A Radial Coordinatization of the Triangular-Fractional Grid: Integer Boundaries, a Phase-Pole Center, and a Tiling–Phase Obstruction

We give a radial (square-spiral) coordinatization of the triangular-fractional grid
$a(n,k) = n-1 + k/n$, $n \ge 1$, $1 \le k \le n$, in which chamber $n$ occupies one arm of a
square spiral and the integer endpoint $a(n,n)=n$ lies at the turn. Three structural observations
follow. First, a reframe: because the integer $r$ occurs only as the endpoint of chamber $r$
(parent paper, Cor.~3.3), the integers act as \emph{boundaries} between chambers rather than as
interior cells; under a mild index extension the construction reads as a total classifier that
sends any positive real to the (arm, cell) address it would occupy, reachable in closed form by
projection rather than by traversal. Second, a phase obstruction at the center: a single-valued
global rotational phase cannot be assigned, because the phase winds nontrivially around the origin
where all arms meet; we mark the center accordingly as a phase pole $e^{i\theta},\, r=0$. We situate
this as an instance of the winding-number / phase-singularity picture of singular optics
(Nye--Berry~1974) and the screw-dislocation picture of crystallography, realized here in a discrete
rational grid. We do not claim the spiral tiles the plane; a single one-turn spiral is not
space-filling. Third, an experimental observation on reduced-numerator primality, together with one
proven exclusion (the composite collar): the entry immediately before each integer has numerator
$n^2-1=(n-1)(n+1)$, composite for $n\ge 3$, so a prime numerator is structurally excluded from that
position. Throughout we distinguish the parent paper's proven arithmetic (which we reuse, not
re-prove) from the present geometric framing, and we record fenced non-results consistent with the
parent paper's methodology.