  Answering a pair of questions of Conrey, Gabbard, Grant, Liu, and Morrison,
we prove that a triplet of dice drawn from the multiset model are intransitive
with probability $1/4+o(1)$ and the probability a random pair of dice tie tends
toward $\alpha n^{-1}$ for an explicitly defined constant $\alpha$. This
extends and sharpens the recent results of Polymath regarding the balanced
sequence model. We further show the distribution of larger tournaments
converges to a universal tournamenton in both models. This limit naturally
arises from the discrete spectrum of a certain skew-symmetric operator (given
by the kernel in the title acting on $L^2([-1,1])$). The limit exhibits a
degree of symmetry and can be used to prove that, for instance, the limiting
probability that $A_i$ beats $A_{i+1}$ for $1\le i\le 4$ and that $A_5$ beats
$A_1$ is $1/32+o(1)$. Furthermore, the limiting tournamenton has range
contained in the discrete set $\{0,1\}$. This proves that the associated
tournamenton is non-quasirandom in a dramatic fashion, vastly extending work of
Cornacchia and H\k{a}z{\l}a regarding the continuous analogue of the balanced
sequence model.
  The proof is based on a reduction to conditional central limit theorems
(related to work of Polymath), the use of a "Poissonization" style method to
reduce to computations with independent random variables, and the systematic
use of switching-based arguments to extract cancellation in Fourier estimates
when establishing local limit-type estimates.
