Mathematics Cubic Symmetry Group From Rs Rank Sum

The rank sum

The declaration rank_sum is a formal theorem stating that the list of numbers [3, 2, 1] has a sum of 6. In plain terms, it proves that 3 + 2 + 1 = 6. This is not a discovery about arithmetic; it is a machine-checked certification of a specific structural fact used in the Recognition Science framework. The list [3, 2, 1] is called the rank decomposition, and it records the subgroup ranks of the symmetry group of a three-dimensional cube.

The symmetry group of a cube, known as the hyperoctahedral group B₃, has 48 elements. This number factors as 2³ × 3! = 8 × 6 = 48. The rank decomposition [3, 2, 1] reflects the nested subgroup structure: a cube's symmetries can be built from a chain of subgroups of ranks 3, 2, and 1. The theorem rank_sum confirms that the sum of these ranks is 6, which equals the dimension of the Lie algebra associated with the group. The formal library states this as a definition and proves it by direct computation.

In Recognition Science, this rank sum appears as part of a certificate that the cube's symmetry structure is internally consistent. The certificate bundles three facts: the order of B₃ is 48, the hyperoctahedral order at dimension 3 is 48, and the rank sum is 6. Each fact is proved with zero axioms and zero unresolved proofs, meaning the library accepts them as fully verified. The rank sum itself is the simplest of the three, a bookkeeping check that the decomposition adds up as intended.

What rank_sum does not claim is broader significance. It does not assert that three-dimensional space is forced by this arithmetic, nor that the number 6 has physical meaning by itself. The declaration only certifies the sum of a specific list. The framework's larger claims about why three dimensions arise come from other theorems in the library, not from this single line. The rank sum is a supporting detail, not a standalone conclusion.

theorem rank_sum : rankDecomposition.sum = 6 := by decide

theorem b3Order_eq_48 : b3Order = 48 := by decide

/-- The (3,2,1) subgroup rank structure. -/
def rankDecomposition : List ℕ := [3, 2, 1]

What this page does not claim

The rank sum alone does not force three-dimensional space. The number 6 has no standalone physical interpretation from this declaration. The theorem does not describe the geometric rotations of a cube, only the group order and rank arithmetic.

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/Mathematics/CubicSymmetryGroupFromRS.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: