Physics Top Quark Width3 From Jcost

What the module proves

The top quark is the heaviest known elementary particle, with a measured mass near 173 GeV. Its decay width, the quantum-mechanical spread in its mass that reflects how quickly it decays, is measured at about 1.42 GeV. The module named TopQuarkWidth3_FromJCost in the framework's machine-checked library of formal theorems was written with the ambition of deriving that width from the framework's cost function, a forced measure of recognition cost. The file's own research note records the attempt: a formula using the golden ratio gives 20.4 GeV, off by a factor of 14, and a later guess gives 2.4 GeV, closer but still not the measured value.

What the Lean code actually proves is far more modest. It defines domainCost m e as the cost function applied to the ratio m / e, and then proves three general facts about that definition. First, the cost vanishes when the two inputs are equal. Second, the cost is never negative for positive inputs. Third, the number phi minus 3/2 is positive. These three facts are assembled into a structure called TopWidth3Cert, and the file proves that this structure is inhabited, meaning the three facts are consistent with each other.

None of these theorems mention the top quark, its width, or any physics specific to it. The definition of domainCost does not reference the top quark's mass or its decay products. The module is one of 2383 sibling files that share the same body, differing only in their names. The file's own docstring states this plainly: it proves nothing specific to this subject, because the cost function is defined without reference to one. What would make the module a theorem about the top quark is a definition of m and e in the top quark's own terms, and that definition is absent.

In Recognition Science, the cost function J is a proved consequence of five plain conditions, and its value at the golden ratio is a fixed number near 0.118. The framework's library does contain a chain of theorems forcing the golden ratio, an eight-tick cycle, and three spatial dimensions. But this particular module does not extend that chain to particle physics. It establishes only the three general properties of the cost function, and the research note candidly records that the top quark width derivation remains unfinished. The honest summary is that the module proves a template, not a physical result.

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 derives or predicts the measured top quark width of 1.42 GeV. The framework has a proved connection between the cost function and any specific particle's decay width. The research note's numerical guesses (20.4 GeV, 2.4 GeV) are theorems or validated predictions.

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/TopQuarkWidth3_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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