Mathematics Abstract Algebra From Rs Q3 Size Eq 8

The size of Q₃

In abstract algebra, a group is a set with an operation that combines any two elements to make a third, following rules like associativity and the existence of an identity element. The group (ℤ/2)³, read as "Z mod 2 cubed," is built from three independent binary choices, each either 0 or 1. Its eight elements are all possible triples of bits: 000, 001, 010, 011, 100, 101, 110, and 111. The group operation is addition modulo 2, applied separately to each coordinate, so that 1 plus 1 equals 0. This group is abelian, meaning the order of addition does not matter, and every element has order 1 or 2, since adding any element to itself always gives the identity 000.

The Recognition Science framework models a recognition lattice, a discrete record of events, and names a particular object Q₃. Within this framework, Q₃ is identified with the group (ℤ/2)³. The machine-checked library of formal theorems proves the statement q3Size_eq_8: the size of Q₃ equals 8, which is 2³. This theorem is verified by direct computation in the library, with no unproved assumptions and no axioms beyond the standard logical rules. The proof is a simple evaluation: the definition of q3Size is 2 ^ 3, and the theorem confirms that this evaluates to 8.

The same library also proves that Q₃ has exponent 2, meaning every element squared gives the identity, and that there are exactly five canonical algebraic structures: group, ring, field, module, and algebra. These facts together describe the algebraic character of Q₃ as a small, well-understood object. The theorem q3Size_eq_8 does not claim that Q₃ is the only group of size 8, nor does it assert anything about how Q₃ arises from physical recognition events. It establishes a cardinality fact within a formal algebraic setting, nothing more.

theorem q3Size_eq_8 : q3Size = 8 := by decide

theorem q3Exponent_eq_2 : q3Exponent = 2 := rfl

theorem algebraicStructureCount : Fintype.card AlgebraicStructure = 5 := by decide

What this page does not claim

The theorem does not claim that Q₃ is the only group of size 8. The theorem does not claim any physical interpretation of Q₃ or its size. The theorem does not assert that the recognition lattice is a group in any empirical sense.

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/AbstractAlgebraFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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