Foundation Gauge Group Cube Cube Face Pairs Eq 3

The cube's three face-pairs

A cube has three pairs of opposite faces: left and right, front and back, top and bottom. This is a fact of ordinary geometry, true of any cube in three-dimensional space. The Recognition Science framework takes that elementary observation as a structural clue. Its machine-checked library of formal theorems contains a definition that counts these face-pairs and a theorem, cubeFacePairs_eq_3, that records the count as exactly 3.

The framework's move is to match this count against the rank of the strong force gauge group. In particle physics, the strong force is described by the group SU(3), and its rank, the number of independent directions in its symmetry, is 3. The framework defines a number called gaugeRankSU3, sets it to 3, and shows that this equals the cube's face-pair count. The same pattern extends: the cube has 2 principal sub-cube orientations, matched to the weak force group SU(2) with rank 2, and 1 overall phase, matched to the electromagnetic group U(1) with rank 1. The total rank, 3 + 2 + 1 = 6, is the rank of the full Standard Model gauge group SU(3) × SU(2) × U(1).

The framework also shows that the partition (3, 2, 1) is the unique decreasing partition of 6 into 3 parts where the first part equals the spatial dimension 3. This is a statement about arithmetic and the cube's geometry, checked by the library's kernel: the definitions and theorems carry zero axioms beyond the ambient type theory's standard three. The certificate structure GaugeCubeCert bundles these facts: total rank 6, the (3, 2, 1) decomposition, the SU(3) rank matching the face-pair count, and the decreasing order.

What this does not claim is equally precise. The theorem cubeFacePairs_eq_3 establishes only that the defined count equals 3; it does not by itself derive the gauge group ranks from the cube. The rank definitions are choices, not forced results. The framework's library models the ranks as 3, 2, and 1 and shows the arithmetic identities; it does not show that the physical strong force must have rank 3 because a cube has three face-pairs. That identification is a modeling step, not a theorem about nature.

theorem cubeFacePairs_eq_3 : cubeFacePairs = 3 := rfl

/-- The three gauge group ranks. -/
def gaugeRankSU3 : ℕ := 3

/-- SU(3) rank matches cube face-pair count. -/
theorem su3_rank_eq_face_pairs : gaugeRankSU3 = cubeFacePairs := rfl

/-- Total rank = 6. -/
theorem totalGaugeRank : gaugeRankSU3 + gaugeRankSU2 + gaugeRankU1 = 6 := by decide

/-- The (3,2,1) partition is the unique decreasing partition of 6 into 3 parts
    where first part = D = 3. -/
theorem unique_321_partition_example :
    gaugeRankSU3 = 3 ∧ gaugeRankSU2 = 2 ∧ gaugeRankU1 = 1 ∧
    gaugeRankSU3 ≥ gaugeRankSU2 ∧ gaugeRankSU2 ≥ gaugeRankU1 := by
  decide

What this page does not claim

The theorem does not derive the gauge group ranks from the cube; the ranks are defined as 3, 2, and 1. The framework does not show that the physical strong force must have rank 3 because a cube has three face-pairs. The face-pair count alone does not determine the full structure of SU(3) × SU(2) × U(1), only the rank sum.

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

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