Foundation Gauge Group Cube

The gauge cube

The gauge group of the Standard Model of particle physics, the group whose symmetries govern the strong, weak, and electromagnetic forces, is the product SU(3) × SU(2) × U(1). Each factor has a rank, the number of independent directions in its symmetry space: SU(3) has rank 3, SU(2) has rank 2, and U(1) has rank 1. Their sum, 6, is the total rank of the Standard Model's gauge group. This much is standard physics, taught in every graduate course.

The recognition framework, a system that derives physical structure from a discrete record of events, offers a strikingly concrete picture of where those numbers come from. Consider an ordinary three-dimensional cube, the set of eight corner points {0,1}³. The cube has three pairs of opposite faces, one pair for each spatial direction. It has two principal sub-cube orientations, the ways to slice it into smaller cubes. And it has one overall phase, a global rotation or reflection. Count them: 3 face-pairs, 2 orientations, 1 phase. The framework identifies these three counts with the ranks of SU(3), SU(2), and U(1), respectively.

In the framework's machine-checked library of formal theorems, this identification is not a metaphor but a definition. The module defines three numbers: gaugeRankSU3 = 3, gaugeRankSU2 = 2, and gaugeRankU1 = 1. It then proves, by direct computation, that their sum is 6, that the triple (3, 2, 1) is decreasing, and that the SU(3) rank equals the cube's face-pair count. The theorem states plainly: gaugeRankSU3 + gaugeRankSU2 + gaugeRankU1 = 6. The proof is a single line, by decide, meaning the computer checks it by evaluating the definitions.

The framework also asserts that (3, 2, 1) is the unique decreasing partition of 6 into three parts where the first part equals the spatial dimension 3. This is a combinatorial fact about the number 6, not a physical measurement. The module packages all these facts into a single structure called GaugeCubeCert, a certificate that the cube's geometry and the gauge group's ranks align. The certificate is built from the theorems, and the whole file compiles with zero axioms beyond the kernel's standard three and zero unfinished proofs.

What changes if this picture holds? The Standard Model's gauge group, usually taken as an input chosen to fit data, becomes a consequence of living in three spatial dimensions. The cube's face-pairs, sub-cube orientations, and phase are not arbitrary; they are the only counts a three-dimensional cube can offer. The framework's claim is that these counts are exactly the ranks the Standard Model needs. Whether this identification is physically forced, or merely a suggestive coincidence, remains a question the framework itself does not settle.

/-- 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

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

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

What this page does not claim

The module does not derive the values of the gauge couplings, only the ranks of the symmetry groups. The identification of cube face-pairs with SU(3) rank is a definitional choice, not a theorem about physics. The framework does not claim that the cube is the physical cause of the gauge group, only that the counts match.

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

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