Finite exact-density tile families for multiset profile graphs

Let $\Delta_q(d)$ be the set of nonnegative integer vectors with $q$ coordinates summing to $d$, and let $M_q(d)$ be the size of the largest additive checksum fiber, given by the divisibility-regime formula stated in the source. A finite-state program assigns weights to coordinate-symmetric induced templates and certifies upper bounds for $\alpha_q(d)$.

Finite exact-density tile-family conjecture. For every fixed $q$, there is a finite coordinate-symmetric family of induced templates of independence density $1/q$ such that, in every divisibility regime of the formula for $M_q(d)$, the associated finite-state program certifies

$$\alpha_q(d)=M_q(d)$$

for all sufficiently large $d$, apart from finitely many exceptional degrees.

The conjecture proposes a finite-template explanation for eventual optimality of the additive checksum construction in every fixed alphabet size. The paper establishes several cases, including $q=3,4,5,7$, but the asserted statement for every fixed $q$ remains open.