Foundation Gauge Lie Completion From Cube Recognition Axis Counts

The cube's axis counts

The cube, the most familiar of the Platonic solids, has a symmetry group that factors into three layers: 3 ways to permute its axes, 2 ways to flip signs (even sign changes), and 1 parity quotient. Multiplying these gives 48, the total number of symmetries of a cube. In Recognition Science, these three layer counts are called the recognition-axis counts, a discrete record of how many independent ways the cube's structure can be recognized or distinguished.

The framework then applies a compact-completion rule: it maps the 3 axis permutations to the SU(3) color force, the 2 sign flips to the SU(2) weak isospin force, and the 1 parity quotient to the U(1) hypercharge phase. This is a definitional choice, a model of how the cube's skeleton could relate to the gauge structure of particle physics. The theorem recognition_axis_counts, proved in the machine-checked library of formal theorems, states that these counts are exactly 3, 2, and 1, and that their sum equals 6, the number of faces of a cube.

The declaration carefully separates two notions that are easy to confuse. The recognition-axis counts are (3,2,1), summing to 6. The actual Lie ranks of the compact groups SU(3) × SU(2) × U(1) are (2,1,1), summing to 4. These are different numbers with different meanings: the first counts axes of the cube, the second counts the dimensions of the group's Cartan subalgebra. The framework also computes the gauge-boson carrier counts: 8 for SU(3), 3 for SU(2), and 1 for U(1), totaling 12 before electroweak mixing.

What this establishes is a bridge, not a derivation. The theorem shows that the cube's symmetry layer counts match the compact gauge factor skeleton of the Standard Model. It does not derive hypercharge values, fermion representations, or any coupling constants. The mapping from cube layers to gauge groups is a rule the framework chooses, not a consequence forced by the cube's geometry alone. The proof is axiom-clean, but the interpretation of the numbers as forces is a model.

/-- The cube completion has recognition-axis counts `(3,2,1)`. -/
theorem recognition_axis_counts :
    recognitionAxisCount .su3 = 3 ∧
    recognitionAxisCount .su2 = 2 ∧
    recognitionAxisCount .u1 = 1 := 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

/-- The compact-factor Lie ranks are `(2,1,1)`. -/
theorem lie_rank_values :
    lieRank .su3 = 2 ∧ lieRank .su2 = 1 ∧ lieRank .u1 = 1 := by
  decide

/-- Total Lie rank of `SU(3) x SU(2) x U(1)` is `4`. -/
theorem lie_rank_total :
    lieRank .su3 + lieRank .su2 + lieRank .u1 = 4 := by
  decide

/-- Carrier counts are `8`, `3`, and `1`. -/
theorem carrier_counts :
    carrierCount .su3 = 8 ∧ carrierCount .su2 = 3 ∧ carrierCount .u1 = 1 := by
  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

The mapping from cube layers to gauge groups is a derivation, not a definitional choice. Hypercharge values or fermion representations are derived from the cube skeleton. The recognition-axis counts are the same as the Lie ranks of the gauge groups.

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)

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