Foundation Gauge Group Cube Total Gauge Rank

The cube's gauge ranks

The Standard Model of particle physics groups its forces under a combined symmetry, SU(3) × SU(2) × U(1), whose total rank is 6. Rank counts the independent directions in a symmetry group; for the Standard Model, three come from the strong force, two from the weak force, and one from electromagnetism. The Recognition Science framework derives this same (3, 2, 1) split from the geometry of a three-dimensional cube, the set {0,1}³ whose automorphism group is B₃ = (ℤ/2)³ ⋊ S₃.

In this account, the cube's three face-pair directions give the strong force rank 3, its two principal sub-cube orientations give the weak force rank 2, and its single overall phase gives the electromagnetic rank 1. The framework's machine-checked library of formal theorems states this cleanly: totalGaugeRank, a theorem in the file GaugeGroupCube.lean, proves that 3 + 2 + 1 = 6, matching the Standard Model's total gauge rank. The companion theorem rankDecomposition proves the individual values (3, 2, 1) by definition, and unique_321_partition_example shows this is the unique decreasing partition of 6 into 3 parts with first part equal to the spatial dimension 3.

The derivation is axiom-clean: the library reports 0 sorry and 0 axioms, meaning the proof rests only on the ambient type theory's standard postulates, not on framework-specific axioms. The geometric identification is the substance: the cube's face-pair count is 3, and the theorem su3_rank_eq_face_pairs equates the SU(3) rank with that count. This is a structural match, not a numerical coincidence; the framework models the gauge group ranks as consequences of the cube's symmetry.

What the declaration does not claim is equally important. totalGaugeRank establishes the rank arithmetic, not the full gauge group structure. It does not derive the coupling constants, the particle content, or the dynamics of the Standard Model. The match between the cube's geometry and the gauge ranks is a definitional identification within the framework, not an empirical measurement or a derivation of the Standard Model from first principles. The framework's library proves the arithmetic; the physical bridge from cube geometry to actual forces remains a modeling choice.

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

/-- Ranks match spatial dimension, sub-cube, and phase. -/
theorem rankDecomposition :
    gaugeRankSU3 = 3 ∧ gaugeRankSU2 = 2 ∧ gaugeRankU1 = 1 := by
  exact ⟨rfl, rfl, 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 declaration does not prove the full SU(3) × SU(2) × U(1) group structure, only the rank arithmetic. The framework does not derive the coupling constants or dynamics of the Standard Model. The cube-to-gauge identification is a modeling choice, not an empirical measurement.

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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