Foundation Weinberg Angle Rs5

The Weinberg angle in RS

The weak mixing angle, also called the Weinberg angle, is a parameter in the Standard Model of particle physics. It describes how the weak force and electromagnetism mix, and its sine-squared value is measured at about 0.2312 at the Z boson mass scale. The module named Weinberg_Angle_RS5 in the Recognition Science library does not derive this number. Its own docstring states plainly that the Lean code proves only general facts about a cost function, and that the specific physics content is absent.

The module defines a quantity called domainCost, a discrete recognition cost, as the function J applied to the ratio of two real numbers m and e. The three theorems it proves are: this cost is zero when m equals e, it is nonnegative for positive inputs, and the constant phi minus 3/2 is positive. These are general properties of the cost function, true for any positive real numbers, not facts about the Weinberg angle. The module itself says it is shared verbatim with 2383 sibling modules, and that a real theorem about its subject would need a definition of m and e in that subject's own terms.

The research note in the module records the intended idea: the ratio 1 minus M_W squared over M_Z squared, which equals 1 minus phi to the 24 over phi to the 25, or about 0.382. The measured value is about 0.2312, so the simple expression misses by roughly 65 percent. The note suggests a structural interpretation at grand unification scale, but this is a research note, not a proved result. The Lean file proves no statement connecting the cost function to the Weinberg angle.

What the module does establish, in plain language, is a small set of formal facts about a cost function and a threshold constant. It proves that the cost vanishes at equality, that it never goes negative for positive inputs, and that a certain golden-ratio-derived threshold is positive. These are real theorems, machine-checked, but they are about the cost function, not about particle physics. The module is an honest placeholder: it records where an idea was meant to go, and it proves only what the mathematics actually supports.

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 derives the measured value of the Weinberg angle. The cost function theorems are specific to electroweak physics. The research note's structural interpretation is a proved result.

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/Foundation/Weinberg_Angle_RS5.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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