Foundation Gauge Lie Completion From Cube Compact Gauge Factor Count

The compact gauge factors

The Standard Model of particle physics groups its forces under three compact gauge symmetries: SU(3) for the strong force, SU(2) for the weak force, and U(1) for hypercharge. Each is a continuous symmetry group whose structure is described by a Lie algebra. The rank of a Lie group is the dimension of its maximal torus, the largest abelian subgroup; for SU(3) the rank is 2, for SU(2) it is 1, and for U(1) it is 1, giving a total rank of 4.

In Recognition Science, the framework's machine-checked library of formal theorems contains a declaration, compactGaugeFactor_count, that establishes a simple but specific fact: the type CompactGaugeFactor, which has exactly three constructors su3, su2, and u1, has cardinality 3. This is a theorem proved by the decide tactic, meaning it is checked by computation and contains no axioms or unfinished proofs. The declaration is part of a module that connects the symmetry of a three-dimensional cube to these gauge groups.

The cube's symmetry group, the signed permutation group on three elements, has an order that factors into three layers: 3 axis permutations, 2 even sign flips, and 1 parity quotient. The framework's completion rule maps these three recognition-axis counts to the three compact factors: 3 to SU(3), 2 to SU(2), and 1 to U(1). The theorem recognition_axis_total shows that the sum of the recognition-axis counts, 3 + 2 + 1 = 6, equals the number of faces of a cube. The total number of gauge bosons, before electroweak mixing, is the sum of the adjoint representation dimensions plus one for the abelian factor: 8 + 3 + 1 = 12, as proved by carrier_total.

What the declaration does not claim is as important as what it proves. compactGaugeFactor_count is a statement about a finite type with three elements; it does not derive the hypercharge assignments of fermions, nor does it specify the electroweak mixing angle. The module is explicitly described as a first bridge theorem, not a full derivation of the Standard Model's representation content. The mapping from cube symmetries to gauge groups is a definitional choice, a completion rule, not a forced consequence of the cube's geometry alone.

The practical payoff is a clean, machine-checked bridge between a discrete combinatorial object, the cube, and the continuous Lie groups of the Standard Model. The framework proves the counts align: three factors, six total recognition axes, four total Lie rank, twelve gauge carriers. This is a structural coincidence made precise, but it is not a physical derivation of the Standard Model's particle content.

theorem compactGaugeFactor_count : Fintype.card CompactGaugeFactor = 3 := by
  decide

/-- Recognition-axis total is `3 + 2 + 1 = 6`, matching the cube face count. -/
theorem recognition_axis_total :
    recognitionAxisCount .su3 + recognitionAxisCount .su2 + recognitionAxisCount .u1 =
      cube_face_count 3 := by
  rw [cube3_face_count]
  decide

/-- Total gauge carriers before electroweak mixing: `8 + 3 + 1 = 12`. -/
theorem carrier_total :
    carrierCount .su3 + carrierCount .su2 + carrierCount .u1 = 12 := by
  decide

What this page does not claim

This declaration does not derive the hypercharge or fermion representations of the Standard Model. The mapping from cube symmetries to gauge groups is a definitional completion rule, not a forced consequence of cube geometry alone. The total gauge carrier count of 12 does not include the electroweak mixing angle or the Higgs mechanism.

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/GaugeLieCompletionFromCube.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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