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arXiv · 2608.15925

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

Creighton Dement
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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 gamma-cycles, |L_n| = (7^n - 4^n)/3 and |L_n|/|E_n| is asymptotic to 4(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). 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 gamma-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 |AC_n intersect L_n| = (7^n - 1)/6, while nonlocal pairwise-anticommuting examples already occur in order 3.

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Creighton Dement. 2026-08-16. Equilateral Completion in Floretion Triangular Coordinates: Locality, Product Points, and Reflection Symmetry. https://arxiv.org/abs/2608.15925

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