Foundation Gauge From Cube Color From Axis Permutations

The cube's color count

A cube has six faces. You can also rearrange the cube by swapping its three axes, for instance exchanging the roles of up and forward. There are exactly six ways to do this, matching the number of permutations of three objects. The result color_from_axis_permutations in the framework's machine-checked library records that this count, six, is also the number of colors the framework assigns to the strong force. The framework calls this the color layer, and its fundamental representation has dimension three, echoing the three axes.

The result is a piece of finite arithmetic. The library defines an axis permutation as a signed permutation that leaves all signs unchanged, and it proves that in three dimensions the count of such permutations is six. It then defines a color layer with a fundamental representation dimension of three, and the result simply states that the axis permutation count equals the factorial of the number of colors. The proof is a direct computation, not a physical derivation.

In Recognition Science, this is part of a larger model. The full symmetry group of the cube has 48 elements, which the framework factors as 6 times 4 times 2. It labels these factors with the names SU(3), SU(2), and U(1), matching the structure of the Standard Model of particle physics. The framework proves the arithmetic: the cube has 48 symmetries, the axis permutations give 6, the even sign flips give 4, and the parity quotient gives 2, and these numbers sum to the six faces of the cube.

What the declaration does not claim is just as important as what it proves. It does not construct the Lie group SU(3), nor does it build a gauge connection or a Yang-Mills action. The parity quotient of order two is explicitly not the group U(1). The assignment of the names SU(3), SU(2), and U(1) to the finite factors is an identification, a modeling choice, not a proof that the Standard Model gauge group is forced by the cube. The derivation of the actual Standard Model gauge group remains an open target.

/-- Equality between two closed-form counts. It does not construct a color
representation. -/
theorem color_from_axis_permutations :
    axis_perm_count 3 = Nat.factorial (N_colors 3) := by
  simp [axis_perm_count, N_colors, face_pairs]

/-- **THEOREM**: |Aut(Q₃)| = |B₃| = 48.
    The automorphism group of the 3-cube has order 48. -/
theorem cube_aut_order : Fintype.card (SignedPerm 3) = 48 := by
  rw [signed_perm_card]
  norm_num

/-- The factorization in terms of the Standard Model structure. -/
theorem sm_factorization :
    (48 : ℕ) = 6 * 4 * 2 := by norm_num

/-- The three MODEL-labeled records. The names and dimensions are definitions,
not consequences of the finite groups. -/
def color_layer : GaugeLayer :=
  { name := "SU(3) color"
    fund_rep_dim := 3
    discrete_order := 6 }

def weak_layer : GaugeLayer :=
  { name := "SU(2) weak"
    fund_rep_dim := 2
    discrete_order := 4 }

def hypercharge_layer : GaugeLayer :=
  { name := "U(1) hypercharge"
    fund_rep_dim := 1
    discrete_order := 2 }

What this page does not claim

The result does not construct the Lie group SU(3) or any continuous gauge group. The parity quotient of order two is not claimed to be the group U(1). The framework does not claim to have derived the Standard Model gauge group from the cube.

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/GaugeFromCube.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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