Foundation Gauge Lie Completion From Cube Lie Rank Total

The rank total

In the mathematics of particle physics, the Standard Model's gauge symmetries are described by the group SU(3) × SU(2) × U(1). Each factor has a Lie rank, a count of how many independent directions its continuous symmetry possesses. For SU(3) the rank is 2, for SU(2) it is 1, and for U(1) it is 1. Adding these gives 2 + 1 + 1 = 4, a fact that is elementary for anyone who knows the definitions.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration, lie_rank_total, that proves exactly this sum. The theorem states that the Lie ranks of the three factors add to 4. It is a theorem in the Lean proof assistant, meaning it has been verified by a computer from the definitions, with no gaps or unproved assumptions. The proof itself is a simple computation, but its significance lies in what it connects.

In Recognition Science, the framework models physical structure as arising from a discrete record of recognition events, a ledger. Earlier work in the framework derives, from the geometry of a three-dimensional cube, a set of axis counts: 3 for axis permutations, 2 for even sign flips, and 1 for the parity quotient. The framework's compact-completion rule maps these counts to the Standard Model factors: 3 to SU(3), 2 to SU(2), and 1 to U(1). The lie_rank_total theorem confirms that the Lie ranks of these factors, (2,1,1), sum to 4, a number distinct from the axis-count total of 6.

What this theorem does not claim is just as important as what it proves. It does not derive the full hypercharge assignments or fermion representations of the Standard Model; those remain targets for future work. It does not establish that the mapping from cube counts to gauge factors is physically necessary. The theorem only certifies an arithmetic identity about the chosen factors. It is a bridge from one skeleton to another, not a complete derivation of particle content.

The practical upshot is that the framework now has a machine-checked guarantee that its cube-derived counts are consistent with the rank structure of the Standard Model's gauge group. A reader can trust that this particular number, 4, follows from the framework's definitions without error. The larger question of whether the cube truly forces the Standard Model's full structure remains open, but this small piece is settled.

/-- 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

/-- 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

What this page does not claim

This theorem does not derive the hypercharge values or fermion representations of the Standard Model. This theorem does not prove that the mapping from cube counts to gauge factors is physically necessary. This theorem does not establish the full gauge boson content beyond the carrier counts of 8, 3, and 1.

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.

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