Physics Feynman Diagrams From Rs

Feynman diagrams in the framework

Feynman diagrams are the standard pictorial shorthand for particle physics. Each diagram shows particles arriving, meeting at a vertex, and leaving, and the whole picture stands for one term in a perturbative expansion of the S-matrix, the object that turns an initial set of particles into a final set. In the Standard Model, the vertices come in a small number of canonical types. The strong force contributes a three-gluon vertex and a four-gluon vertex; the weak force contributes a W-fermion vertex; the Higgs mechanism contributes a Higgs-fermion vertex; and the strong force also contributes a quark-gluon vertex. That list contains five vertex types in total.

The reason the gluon vertices exist at all is that the strong force's symmetry group, SU(3), is non-Abelian: its transformations do not commute. Richard Feynman introduced the diagrams in 1948 as a bookkeeping device for quantum electrodynamics, and the non-Abelian case, quantum chromodynamics, was developed in the early 1970s. A non-Abelian gauge group means the force carriers themselves carry the charge, so they interact with each other, which is exactly what the three-gluon and four-gluon vertices depict. In a commuting, Abelian group like the one for electromagnetism, the photon does not couple to itself, and those two vertices would be absent.

In Recognition Science, the framework models each vertex as a cost coupling event in a discrete ledger, a record of recognition events where the cost of recognition is forced by a proved functional equation. The framework's machine-checked library of formal theorems contains a module, FeynmanDiagramsFromRS, that certifies three facts about this picture. First, it proves that the number of canonical Standard Model vertex types is exactly five. Second, it defines the rank of SU(3) as 3 and proves that this equals the framework's spatial dimension D, also 3. Third, it establishes that the number of non-Abelian vertices, the three-gluon and four-gluon ones, is 2.

The theorem vertexTypeCount proves the five-vertex count by direct computation. The theorem su3Rank_eq_D states the equality of the SU(3) rank with the dimension 3 by definitional reflexivity. The theorem totalVertices proves that the 2 non-Abelian vertices plus the 3 remaining vertices sum to 5. These three results are packaged into a structure called FeynmanCert, and a single definition, feynmanCert, assembles the proofs. The verification status is clean: 0 sorry, 0 axiom.

What the module does not do is derive the Feynman rules from first principles or compute any scattering amplitude. It takes the Standard Model's vertex inventory as given and certifies that the count is 5, that the SU(3) rank equals the framework's dimension, and that the non-Abelian vertex count is 2. The physical bridge from the recognition ledger to the actual Feynman rules remains an open target. The value of the module is narrower and precise: it shows that a specific combinatorial fact about the Standard Model, the number of vertex types, is consistent with the framework's forced dimension, and it does so with machine-checked proofs.

theorem vertexTypeCount : Fintype.card VertexType = 5 := by decide

theorem su3Rank_eq_D : su3Rank = 3 := rfl

/-- Non-Abelian vertex count = 2 (3-gluon, 4-gluon). -/
def nonAbelianVertices : ℕ := 2

def feynmanCert : FeynmanCert where
  five_vertices := vertexTypeCount
  su3_rank_D := su3Rank_eq_D
  non_abelian := rfl

What this page does not claim

The module does not derive the Feynman rules from first principles. The module does not compute any scattering amplitude or cross-section. The physical bridge from the recognition ledger to Feynman diagrams is not established by this module.

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/FeynmanDiagramsFromRS.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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