axis3
axis3 supplies the third standard basis vector in the elementary abelian 2-group of rank 3. Researchers encoding Booker's seven plot families cite the definition when mapping the comedy family to the value-alignment axis. The declaration is a direct one-line constructor that populates the Fin 3 to Bool function with the pattern false, false, true.
claimLet $e_3$ be the element of the vector space $V = (Fin 3) → Bool$ (with pointwise XOR) whose components are $false, false, true$.
background
The module defines F2Power D as the type Fin D → Bool equipped with pointwise XOR, forming the elementary abelian 2-group of rank D. For D = 3 the group has order 8; the nonzero elements number exactly 7 by the theorem nonzero_card_three, and the elements decompose by Hamming weight into the classes 1 + 3 + 3 + 1. axis3 is the unique weight-1 vector with support only on the third coordinate.
proof idea
The declaration is a one-line definition that directly constructs the element of F2Power 3 by the array literal ![false, false, true]. No lemmas are invoked; the type checker confirms the codomain.
why it matters in Recognition Science
The definition is invoked by plotEncoding to realize the mapping from the comedy plot family to the third axis, and by axis3_weight to establish its Hamming weight equals 1. It participates in the weight decomposition that yields the seven 1-dimensional subgroups, confirming the count asserted in Seven_Plots_Three_Dimensions.tex. The construction supplies one of the three axes used in the structures RSIndependentTriple and RSDisjointSum3.
scope and limits
- Does not prove any algebraic identity satisfied by the vector.
- Does not compute its Hamming weight or relations to axis1 and axis2.
- Does not generalize the construction to arbitrary rank D.
- Does not supply the group operation or Fintype instance.
formal statement (Lean)
205def axis3 : F2Power 3 := ![false, false, true]
proof body
Definition body.
206