Lemma 9. For every sufficiently large prime s≡23 , there is a set Y⊆_s^2 with the following properties. For each d∈_s^* , there is an affine line ℓ_d of slope d , and Y=_s^2⋃_ d∈_s^* ℓ_d. For each d∈_s^* , every line of slope d other than …

Let s tend to infinity through primes congruent to 23 . Since (3,s-1)=1 , cubing permutes _s^* and fixes zero. Thus f is bijective. It maps horizontal lines to horizontal lines and vertical lines to vertical lines. We first prove the two survival properties simultaneously with high probability, then use them to classify mixed covers satisfying. Survivors in the original coordinates. Choose the lines ℓ_d , independently for the s-1 values of d , uniformly within their parallel classes. Fix a line ℓ of finite nonzero slope d , and temporarily ignore the choice ℓ_d . Each of the other s-2 choices deletes a uniformly distributed point of ℓ . Let Z_ℓ count the surviving points on ℓ . Its expectation is μ:= Z_ℓ=s(1-1/s)^ s-2 . Changing one of these choices changes the survivor count by at most two. Apply the independent-trial jump bound and bounded-drift concentration estimate of to the exposure martingale, with T=s-2 , L=2 , and u=μ/2 . The martingale has zero drift, and hence (Z_ℓ<μ/2) ≤2(-(μ^2)/(128(s-2))) =(-Ω(s)). There are fewer than s^2 lines under consideration. A union bound, together with μ/2≥5 for sufficiently large s , shows that all these counts are at least five with probability tending to one. If ℓ≠ℓ_d , reinstating the deletion by ℓ_d changes no point on ℓ , since the two lines are parallel. Thus every required line in the original coordinates has at least five points of Y . Survivors and almost-pencil covers in the new coordinates. In the new coordinates, the possible deletion shapes are F_ d,z = (x^3,dx+z):x∈_s , d∈_s^*, z∈_s. There are s(s-1) such shapes, each of size s , and each of the s-1 independent trials chooses one of them. Every point is deleted by a specified trial with probability 1/s , so its overall survival probability is the uniform value δ_s=(1-1/s)^ s-1 ≥14 for all sufficiently large s . Each shape meets every affine line in at most three points. For a vertical line this follows from bijectivity of cubing; for a horizontal line it follows from d≠0 . For a line of finite nonzero slope m , the intersection is determined by dx+z=mx^3+u, a nonzero polynomial equation of degree three. In particular, no shape is itself a line when s>5 . All the hypotheses of Lemma 7 ([Line covers after random deletions]) hold with field size s , a fixed integer constant, for example D=3 , survival lower bound δ=1/4 , and ρ_0=1/100 . The exact-line occurrence probabilities are zero, and the subfield restriction is automatic for a prime field. With probability tending to one, every line consequently contains at least s/8 points of f(Y) , and its almost-cover conclusion holds for f(Y) as well. We now combine that conclusion with the lower bound on every line to replace an almost pencil by a complete one. From an almost pencil to a full parallel class. Assume these events hold, and consider a cover satisfying . Repeated resources can be removed; alternatively, one may keep their original counts, since removal only reduces the budget. The line portion has at most s distinct lines and leaves at most b≤2s points of f(Y) uncovered. Lemma 7 ([Line covers after random deletions]) supplies at least (1-ρ_0)s distinct lines in this portion through a common projective point p . There are at most ρ_0s other lines, and b≤2(s-a)≤2ρ_0s. If an affine line through p were absent from the cover, all the chosen lines through p would cover at most one of its affine points. The other lines would cover at most ρ_0s further points, and the individual point resources at most 2ρ_0s . Thus at most 1+3ρ_0s points of this missing line would be covered. For sufficiently large s , this contradicts s8>1+3ρ_0s. Every affine line through p must therefore be present. A finite p requires s+1 lines, which is impossible. An infinite p requires all s lines in its parallel class, and then forces a=s and b=0 . This also rules out repetitions in a cover attaining the budget. The events used in the two coordinate systems hold simultaneously with probability tending to one. Consequently every sufficiently large prime s≡23 admits a successful configuration Y . A mixed cover of cost less than s would satisfy but not its necessary equality conclusion. The two stated consequences follow.