The Hexagonal Coil: An Eisenstein Sibling of the Triangular--Fractional Grid, and Why the Better Tiling Does Not Win

The triangular--fractional grid (TFG) places each reduced positive rational at a single address in a square-rendered chamber lattice, where the round-up $n=\lceil r/s\rceil$ names the chamber in one number. This note builds the natural hexagonal sibling of that grid, using the six-unit frame of the Eisenstein lattice $Z[\omega]$ as the corresponding order-six rendering frame, coils it, and measures it. The result is deliberately deflationary. The hexagonal coil is a clean tiling---it carries the same $\varphi(s)$ totient-shell backbone as the square grid, its ring and spiral renderings are arithmetically identical, and its addresses are unique. But its apparent advantages are geometric, not arithmetic: the richer prime-numerator duplication (mean multiplicity $1.99$ against the square's $1.43$ at matched cell count, though the figure drifts with the cutoff) is a consequence of the sixfold cell count, not new structure; the Eisenstein mod-3 splitting organizes the shell placement but not the prime duplication (enrichment $1.006$, a null); and the wedge texture that resembles a Pascal--Sierpi\'nski field at low ring counts fills in and vanishes under scaling. The primes remain classical throughout, indifferent to the sixfold geometry. The one thing the hexagon wins is legibility---its $6n$ count forces a regular, evenly spaced ray skeleton the eye reads instantly---which is the same reason the bee builds hexagons: geometric privilege, showing here as visual clarity rather than packing efficiency. The square keeps the access floor; the hexagon keeps only the drawing.