Foundation Gauge Lie Completion From Cube

The gauge bridge

The Standard Model of particle physics groups its forces by symmetry. The strong force uses the group SU(3), the weak force uses SU(2), and the electromagnetic force uses U(1). These three groups are not arbitrary; their sizes and structures follow from the symmetries they describe. The module foundation gauge lie completion from cube in the Recognition Science framework shows that these same three groups emerge from counting the symmetries of a three-dimensional cube.

A cube has a natural set of recognition axes, the framework's term for the independent directions along which a discrete record of events can be organized. Counting these axes for a cube gives three axis permutations, two even sign-flip completions, and one parity quotient. The module defines a completion rule that maps these counts to the Standard Model factors: the 3 becomes SU(3) color, the 2 becomes SU(2) weak isospin, and the 1 becomes U(1) hypercharge. This is a definitional choice, a model, not a derivation of the forces themselves.

The module keeps two distinct notions separate. The recognition-axis count is (3, 2, 1), totaling 6, which matches the cube's face count. The Lie rank, a classical measure of a group's dimension, is (2, 1, 1), totaling 4. The module proves these values with a machine-checked library of formal theorems: the axis counts, the Lie ranks, and the carrier counts of 8, 3, and 1 for the gauge bosons, summing to 12 before electroweak mixing. These are exact arithmetic facts, verified by the library's kernel.

This is not yet the full derivation of hypercharge or fermion representations. The module is the first clean bridge from the cube's layer skeleton to the compact gauge-factor skeleton. It establishes that the cube's symmetry structure, when completed by this rule, carries the same numerical skeleton as the Standard Model's force groups. The physical bridge from recognition to linking, which would connect this mathematical structure to actual spacetime, remains an open target.

/-- The cube completion has recognition-axis counts `(3,2,1)`. -/
theorem recognition_axis_counts :
    recognitionAxisCount .su3 = 3 ∧
    recognitionAxisCount .su2 = 2 ∧
    recognitionAxisCount .u1 = 1 := by
  decide

/-- Recognition-axis total is `3 + 2 + 1 = 6`, matching the cube face count. -/
theorem recognition_axis_total :
    recognitionAxisCount .su3 + recognitionAxisCount .su2 + recognitionAxisCount .u1 =
      cube_face_count 3 := by
  rw [cube3_face_count]
  decide

/-- The compact-factor Lie ranks are `(2,1,1)`. -/
theorem lie_rank_values :
    lieRank .su3 = 2 ∧ lieRank .su2 = 1 ∧ lieRank .u1 = 1 := by
  decide

/-- Total Lie rank of `SU(3) x SU(2) x U(1)` is `4`. -/
theorem lie_rank_total :
    lieRank .su3 + lieRank .su2 + lieRank .u1 = 4 := by
  decide

/-- Carrier counts are `8`, `3`, and `1`. -/
theorem carrier_counts :
    carrierCount .su3 = 8 ∧ carrierCount .su2 = 3 ∧ carrierCount .u1 = 1 := by
  decide

/-- Total gauge carriers before electroweak mixing: `8 + 3 + 1 = 12`. -/
theorem carrier_total :
    carrierCount .su3 + carrierCount .su2 + carrierCount .u1 = 12 := by
  decide

What this page does not claim

This module does not derive the hypercharge values or fermion representations. The mapping from cube counts to gauge groups is a definitional model, not a forced derivation. The physical recognition-to-linking bridge that would connect this to spacetime is not established here.

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/GaugeLieCompletionFromCube.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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