Foundation Smgauge Algebra Factor Count

The gauge factor count

The Standard Model of particle physics describes three of the four fundamental forces through gauge theories. Each force has its own symmetry group, and the number of independent directions in that group is called its rank. For the strong force the group is SU(3), for the weak force SU(2), and for the hypercharge force U(1). A single theorem in the machine-checked library of formal theorems, named factor_count, proves that the number of these gauge factors is exactly 3. The proof is a direct computation: the library defines a type with three elements, one for each force, and verifies that the type has cardinality 3.

The count of 3 is not an accident of the library's design. It matches a deeper structural claim in the Recognition Science framework. The framework models physical structure as a discrete record of events, called a ledger, and it derives the gauge group from the symmetries of a cube. The cube has a three-layer decomposition, and the framework's library proves that this decomposition forces the gauge group to have exactly three factors. The theorem factor_count is the final step: it takes the three factors and counts them, closing the chain from cube symmetry to the Standard Model's gauge structure.

The same module also proves the individual generator counts. SU(3) has 8 generators, SU(2) has 3, and U(1) has 1. These are the dimensions of the Lie algebras, and they sum to 12, the total number of gauge bosons in the Standard Model. The library proves this sum as well. The structural prediction is precise: any gauge group derived from the same cube-automorphism decomposition must have exactly 12 generators. A deviation from this count would falsify the identification of the gauge group with the cube's symmetry.

What the theorem does not claim is equally important. It does not prove that the Standard Model is the only possible gauge theory, nor does it derive the values of the coupling constants. It does not even prove that the cube-automorphism decomposition is the correct physical description of the gauge group; that identification is a separate claim, tagged as a structural prediction. The theorem only establishes the arithmetic fact that the three factors, once defined, number exactly 3.

/-- Number of SM gauge factors = 3, matching the cube-automorphism
three-layer decomposition. -/
theorem factor_count : Fintype.card SMGaugeFactor = 3 := by decide

/-- Number of SM gauge factors = 3, matching the cube-automorphism
three-layer decomposition. -/
theorem factor_count : Fintype.card SMGaugeFactor = 3 := by decide

theorem strong_gen_count : factorGenCount .strong = 8 := by decide

theorem weak_gen_count : factorGenCount .weak = 3 := by decide

theorem hyper_gen_count : factorGenCount .hyperY = 1 := by decide

theorem sm_total_gen_count : smTotalGenCount = 12 := by decide

What this page does not claim

The theorem does not prove that the Standard Model is the only possible gauge theory. The theorem does not derive the values of the coupling constants. The theorem does not establish that the cube-automorphism decomposition is the correct physical description of the gauge group.

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

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