Strict-capacity weighted bounded representatives

Let $d$ be admissible fixed determinant data with $m=d.m$ logarithmic jet directions, let $R<S$ be real scales, let $e_i>0$ be multiplicities and $T_i$ truncation orders for the logarithmic jet coordinates. Assume the common-scale identity $S = e_i\,w^{\mathrm{row}}_i$ for all $i$, the strict capacity inequality $2R^{m+1}>S^{m+1}$, and the strict log-cutoff inequality $w^{\mathrm{row}}_{i+1}<T_i\,v_0$. Write $I$ for the simultaneous logarithmic jet ideal at the rational centres and $\mathcal W$ for the column weights of the ambient interpolation matrix.

Then there are a representative slope $\rho$ with $0<\rho<R$ and an additive overhead $B\ge 0$ such that for every $k\ge 1$ and every polynomial $P\in\mathbb{C}[x_0,\dots,x_m]$ there is a polynomial $Q$ satisfying

where $\deg_j(Q)$ is the exponent of $x_j$ in a monomial of the support of $Q$ (the inequality is required for every monomial in the support).

This is the weighted bounded-representative statement with strict capacity: it says that classes modulo the $k$-th power of the simultaneous jet ideal have representatives whose column-weighted size grows at most linearly in $k$ with slope strictly below the scale $R$, uniformly in the polynomial and in $k$ up to a constant overhead. It is the analytic heart of the strict-capacity pillar: the capacity inequality $2R^{m+1}>S^{m+1}$ is exactly the statement that the volume of the column-weight simplex of slope $R$ exceeds (twice) the volume of the row-weight simplex of height $S$, which is what provides the necessary room for the representative. Together with the elementary construction converting bounded representatives into monic weighted relations, it implies the strict-capacity weighted monic relations statement.