machine, but they are about the abstract cost function, not about particle physics.

The W boson width remains a target for the framework, not a derived result. The module is an honest placeholder: it records where the derivation was meant to go and proves only the generic scaffolding that would support it. A reader looking for a prediction of the W boson's width will not find one 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]

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

What this page does not claim

This module does not derive the W boson decay width from the cost function. The numerical comparisons in the research note are not proved theorems. The framework does not yet provide a physical bridge from recognition cost to particle decay widths.

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/WBoson_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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