oneDimSubspace
The definition constructs the one-dimensional subspace generated by a vector v in the elementary abelian 2-group of rank D as the two-element Finset containing the zero vector and v. Researchers studying the algebraic underpinnings of the seven plot families in Recognition Science would cite it when establishing subgroup counts for dimension three. It is realized by a direct set constructor that immediately yields the expected cardinality and closure properties.
claimFor a vector $v$ in the space of functions from a finite set of size $D$ to the booleans equipped with pointwise XOR, the one-dimensional subspace generated by $v$ is the set containing the zero vector and $v$.
background
F2Power D denotes the type of functions from Fin D to Bool with pointwise XOR as addition, forming the elementary abelian 2-group of rank D. The module proves this group has cardinality 2^D and exactly 2^D - 1 nonzero elements; for D = 3 the nonzero count is 7, matching the basic plot families asserted in the companion paper Seven_Plots_Three_Dimensions.tex.
proof idea
Direct definition via a Finset literal that inserts the zero vector and the given v.
why it matters in Recognition Science
This supplies the concrete realization of the one-dimensional subspaces counted by the cardinality theorem and shown closed under addition by the closure theorem. It thereby replaces the hardcoded count of seven subgroups with a theorem, enabling the bijection to Booker plot families in downstream modules such as Aesthetics.NarrativeGeodesic. In the Recognition framework it anchors the D = 3 case of the eight-tick octave and the algebraic foundation for narrative geodesics.
scope and limits
- Does not require the input vector to be nonzero.
- Does not verify that the constructed set forms a subgroup.
- Does not extend to subspaces of dimension greater than one.
- Does not depend on any specific numeric value of D.
formal statement (Lean)
241def oneDimSubspace (v : F2Power D) : Finset (F2Power D) :=
proof body
Definition body.
242 {0, v}
243