Equilateral Completion in Floretion Triangular Coordinates: Locality, Product Points, and Reflection Symmetry

Abstract

We study unordered triples of order-$n$ floretion base vectors whose tile centroids form nondegenerate equilateral triangles. A scaled integer centroid map turns Euclidean completion into exact arithmetic on a triangular lattice, and a residue obstruction modulo $3$ shows that every equilateral centroid triangle uses three tiles of one orientation. Combined with finite triangular-lattice completion counts, this gives $|E_n|=4^n(4^n-1)/12$. For synchronized local $γ$-cycles, $|L_n|=(7^n-4^n)/3$ and $|L_n|/|E_n|\sim4(7/16)^n$, while on the no-$e$ support $S_n=\{i,j,k\}^n$ locality is exhaustive and $|E_n^S|=|L_n^S|=(2^n-1)3^{n-1}$. The union of the three main axes supports exactly $|E_n^{\rm ax}|=4^{n-1}+2^n-2$ equilateral triangles, split into the branches $x=y=z$ and $x+y+z=0$. For $T\in E_n$, the unsigned vertex product defines a product point $C_T$; a digitwise parity criterion characterizes $C_T=Q_T$ on local cycles and yields Fibonacci subfamilies. Multiplication-generation is equivalent to $p(T)=e_n$, hence $C_T=0$; locally this gives exactly the nontrivial global $γ$-orbits, and exact enumeration through order $6$ finds no nonlocal example. Retaining the signs discarded by the unsigned product gives a second classifier: a triangle has scalar vertex-sum square exactly when its three vertices pairwise anticommute. For local cycles this occurs exactly when $|S|$ is odd, giving $|\mathrm{AC}_n\cap L_n|=(7^n-1)/6$, while nonlocal pairwise-anticommuting examples already occur in order $3$.