Coordinate projections of $c$-vectors of cluster algebras from the annulus

For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real
Schur roots of the associated root system. We study the two-coordinate
projections $(c_v, c_w)$ of this configuration: when the difference
$c_v - c_w$ is bounded the image lies in a band of lattice lines, and we ask
when the projection fills that band. A band-existence dichotomy, valid in
every acyclic type, shows the difference is bounded if and only if the null
root satisfies $\delta_v = \delta_w$. For affine type $\widetilde{A}_n$ (the
annulus), in the source-sink orientation, we resolve the filling question
completely: every coordinate projection fills its band except along the
source-sink diagonal, which carries only the finite regular part. The
obstruction is the Auslander--Reiten defect, which a projection sees on its
diagonal exactly when the defect is a coordinate difference; the only such
pair is the source-sink pair of $\widetilde{A}_n$, so the pattern depends on
the chosen seed. More generally, every banded pair of null-root coefficient
one fills, except these diagonals. Off the diagonal a banded pair in
$\widetilde{E}_7$ fails to fill, so non-filling is not confined to type
$\widetilde{A}_n$; a computation classifies the pairs of coefficient at least
two over a range of affine types, where this $\widetilde{E}_7$ pair is the
only further failure, and the general classification remains open.
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