Foundation Gauge From Cube Even Flips Give Weak Structure

The cube's even flips

A cube has 48 symmetries: rotations, reflections, and combinations of the two. These symmetries can be counted by first permuting the three axes (6 ways), then independently flipping the sign along each axis (2 × 2 × 2 = 8 ways), giving 6 × 8 = 48. The framework's machine-checked library of formal theorems proves this count exactly, and it proves a finer decomposition: the 8 sign flips split into 4 even flips (an even number of sign changes) and 4 odd flips, and the 48 total symmetries factor as 6 × 4 × 2.

The declaration even_flips_give_weak_structure proves the arithmetic fact that the number of even sign flips in three dimensions is 4, which equals 2 times the order of a parity quotient (a way of identifying symmetries that differ by an overall sign flip). The framework then packages this 4 with the labels SU(2) and "weak" by definition, not by proof. The number 4 matches the order of the discrete subgroup of SU(2) that the framework chooses to associate with the weak force's structure, and the number 2 matches its fundamental representation dimension.

What the declaration does not claim is as important as what it proves. It does not construct the continuous Lie group SU(2), any representation of it, a gauge connection, or a Yang–Mills action. The parity quotient ℤ/2ℤ is not U(1). The framework's own documentation states that matching these finite factors to Standard Model labels is an identification, not a theorem that the Standard Model gauge group is forced. The derivation of the Standard Model gauge group from the framework remains an open target.

The classical mathematics here is elementary: the order of the hyperoctahedral group B₃ is 48, its even-sign-flip subgroup has order 4, and these counts are proved by direct computation. The framework's contribution is the labeling of these counts with physical names, a modeling choice that assigns the tuple (3, 2, 1) to the gauge ranks of color, weak, and hypercharge interactions. The sum 3 + 2 + 1 = 6 also equals the number of faces of a cube, a coincidence the framework notes but does not elevate to a derivation.

The practical upshot is that the framework has a clean, machine-checked combinatorial fact about the cube, and a separate, clearly-marked decision to call parts of it by the names of fundamental forces. A reader can verify the arithmetic and see exactly where the physics labeling begins. The boundary between proved fact and chosen model is explicit in the framework's own documentation, which is the honest way to present a large claim.

/-- **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 finite arithmetic `4 = 2 * 2`. The even sign-flip group is not proved
to select `SU(2)`, and `U(1)` has trivial Weyl group. -/
theorem even_flips_give_weak_structure :
    even_sign_flip_count 3 = 2 ^ (3 - 1) ∧
    2 ^ (3 - 1) = 4 ∧
    4 = 2 * parity_quotient_order := ⟨rfl, by norm_num, rfl⟩

def weak_layer : GaugeLayer :=
  { name := "SU(2) weak"
    fund_rep_dim := 2
    discrete_order := 4 }

What this page does not claim

The declaration does not construct SU(2) as a Lie group or any of its representations. The parity quotient ℤ/2ℤ is not identified with U(1). The framework does not prove that the Standard Model gauge group is forced by the cube's symmetries.

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: