The Covering Number C(12,6,4) is 41

A t-(v, k, λ) covering is a collection of k-subsets (blocks) of a v-set such that everyt-subset of points lies in at least λ blocks; the covering number Cλ(v, k, t) is the leastnumber of blocks in such a collection, and one writes C(v, k, t) when λ = 1. The recordedbounds for C(12, 6, 4) have been 40 ≤ C(12, 6, 4) ≤ 41. We show that no 4-(12, 6, 1)covering with 40 blocks exists, and hence that C(12, 6, 4) = 41. A counting argumentshows that in a hypothetical 40-block covering every point lies in exactly 20 blocks, thelink of every point is an optimal 3-(11, 5, 1) covering with a forced degree sequence, andthe six pairs of points of degree 10 form a perfect matching; an exhaustive case analysisover the orbits of a group of order 3840, carried out by satisfiability solving, then showsthat no optimal 3-(11, 5, 1) covering occurs as such a link. Each of the 81 formulas inthe primary proof has an unsatisfiability certificate checked by drat-trim and by theformally verified checker cake_lpr; two additional cross-encoding certificates are checkedby the same pipeline. The lower-bound argument uses no tabulated covering number: itsonly numerical input, C(10, 4, 2) ≥ 9, is itself certified. As a by-product the certificatesyield a self-contained certified proof that the optimal 3-(11, 5, 1) covering is unique up toisomorphism. Equivalently, the Turán number T (12, 8, 6) is 41; the new value propagatesto improved lower bounds for C(13, 7, 5), C(14, 8, 6), C(15, 9, 7) and C(16, 10, 8).