Independent Sets in Multiset Profile Graphs via Weighted Local Covers

Let $G_q(d)$ be the unit-transfer graph on the nonnegative integer vectors whose $q$ coordinates sum to $d$, equivalently on the multiplicity profiles of size-$d$ multisets over $q$ symbols. The prime-checksum conjecture predicts that, for prime $q$ and all sufficiently large $d$, a largest independent set is a fiber of the natural cyclic checksum. We introduce weighted local covers of $G_q(d)$ by translated induced subgraphs. For fixed $q$, capped anchor profiles reduce the covering conditions for infinitely many degrees to a finite rational linear system. This method gives new proofs of the known cases $q=3$ and $q=4$ and determines $\alpha (G_q(d))$ exactly for $q=5$ and $q=7$ in every degree, thereby proving the next two odd-prime cases of the conjecture. In the complementary regime where $d$ is fixed and $q$ grows, a partition-orbit reduction solves degree five for $q\ge 7$, gives exact power-of-two families in degrees six, eight, and ten, and yields an asymptotically sharp upper bound through three terms for every fixed $d\ge 7$. All computer-assisted assertions reduce to finite rational or integer systems and are supported by independently checkable certificates.