A finite-board reduction for the ErdH{o}s Matching Conjecture and the 4-uniform case via exact certificates

We prove the 4-uniform Erd\H{o}s Matching Conjecture for every matching number $s\ge 6961$. The proof has two parts. First, building on ideas from Frankl--R\"odl--Ruci\'nski, we formulate a general finite-board criterion for the $r$-uniform conjecture. The criterion has two assumptions: the $(r-1)$-uniform cover-side bound for links with matching number at most $t$ holds at every $m\ge n_r(t)$, and a finite optimization problem for mixed-size trace configurations on an $(r^2+r-1)$-vertex board. Together with the corresponding lower-uniformity input, this finite-board optimization implies the Erd\H{o}s Matching Conjecture with explicit large-matching thresholds. Second, we verify the finite-board assumption for $r=4$. The local board has 19 vertices, and the required inequality is decomposed into three weighted local inequalities: a leading wide layer, a 15-board layer, and an 11-board layer. The verification is reduced to exact finite optimization and certificate-validation problems: Ferrers down-set enumerations for pair and triple traces, rational Farkas-dual certificates for the top-star branch, integer branch-and-bound up-set hitting and pattern searches for the no-top-star branch, and residual-cut dual certificates for the 15-board and 11-board layers.