Mathematics Linear Algebra From Rs Linear Algebra Cert

What the certificate certifies

Linear algebra is the branch of mathematics that studies vectors: objects that can be added together and multiplied by numbers. A vector space over a field is a set with two operations, vector addition and scalar multiplication, that obey a short list of axioms such as associativity and distributivity. The real plane ℝ² and three-dimensional space ℝ³ are the familiar examples. A basis of a vector space is a minimal set of vectors that can be combined to produce every vector in the space; the number of vectors in any basis is the dimension. For ℝ³ the dimension is 3, and the standard basis has three vectors: (1,0,0), (0,1,0), and (0,0,1).

The theory of vector spaces over finite fields is a standard and older subject. For a prime power q, the vector space F_qⁿ has qⁿ elements. When q=2 and n=3, the space F₂³ has 2³=8 elements, each a triple of bits. This space is small enough to enumerate by hand: its elements are 000, 001, 010, 011, 100, 101, 110, and 111. The finite field F₂ has just two elements, 0 and 1, with arithmetic modulo 2. This space is a standard object in coding theory and combinatorics, where it is often called the three-dimensional binary cube.

In Recognition Science, the framework models its recognition lattice as a three-dimensional vector space over F₂, and it defines a certificate called LinearAlgebraCert that packages three numerical facts. The certificate states that the number of canonical linear algebra operations (addition, scalar multiplication, inner product, outer product, tensor product) is exactly five; that the recognition dimension is exactly three; and that the size of the F₂³ cube is exactly eight, which equals 2³. These facts are established by a machine-checked library of formal theorems, meaning that a computer verified the proofs step by step from definitions. The certificate is a structure that bundles these three verified facts into one object, so that later theorems can cite the certificate as a whole.

The certificate does not define linear algebra from scratch. It takes the standard definitions of vector space, basis, and dimension as given, and it observes that the recognition lattice satisfies them. It does not prove that the dimension must be three from first principles; the value 3 is built into the definition rsDimension := 3. The certificate also does not claim that the five operations are the only possible operations, merely that these five are the canonical ones in the framework. It is a consistency check, not a derivation of linear algebra from deeper axioms.

What the certificate changes is the workflow: instead of re-checking these three facts in every later proof, a theorem can require a LinearAlgebraCert as a parameter and trust that the facts hold. That is the practical payoff of a certificate in a machine-checked library: it groups verified facts so that downstream mathematics can be written more cleanly. The certificate itself is a small object, but it is the kind of small object that makes larger formal developments tractable.

theorem linearAlgebraOpCount : Fintype.card LinearAlgebraOp = 5 := by decide

theorem rsDimension_eq_3 : rsDimension = 3 := rfl

theorem f2CubeSize_eq_8 : f2CubeSize = 8 := by decide

What this page does not claim

The certificate does not prove that the dimension must be three; it assumes the value 3 as a definition. The certificate does not define the standard axioms of a vector space; it uses them as background. The certificate does not claim that the five operations are the only possible linear algebra operations.

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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