Nonzero weighted auxiliary polynomial with vanishing jet packets

Let $K\ge 0$, let $w_0, v_0, \theta$ be real parameters, let $w = (w_1,\dots,w_m)$ be the finite logarithmic weight profile, and let $r = (r_1,\dots,r_m)$ be the rational centers of the determinant construction. Write $\mathcal W$ for the column weights $W_0 = w_0$, $W_i = w_i$, and $\mathcal V$ for the row weights $V_0 = v_0$, $V_i = w_i/\theta$. For a real scale $a>0$ suppose the strict volume inequality

where $\mathbb{K}$ is the cardinality of the finite center index set, and suppose that for all sufficiently large real heights $N$ the auxiliary cardinality comparison

holds, where the right-hand count is the number of exponent vectors $u$ with $\sum_i \mathcal W_i u_i \le N$ and the left-hand count is $\mathbb{K}$ times the number of row-weight indices of the strict simplex $\sum_i \mathcal V_i u_i < aN$.

Then for all sufficiently large $N$ there is a nonzero polynomial $P$ in $m+1$ complex variables whose monomials all satisfy the weighted degree bound $\sum_i \mathcal W_i \deg_i(P) \le N$, and whose raw logarithmic jet packet at scale $aN$ vanishes at every center and every strict row index: $\mathrm{rawPacketMap}_{\mathcal V}(aN)(P) = 0$.

This is the auxiliary-polynomial kernel lemma of the strict-capacity argument: the monomial count at slope $1$ strictly exceeds the number of jet conditions at slope $a$, so the linear "jet evaluation at height $aN$" map from the weighted monomial space into the packet space has a nontrivial kernel; any nonzero kernel element is an auxiliary polynomial of bounded weighted degree that vanishes on all the $aN$-jets. It is the algebraic counting input that, together with the padded packet-kernel inclusion and the quotient-representative machinery, produces the bounded representatives modulo powers of the simultaneous logarithmic jet ideal required by the strict-capacity pillar.