Foundation Gauge Lie Completion From Cube Recognition Axis Total

The axis total

A cube is the everyday object with six faces. Recognition Science, a framework that derives physical structure from a forced accounting of recognition events, has a formal library of machine-checked theorems. One theorem in that library, called recognition_axis_total, states a simple arithmetic fact: the three axis counts 3, 2, and 1 add to 6, which equals the number of faces on a three-dimensional cube.

Those axis counts come from a completion rule. The framework's cube work proves a forced layer structure with three kinds of counts: axis permutations (3), even sign-flip completions (2), and a parity quotient (1). The completion rule sends these counts to the compact gauge factors of the Standard Model: 3 maps to SU(3) color, 2 maps to SU(2) weak isospin, and 1 maps to U(1) hypercharge phase. The theorem recognition_axis_total is the bridge that connects the cube layer skeleton to this compact gauge-factor skeleton.

The theorem keeps two separate notions apart. The recognition-axis count is (3,2,1) with total 6. The Lie rank, a different measure of each factor's internal structure, is (2,1,1) with total 4. The carrier counts, which count gauge bosons in the adjoint or phase sector, are 8, 3, and 1 with total 12. These three totals answer different questions and should not be confused.

What recognition_axis_total does not claim is just as important. It is not the full derivation of hypercharge or fermion representations. It is the first clean bridge theorem from the cube layer skeleton to the compact gauge-factor skeleton, nothing more. The theorem is machine-checked with zero axioms beyond the standard three, but it establishes a numerical identity, not a physical derivation of the Standard Model's forces.

/-- 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

/-- Recognition-axis count carried by each factor. -/
def recognitionAxisCount : CompactGaugeFactor -> ℕ
  | .su3 => 3
  | .su2 => 2
  | .u1 => 1

/-- 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

What this page does not claim

This theorem does not derive the physical forces of the Standard Model from first principles. This theorem does not establish hypercharge values or fermion representations. This theorem does not claim the axis total 6 has any meaning beyond matching the cube face count.

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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