Mathematics Linear Algebra From Rs F2 Cube Size Eq 8

The eight-element cube

In linear algebra, a vector space over the field F₂ (whose only elements are 0 and 1) has exactly 2ⁿ elements when its dimension is n. The three-dimensional case F₂³ therefore has 2³ = 8 elements. This is the classical fact that the Recognition Science declaration f2CubeSize_eq_8 records: it states, as a theorem, that the size of the cube F₂³ is 8.

The declaration is part of a small machine-checked library of formal theorems. The library defines a dimension constant, sets it to 3, and then establishes the size result by direct computation. The proof is a recognition (a discrete record of events) lattice fact: the three-dimensional recognition space is identified with F₂³, and the count of its elements is the recognition period. The theorem does not derive the number 8 from first principles; it assumes the dimension is 3 and computes the size of the cube.

What the theorem does not claim is broader. It does not assert that physical space has three dimensions; that claim belongs to a separate forcing chain in the framework, not to this declaration. It does not say that the number 8 is physically meaningful on its own; the theorem merely computes a cardinality. It does not establish that the five linear algebra operations (addition, scalar multiplication, inner product, outer product, tensor product) are the only ones possible; that count is a separate theorem about a finite list, not a claim about all possible operations.

The practical content is narrow but precise: within the framework's model of recognition space, the three-dimensional binary cube has exactly eight points. A reader who wants to know why three dimensions arise, or why eight matters physically, must look to other results in the framework, not to this declaration.

theorem f2CubeSize_eq_8 : f2CubeSize = 8 := by decide

def rsDimension : ℕ := 3  -- D = 3

theorem rsDimension_eq_3 : rsDimension = 3 := rfl

What this page does not claim

The theorem does not assert that physical space has three dimensions. The theorem does not establish that the number 8 is physically meaningful on its own. The theorem does not prove that the five listed operations are the only possible ones.

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/LinearAlgebraFromRS.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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