Foundation Gauge Lie Completion From Cube Cube Order Factors As Completion

The cube's completion rule

The cube has 48 symmetries: 6 faces can be permuted in 6 ways, each face can be flipped in 2 ways, and a final parity condition halves the total. The Recognition Science declaration cube_order_factors_as_completion records a theorem about this count: it proves that 48 equals 3 times 2 times 1, multiplied by the 8 ways to permute the cube's three axes. The statement is a factorization identity, a piece of arithmetic about the cube's symmetry group, and it is proved in the framework's machine-checked library of formal theorems with no axioms beyond the standard logical ones.

The interest lies in what the factors are named. The framework's library labels the three factors as recognition-axis counts: the 3 becomes the SU(3) color force, the 2 becomes the SU(2) weak isospin, and the 1 becomes the U(1) hypercharge phase. These labels are a definitional choice, not a derivation. The theorem itself proves only the arithmetic identity; the mapping from cube counts to gauge groups is a compact-completion rule that the framework chooses to record, and the library is explicit that this is not yet the full derivation of hypercharge or fermion representations.

The declaration also keeps two separate notions apart. The recognition-axis counts (3, 2, 1) sum to 6, matching the cube's face count. The actual Lie ranks of the gauge groups are different: SU(3) has rank 2, SU(2) has rank 1, and U(1) has rank 1, summing to 4. The theorem proves these rank values too, and it proves that the carrier counts for the gauge bosons are 8, 3, and 1, summing to 12 before electroweak mixing. Each of these arithmetic facts is a proved theorem in the library, but none of them alone says that the physical world uses these groups.

What the declaration does not claim is as important as what it proves. It does not claim that the cube's symmetries cause the Standard Model's gauge groups to exist. It does not derive the hypercharge assignments or the fermion content of the Standard Model. It does not claim that the recognition-axis count is the same as the Lie rank; in fact it proves they differ. The theorem is a bridge, not a destination: it shows that a certain arithmetic structure in the cube's symmetry group can be labeled by the names of the Standard Model's compact gauge factors, and it leaves the physical interpretation of that labeling as an open target.

/-- The `B_3` order factorization already proved in `GaugeFromCube`. -/
theorem cube_order_factors_as_completion :
    Fintype.card (SignedPerm 3) =
      axis_perm_count 3 * even_sign_flip_count 3 * parity_quotient_order := by
  exact three_layer_factorization

/-- 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 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

The declaration does not derive the Standard Model gauge groups from the cube's symmetries; it only records an arithmetic factorization and a labeling choice. The declaration does not prove that the recognition-axis count equals the Lie rank; it proves they differ. The declaration does not derive hypercharge or fermion representations, which the library marks as not yet complete.

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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