Mathematics Cubic Symmetry Group From Rs B3 Order Eq 48

The cube's symmetry count

A cube has 48 symmetries: rotations, reflections, and combinations of the two that map the cube onto itself. The count comes from a standard formula. For a D-dimensional hypercube, the symmetry group has order 2^D × D!, and at D = 3 this is 2^3 × 3! = 8 × 6 = 48. The 8 accounts for the sign flips of the three coordinate axes, and the 6 accounts for the permutations of those axes. This is classical group theory, known since the 19th century through the work of mathematicians like Arthur Cayley on permutation groups.

The Recognition Science framework's machine-checked library of formal theorems contains a declaration named b3Order_eq_48. This theorem states that the defined value b3Order, which the framework sets as 2^3 × 3!, equals 48. The proof is by direct computation, marked with zero unresolved proofs and zero axioms. The framework also defines a general hyperoctahedralOrder for any dimension D as 2^D × D!, and proves the D = 3 case separately. A structure called CubicSymmetryCert bundles these facts together: the order equals 48, the hyperoctahedral formula gives 48, and a rank decomposition [3, 2, 1] sums to 6 with length 3.

In Recognition Science, this count is a structural anchor. The framework models its recognition lattice using the 3-cube Q₃, and the symmetry group of that cube is the hyperoctahedral group B₃. The (3,2,1) rank decomposition reflects the subgroup structure of B₃, where a point stabilizer chain reduces dimension step by step. The theorem b3Order_eq_48 is what the framework's library uses to certify that this symmetry count is correct within its formal system.

What the declaration does not claim is broader. It does not prove that the cube's symmetry group is 48 in the physical world; that is a mathematical fact established independently. It does not derive the number 48 from Recognition Science's forcing chain or from its cost function J(x). The declaration is a definitional choice plus a computational verification: the framework defines b3Order as 2^3 × 3!, then proves the arithmetic. The significance for RS is that its recognition lattice, built on the 3-cube, inherits this classical symmetry count as a certified structural fact, not as a new derivation.

theorem b3Order_eq_48 : b3Order = 48 := by decide

theorem b3Order_eq_48 : b3Order = 48 := by decide

theorem b3Order_eq_48 : b3Order = 48 := by decide

theorem hyperoctahedral_D3 : hyperoctahedralOrder 3 = 48 := by decide

What this page does not claim

The declaration does not derive the number 48 from the framework's cost function or forcing chain. The declaration does not prove physical facts about cube symmetries beyond the mathematical definition. The declaration does not establish the physical recognition-to-linking bridge for three dimensions.

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.

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