Lonely Runners over Function Fields: Quantized Phase--Riesz product

Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over ${\mathbb{F}}_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani’c conjectured that $C_k(q)=1+q+\cdots +q^k$. We disprove the unrestricted conjecture by constructing thirteen monic polynomials over ${\mathbb{F}}_2$ whose $k=3$ kernels cover ${\mathbb{F}}_2^7$; in particular, $C_3(2)\le 13<15$. For a general covering family of size $N=q^k+S$ and ${\mathbb{F}}_q$-linear rank $d$, we prove $S\gg d^{2\mathrm{/}3}{\left(\frac{\log (2q)}{\log (eN{q}^k\mathrm{/}S)}\right)}^{2\mathrm{/}3}$. Consequently, for every fixed $k\ge 2$ and all sufficiently large $q$, $C_k(q)\ge q^k+c_k{q}^{2\mathrm{/}3}$. When $k=2$, an integer-multiplicity refinement of the second-moment covering argument yields ${\mathrm{lim\hspace{0.17em}inf}}_{q\to \mathrm{\infty }}(C_2(q)-q^2)\mathrm{/}q\ge {\tilde{c}}_2$, where ${\tilde{c}}_2$ is an explicit one-variable variational constant with numerical value ${\tilde{c}}_2=0.5829944375\dots$. We also classify triples admitting two independent low-degree polynomial syzygies and prove a conditional packet-free lower bound of size $q^k+(1\mathrm{/}2-o(1))q^{k-1}$.