Foundation Gauge From Cube Parity Gives Hypercharge

Cube parity and hypercharge

Hypercharge is a quantum number in particle physics that helps distinguish different kinds of particles, roughly a measure of their average electric charge within a family. The declaration parity_gives_hypercharge in the Recognition Science framework's machine-checked library of formal theorems proves a small, exact fact: the order of the parity quotient of the cube's symmetry group is 2, and the model layer assigned the name "U(1) hypercharge" has a fundamental representation dimension of 1. In plain terms, the cube's symmetries that flip an odd number of axes form a two-element group, and the framework's bookkeeping labels that two-element group with the name and dimension of the hypercharge part of the Standard Model's gauge structure.

The proof itself is arithmetic. The full symmetry group of a three-dimensional cube, including sign flips of the axes, has 48 elements. The framework proves this group factors into three pieces: 6 for permuting the axes, 4 for flipping an even number of signs, and 2 for the parity quotient that distinguishes even from odd flips. The theorem parity_gives_hypercharge states that this final factor of 2 matches the assigned dimension of 1 for the hypercharge layer. The matching is a definitional choice, recorded in the structure hypercharge_layer, not a forced consequence. The framework's own documentation is explicit that the parity quotient ℤ/2ℤ is not the continuous group U(1).

What the declaration does not claim is as important as what it proves. It does not construct the Lie group U(1), a representation of it, a gauge connection, or any map from the finite quotient to a continuous gauge group. It does not prove that the Standard Model's gauge group is forced by cube symmetries. The framework's registry item P-014 remains open as a derivation of the Standard Model gauge group. The proved content is a finite group factorization: 48 = 6 × 4 × 2, with the final 2 assigned the label "U(1) hypercharge" by definition. The reader can verify the arithmetic and the group-theoretic decomposition, but the naming is a model, and the framework says so plainly.

/-- The parity quotient has order two, and the MODEL record assigns dimension
one. A finite `ℤ/2ℤ` quotient is not the continuous group `U(1)`. -/
theorem parity_gives_hypercharge :
    parity_quotient_order = 2 ∧
    hypercharge_layer.fund_rep_dim = 1 := ⟨rfl, rfl⟩

/-- **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 ⊢

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

What this page does not claim

The parity quotient ℤ/2ℤ is not the continuous group U(1). The declaration does not prove the Standard Model gauge group is forced by cube symmetries. No gauge connection or Yang-Mills action is constructed from the finite model.

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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