Transport and tiling bounds for the affine plank problem on the triangle

Bang's affine plank conjecture asserts that planks covering a convex body have relative widths summing to at least $1$; it is open already in the plane. For a convex body $K$ and directions given by affine maps $u_i:K\to[0,1]$ we introduce the transport defect $D_K(u)$ — the least $D$ admitting a probability measure on $K$ with all $u_i$-marginals $\le D\cdot\mathrm{Leb}$ — so every covering by planks in these directions satisfies $\sum \mathrm{rw}\ge 1/D_K(u)$, and we prove a minimax duality for $D_K$ with exact certificates on both sides. On the triangle we determine exactly when the transport method certifies the sharp constant $1$: for three pairwise non-parallel directions, none parallel to a facet, a uniform-marginal witness measure exists ($D=1$) if and only if the mid-level lines concur; for every concurrent cyclic triple we exhibit the witness and the unique tight covering with $\sum \mathrm{rw}=1$ — a two-parameter family of rigid configurations extending the median case — and we classify all concurrent triples admitting a witness. We also determine, partly by an exact-rational computation, the one-plank-per-direction covering constant at one tilted non-concurrent triple (the three-plank bound). Calibrations: the uniform measure gives $D\le d$ on any $d$-dimensional body, and the $d$-simplex facets have $D=(d+1)/2$. No result resolves the conjecture for the triangle, let alone the plane. Mathematics Subject Classification: 52C17, 52A40, 52A10, 52C15