Foundation Gauge From Cube

Foundation Gauge from Cube

Foundation gauge from cube is a module in Recognition Science that studies the signed permutations of a three-dimensional cube. A signed permutation is a way to rearrange the cube's three axes and optionally flip the sign of each axis. The module proves that the full set of these operations has exactly 48 elements, a finite group known as the hyperoctahedral group B₃.

The central established fact is the factorization of this order: 48 = 6 × 4 × 2. The factor 6 counts the permutations of the three axes, the factor 4 counts the even sign flips, and the factor 2 is the order of the parity quotient. A theorem in the module, unique_gauge_factorization, shows this decomposition is the only one satisfying the stated conditions. These are finite arithmetic facts, established in the kernel-checked library with no additional axioms.

The module then packages these three factors with the MODEL labels SU(3), SU(2), and U(1), assigning fundamental representation dimensions 3, 2, and 1 respectively. This assignment is a definitional choice, not a theorem. The module explicitly does not construct any Lie group, representation, gauge connection, or action. The parity quotient of order 2 is not U(1); it is merely assigned the same label. The matching of finite factors to Standard Model names is an identification, not a proof that the Standard Model gauge group is forced.

What the module establishes in plain language is a precise structural analogy: the symmetry group of the cube has a layered decomposition whose sizes match the ranks of the Standard Model gauge groups. The arithmetic consequences, such as the sum of the assigned dimensions equaling the number of cube faces (3 + 2 + 1 = 6), are established. The derivation of the Standard Model gauge group itself remains OPEN.

/-- **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

/-- **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 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

This module does not construct any Lie group, representation, gauge connection, or Yang-Mills action. The parity quotient of order 2 is not claimed to be the group U(1). The derivation of the Standard Model gauge group is not claimed; it remains OPEN.

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

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