Mathematics Linear Algebra From Rs Rs Dimension Eq 3

The dimension declaration

In linear algebra, a vector space has a dimension: the number of independent directions needed to reach any point in it. The Recognition Science framework models its recognition lattice, a discrete record of events, as such a space. Its declaration recognition lattice, the framework's term for its discrete event structure, sets that dimension to exactly 3. The theorem rsDimension_eq_3 proves this by definition: the dimension is 3 because the framework defines it so, not because a longer argument forces it.

The declaration also fixes the size of the space. With dimension 3 over the two-element field F₂, the space has 2³ = 8 points. This matches the framework's recognition period of eight ticks. The same file proves the count of five canonical linear algebra operations, addition, scalar multiplication, inner product, outer product, and tensor product, equals 5. These numbers, 3 and 8 and 5, are the framework's internal constants, each defined in the machine-checked library of formal theorems.

What the declaration does not claim is the physical story. It does not prove that physical space has three dimensions. The framework's separate forcing chain, which derives three spatial dimensions from recognition costs, is a different result in a different file. The declaration here only fixes the dimension of the framework's own algebraic object. It is a definitional choice, not a physical derivation. The bridge from recognition to physical space remains open.

The practical consequence is modest but real: within the framework, the linear algebra is consistent. The dimension, the cube size, and the operation count all cohere. A reader can trust that the framework's internal geometry is not ad hoc, but the declaration itself does no physical work. It is a foundation stone, not a building.

theorem rsDimension_eq_3 : rsDimension = 3 := rfl

theorem f2CubeSize_eq_8 : f2CubeSize = 8 := by decide

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

What this page does not claim

The declaration does not prove that physical space has three dimensions. The declaration does not derive the number 3 from any cost function or forcing argument. The declaration does not establish that the framework's recognition lattice is the only possible discrete structure.

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)

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