Exact and Asymptotic Values for Weak Limited Augmented Zarankiewicz Numbers in the mtimes 3 Case

We determine the exact weak limited augmented Zarankiewicz numbers z_{wL}(m,3) for all mge 3: \[ z_{wL}(m,3)= cases m+3+\left\lceil m{2}\right\rceil+1, & 9\le m\le 15,\\[2mm] m+3+\left\lfloor 2m-4{3}\right\rfloor, & m\ge 16, cases \] with z_{wL}(3,3)=6, z_{wL}(4,3)=8, and z_{wL}(m,3)=2m for 5le mle 9. In particular, \[ \lim_{m\to\infty} z_{wL(m,3)}{m} = 5{3}. \] The proof is fully analytic, relying on a uniform base classification, two constructive lower-bound families (staircase and 5m/3), and a sharp upper-bound argument based on a peeling lemma and the analysis of two W2-sensitive boundary cases. Numerical MILP computations were used only as proof-mining tools to identify the structural lemmas; the final theorem is unconditional. We also extend the known range of the original limited numbers z_L(m,3) through m=13, where the gap to z_{wL}(m,3) is only 2 or 3.