  We propose that the symbol alphabet for classes of planar,
dual-conformal-invariant Feynman integrals can be obtained as truncated cluster
algebras purely from their kinematics, which correspond to boundaries of
(compactifications of) $G_+(4,n)/T$ for the $n$-particle massless kinematics.
For one-, two-, three-mass-easy hexagon kinematics with $n=7,8,9$, we find
finite cluster algebras $D_4$, $D_5$ and $D_6$ respectively, in accordance with
previous result on alphabets of these integrals. As the main example, we
consider hexagon kinematics with two massive corners on opposite sides and find
a truncated affine $D_4$ cluster algebra whose polytopal realization is a
co-dimension 4 boundary of that of $G_+(4,8)/T$ with 39 facets; the normal
vectors for 38 of them correspond to g-vectors and the remaining one gives a
limit ray, which yields an alphabet of $38$ rational letters and $5$ algebraic
ones with the unique four-mass-box square root. We construct the space of
integrable symbols with this alphabet and physical first-entry conditions,
whose dimension can be reduced using conditions from a truncated version of
cluster adjacency. Already at weight $4$, by imposing last-entry conditions
inspired by the $n=8$ double-pentagon integral, we are able to uniquely
determine an integrable symbol that gives the algebraic part of the most
generic double-pentagon integral. Finally, we locate in the space the $n=8$
double-pentagon ladder integrals up to four loops using differential equations
derived from Wilson-loop $d\log$ forms, and we find a remarkable pattern about
the appearance of algebraic letters.
