Foundation Gauge Group Cube Gauge Cube Cert

The cube certificate

The standard model of particle physics groups its forces under three symmetries: SU(3) for the strong force, SU(2) for the weak force, and U(1) for electromagnetism. Each symmetry has a rank, a count of independent directions it can rotate. SU(3) has rank 3, SU(2) has rank 2, and U(1) has rank 1. Their total rank is 6.

Recognition Science, a framework that derives physical structure from the cost of storing a discrete record of events, offers a geometric reading of those numbers. Its machine-checked library of formal theorems contains a declaration called GaugeCubeCert, a certificate that packages four facts about the ranks. The certificate states that the three ranks sum to 6, that they are exactly 3, 2, and 1, that the SU(3) rank equals the number of face-pair directions of a three-dimensional cube, and that the ranks form a decreasing sequence. A cube has three pairs of opposite faces, which gives the 3; its two principal sub-cube orientations give the 2; and a single overall phase gives the 1.

The certificate does not derive the existence of the gauge groups or their interactions. It takes the ranks as defined constants and proves the arithmetic and geometric identities among them. The uniqueness claim it carries is narrow: (3, 2, 1) is the only decreasing partition of 6 into three parts whose first part equals the spatial dimension 3. That is a statement about integer partitions, not about the physical content of the strong, weak, and electromagnetic forces.

What the certificate establishes is consistency. It shows that the rank structure of the standard model's gauge groups can be read off the symmetries of a cube, within the framework's own definitions. It does not predict the existence of those forces, nor does it explain why the gauge group is SU(3) × SU(2) × U(1) rather than some other group with the same ranks. The certificate is a formal bookkeeping check, not a physical derivation.

structure GaugeCubeCert where
  total_rank : gaugeRankSU3 + gaugeRankSU2 + gaugeRankU1 = 6
  decomp : gaugeRankSU3 = 3 ∧ gaugeRankSU2 = 2 ∧ gaugeRankU1 = 1
  su3_from_cube : gaugeRankSU3 = cubeFacePairs
  decreasing_partition : gaugeRankSU3 ≥ gaugeRankSU2 ∧ gaugeRankSU2 ≥ gaugeRankU1

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

/-- 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 certificate does not derive the existence of the gauge groups or their interactions. It does not predict the gauge group SU(3) × SU(2) × U(1) as opposed to any other group with the same ranks. The certificate is not a physical derivation of the strong, weak, or electromagnetic forces.

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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