Foundation Gauge From Cube Unique Gauge Factorization

The cube's unique arithmetic

A cube has 48 symmetries: 6 ways to permute its three axes, 8 ways to flip signs on coordinates, and their combinations. The number 48 factors as 6 × 4 × 2, and the framework's declaration unique_gauge_factorization proves this is the only way to split 48 into those three prescribed parts. The theorem states: if three natural numbers multiply to 48, the first equals 3 factorial (which is 6), the second is a power of 2 with exponent 2 (which is 4), and the third is 2, then the triple must be exactly (6, 4, 2).

The framework then attaches model labels to these factors: 6 for SU(3) color, 4 for SU(2) weak, and 2 for U(1) hypercharge. This is a definitional choice, not a derivation. The declaration gauge_rank_match verifies that the assigned dimensions are indeed 3, 2, and 1, and dimension_sum checks that 3 + 2 + 1 equals the cube's face count of 6. These are arithmetic facts about the chosen labels, not statements about physical gauge groups.

The framework's library explicitly states what this result does not do. It does not construct the Lie groups SU(3), SU(2), or U(1), nor any representation, gauge connection, Maxwell action, or Yang–Mills action. The parity quotient ℤ/2ℤ is explicitly not U(1). Matching the finite factors to Standard Model names is an identification, not a theorem that the Standard Model gauge group is forced. The registry item P-014 remains OPEN as a derivation of the Standard Model gauge group.

What the declaration genuinely establishes is a uniqueness result in finite arithmetic. Given the constraints that one factor is 3 factorial, another is a power of 2 with exponent plus one equal to 3, and the third is 2, the factorization 48 = 6 × 4 × 2 is the only possibility. This is a proved theorem in the machine-checked library of formal theorems. The framework's contribution is to observe that this unique arithmetic split mirrors the rank structure of the Standard Model's gauge groups, but the bridge from finite combinatorics to continuous physics remains a labeled model, not a proved consequence.

/-- **THEOREM (Unique Factorization under fixed hypotheses)**:
    The ONLY way to decompose 48 = |B₃| as an ordered product a × b × c
    where `a = 3!`, `b = 2^k` with `k+1=3`, and `c=2` is `6 × 4 × 2`.
    The hypotheses already prescribe the factor shapes; no gauge group follows. -/
theorem unique_gauge_factorization :
    ∀ a b c : ℕ,
      a * b * c = 48 →
      a = Nat.factorial 3 →
      (∃ k, b = 2 ^ k ∧ k + 1 = 3) →
      c = 2 →
      a = 6 ∧ b = 4 ∧ c = 2 := by
  intro a b c habc ha hb hc
  subst ha; subst hc
  obtain ⟨k, hbk, hk3⟩ := hb
  have hk : k = 2 := by omega
  subst hk
  simp at hbk
  subst hbk
  norm_num at habc ⊢

/-- 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 }

/-- The assigned record fields are `(3,2,1)` by definition. -/
theorem gauge_rank_match :
    color_layer.fund_rep_dim = 3 ∧
    weak_layer.fund_rep_dim = 2 ∧
    hypercharge_layer.fund_rep_dim = 1 := ⟨rfl, rfl, rfl⟩

/-- The assigned integers satisfy `3 + 2 + 1 = 6`, the closed-form face count
at `D = 3`. No physical decomposition of faces is constructed. -/
theorem dimension_sum :
    color_layer.fund_rep_dim + weak_layer.fund_rep_dim + hypercharge_layer.fund_rep_dim
    = cube_face_count 3 := by
  native_decide

What this page does not claim

The declaration does not construct the Lie groups SU(3), SU(2), or U(1). The parity quotient ℤ/2ℤ is not U(1). The Standard Model gauge group is not forced by this theorem.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: