Standard Model Weinberg Angle Exact Rs

The Weinberg angle in RS

The Weinberg angle, usually written θ_W, is the angle that rotates the neutral electroweak fields into the photon and the Z boson. It is a measured parameter of the Standard Model, not a predicted one, and its sine squared is about 0.231 at low energies. The angle controls the relative strength of the electromagnetic and weak interactions, and it is a free input to the theory.

Recognition Science (RS) builds a framework from a single cost function J(x) = (x + 1/x)/2 - 1, which is forced by five plain conditions. The module WeinbergAngle_Exact_RS applies this cost to the ratio of two masses. Its central definition is domainCost, a cost assigned to a pair of masses, which measures how far their ratio is from unity. The module proves that this cost is zero when the two masses are equal, and nonnegative for positive masses.

The module also defines a canonical threshold at phi - 3/2, where phi is the golden ratio. This threshold is proved positive. The module assembles these facts into a certificate that the cost function behaves as expected: it vanishes at equality, stays nonnegative, and has a positive threshold. The certificate is a formal object in the machine-checked library of formal theorems.

What this establishes in plain language is a reference point: the framework's cost function, when applied to a mass ratio, has a built-in threshold that is positive and tied to the golden ratio. It does not derive the measured Weinberg angle from first principles. The module shows that the framework's cost structure is consistent and provides a formal anchor for the angle's place in the theory, but the numerical value of θ_W itself remains a measured input, not a derived output.

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)

def canonicalThreshold : ℝ := phi - 3 / 2

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not derive the numerical value of the Weinberg angle. The module does not connect the threshold to any specific particle mass ratio. The module does not prove that the measured Weinberg angle matches the framework's threshold.

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

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