Foundation Gauge Group Cube Su3 Rank Eq Face Pairs

The cube face-pair theorem

The strong nuclear force is described by the gauge group SU(3), a mathematical object whose rank is 3. The rank of a Lie group is the dimension of its maximal torus, which for SU(3) is 2, not 3. However, in the Recognition Science framework, the rank is defined differently: it is the number of independent directions in a discrete cube. The cube Q₃ = {0,1}³ has three face-pair directions: x, y, and z. Each direction has two opposite faces, giving three face-pair counts.

The theorem su3_rank_eq_face_pairs establishes that the framework's gaugeRankSU3, defined as 3, equals cubeFacePairs, also defined as 3. This is a definitional equality, proved by reflexivity (rfl) in the machine-checked library. The theorem is part of a larger structure that derives the (3, 2, 1) rank decomposition of SU(3)×SU(2)×U(1) from the cube automorphism group B₃ = (ℤ/2)³ ⋊ S₃. The cube has 3 face-pair directions giving SU(3) rank 3, 2 principal sub-cube orientations giving SU(2) rank 2, and 1 overall phase giving U(1) rank 1. The total rank is 6, matching the rank of the Standard Model gauge group.

In Recognition Science, this is presented as a derived result, not a free choice. The framework's library contains the theorem totalGaugeRank : gaugeRankSU3 + gaugeRankSU2 + gaugeRankU1 = 6, proved by decision procedure. The unique_321_partition_example theorem states that (3, 2, 1) is the unique decreasing partition of 6 into 3 parts where the first part equals the spatial dimension D = 3. The cubeFacePairs_eq_3 theorem confirms cubeFacePairs = 3 by reflexivity.

What the theorem does not claim: it does not prove that the actual physical SU(3) gauge group of the Standard Model has rank 3 in the conventional sense. The conventional rank of SU(3) is 2, and the theorem does not challenge that. It also does not derive the gauge couplings or the fine-structure constant. The theorem is a statement about definitions within the Recognition Science framework: given the definitions of gaugeRankSU3 and cubeFacePairs, the equality holds trivially. It does not establish any empirical fact about the universe.

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

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

def gaugeRankSU2 : ℕ := 2

def gaugeRankU1 : ℕ := 1

/-- 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 prove that the conventional rank of SU(3) is 3. The theorem does not derive any gauge coupling constants or the fine-structure constant. The theorem does not establish any empirical fact about the universe.

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)

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: