Foundation Gauge Group Cube Unique 321 Partition Example

The cube's gauge ranks

The Standard Model of particle physics organizes its forces under a symmetry group written SU(3) × SU(2) × U(1). Each factor carries a rank, a whole number counting its independent directions: 3 for the strong force, 2 for the weak force, and 1 for electromagnetism. Together they sum to 6. The Recognition Science framework shows that this specific pattern, the decreasing triple (3, 2, 1), is the unique way to split 6 into three decreasing parts when the first part is fixed at 3. That uniqueness is what the declaration unique_321_partition_example establishes.

The framework reaches this pattern from the geometry of a three-dimensional cube. A cube has three pairs of opposite faces, which gives the number 3. It has two principal ways to slice into smaller sub-cubes, giving 2. And it has one overall phase, a global rotation of the whole object, giving 1. The machine-checked library of formal theorems proves that these three counts are exactly the ranks of the three gauge groups, and that no other decreasing triple of positive integers summing to 6 with first part 3 exists. The proof is a direct computation, checked without gaps or extra assumptions.

What the declaration does not claim is just as important. It does not derive the actual forces or particles of the Standard Model, and it does not predict any coupling constants or masses. The match is structural: the cube's combinatorial counts equal the group ranks. Whether this geometry physically causes the gauge structure, or is merely a striking coincidence, remains an open question. The framework itself treats the connection as a model, a definitional choice, not as a proven physical law.

The practical upshot is a clean numerical fingerprint. Any theory that wants to explain the Standard Model's gauge group from geometry must reproduce this exact partition. The cube provides it, and the proof certifies that no smaller or larger arrangement of the same kind would fit. That is the contribution: a precise, checkable constraint, not a complete physical explanation.

/-- Ranks match spatial dimension, sub-cube, and phase. -/
theorem rankDecomposition :
    gaugeRankSU3 = 3 ∧ gaugeRankSU2 = 2 ∧ gaugeRankU1 = 1 := by
  exact ⟨rfl, rfl, rfl⟩

/-- 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 declaration does not derive the forces or particles of the Standard Model. It does not predict any coupling constants or masses. The physical connection between cube geometry and gauge structure is a model, not a proven law.

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.

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