Physics Weinberg Mixing Angle3 V2

Weinberg angle v2

The Weinberg angle θ_W is a number in the standard model of particle physics that describes how the weak force mixes with electromagnetism. It is usually reported through sin²(θ_W), which experiments measure at about 0.2312. The angle itself is not predicted by the standard model; it is an input measured from particle collisions. The file named WeinbergMixingAngle3_v2 in the Recognition Science framework's machine-checked library of formal theorems does not prove anything about this measured value.

The file defines a cost, a measure of how far one quantity is from another, as J(m/e), where J(x) = (x + 1/x)/2 - 1. It then proves three general facts about this cost: it is zero when the two quantities are equal, it is never negative for positive inputs, and the golden ratio minus 1.5 is positive. These are true for any positive numbers m and e. The file's own docstring states plainly that it proves nothing specific to the Weinberg angle, because the cost is defined without any reference to that physics.

The connection to the Weinberg angle lives in a research note, not in a theorem. The note compares the measured sin²(θ_W) = 0.2312 with a value built from the golden ratio φ: it computes sin² = J(φ) * 2 = 0.236, and observes this is consistent with the measured value within about 2 percent. This is a hand-written observation about numbers, not a derived result. The library file itself contains no definition of m and e in terms of weak boson masses or any other physical quantity, so it cannot make a claim about the Weinberg angle.

What the module does establish is a reusable template. It packages the three general cost facts into a certificate structure, a formal object that bundles them together, and proves that such a certificate exists. This is a small, solid piece of infrastructure: it shows that the cost function has the basic properties one would want from any measure of discrepancy. But the leap from those properties to the Weinberg angle remains entirely open. The file is an honest placeholder, a scaffold waiting for a definition of m and e that would connect it to the weak force.

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]

structure WeinbergMix3v2Cert 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

noncomputable def cert : WeinbergMix3v2Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The module derives or predicts the measured value of the Weinberg angle. The 2 percent agreement is a proved result; it is a hand-written observation in a research note. The certificate structure establishes any physical property of the weak force.

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/WeinbergMixingAngle3_v2.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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