Foundation Weinberg Angle Rs5 Weinberg Angle5 Cert

What the certificate proves

The Weinberg angle is a parameter in the standard model of particle physics that describes how the weak force mixes with the electromagnetic force. Its measured value, often quoted as sin²(θ_W) ≈ 0.2312, is one of the key numbers that any theory of particle physics must eventually explain. The Recognition Science framework contains a declaration called WeinbergAngle5Cert, which is a machine-checked certificate, a formal object that bundles together three proved facts about a specific cost function.

The certificate is built from the framework's cost function, denoted Jcost, which measures the recognition cost between two positive numbers. The first fact it proves is that when the two numbers are equal, the cost is zero. The second is that for any two positive numbers, the cost is never negative. The third is a small technical statement that the golden ratio minus 1.5 is positive. These three facts are proved in the machine-checked library of formal theorems, and the certificate simply packages them together. The declaration domainCost defines the cost as Jcost applied to the ratio of two masses, m divided by e.

What the certificate does not do is connect these facts to the Weinberg angle. The documentation states that it proves nothing specific to this subject, because the cost function is defined without any reference to the weak force or to the W and Z bosons. A research note records where the idea was meant to go: a naive comparison of the framework's mass ratio to the Weinberg angle. The note shows the framework's value for 1 - M_W²/M_Z² would be 1 - φ⁻¹ ≈ 0.382, which is off from the measured 0.2312 by about 65 percent. The note suggests this could be a GUT-scale value that runs down to the observed value, but that is a research idea, not a proved result.

In plain terms, the certificate is a small, correct piece of mathematics about a cost function. It proves that the cost function behaves sensibly: zero at equality, nonnegative everywhere. It does not prove anything about the Weinberg angle, and it does not provide a derivation of that angle from the framework. The certificate is honest about its own scope, and that honesty is what makes it a useful building block rather than an overclaim.

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]

What this page does not claim

The certificate does not derive the Weinberg angle from the framework. The certificate does not prove that the framework's value of 0.382 runs to the measured 0.2312. The certificate does not establish any physical meaning for the ratio m/e.

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)

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.

Derived articles

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