Physics Higgs Decay Width3 From Jcost

A template, not a prediction

The Higgs boson, discovered at CERN in 2012, is an unstable particle. It decays into other particles, and its decay width, a measure of how quickly it does so, is a key number in particle physics. The measured value is about 4.07 MeV. In the Recognition Science framework, a file named HiggsDecayWidth3_FromJCost aims to derive this width from a mathematical cost function. The framework's central object is the cost function, a discrete record of the price of recognition, which is forced by a proved theorem to have the form J(x) = (x + 1/x)/2 - 1.

What the file actually proves, however, is far more general. Its main definition, domainCost, simply applies J to the ratio of two real numbers, m and e. The three theorems it establishes are: this cost is zero when m equals e, it is never negative for positive inputs, and a certain constant involving the golden ratio is positive. These are properties of the cost function itself, not of the Higgs boson. The file's own docstring is explicit: it proves nothing specific to this subject, because the cost is defined without reference to any physical quantity. The module is a template, shared verbatim with 2383 sibling modules, each aiming at a different physical target.

The research note inside the file records the intended application. It tries J(phi) * M_H, where phi is the golden ratio and M_H is the Higgs mass, getting 14750 MeV, which is too large by a factor of about 3600. It then tries J(phi)^3 * M_H, getting 205 MeV, which is still off. The note concludes with the word "Structural," suggesting the approach does not fit. The file does not contain a derivation of the measured width, nor does it claim one. What it does contain is a machine-checked proof of general facts about a cost function, with the physics left as an open problem.

In Recognition Science, this file is an honest negative result. It shows what the framework's cost function alone can and cannot do. The theorems are real, but they are about the cost function, not about the Higgs. The gap between the template and a physical prediction is precisely the missing definition of what m and e mean for a decaying particle. Until that definition exists, the module is a placeholder, not a prediction. The lesson is that a proved theorem about a general function is not automatically a theorem about a specific particle.

structure HiggsDecayWidth3Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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 measured Higgs decay width of 4.07 MeV. The theorems proved here say nothing about the Higgs boson itself. The file does not establish that the cost function approach to particle masses is correct.

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/HiggsDecayWidth3_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.

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