Subsets of F_p with two prescribed moments, Part I: the pair (1,3) — a moment subset sum over F_p: exact counts of totally split squarefree polynomials with a non-consecutive pair of prescribed coefficients, and exactly when those counts are elementary

Fix an odd prime p and let C(n,p) count the n-subsets T of Fp with ∑ t = ∑ t3 = 0; equivalently, for p > 3, the totally split squarefree polynomials of degree n over Fp whose coefficients of Xn−1 and Xn−3 both vanish. For n ≥ 3 this count admits an elementary expression — a polynomial in p with coefficients in quadratic characters of fixed integers — for exactly three degrees, n = 3, 4, 6. This note proves that, and gives the closed forms of C(3,p), C(4,p), C(5,p) and C(6,p); the case n = 5 involves the Frobenius trace of the elliptic curve of conductor 20, so it is exact without being elementary. The mechanism is combinatorial: the partition sieve attaches to each partition of n a weighted hyperplane section of a diagonal cubic whose singular points are the biquanimous splittings of the weights, and the stratum of length exactly four decides, because there the variety is a plane cubic and smoothness forces an elliptic curve. The uniform witness (n−3,1,1,1) settles every n > 6 at once. Two further strata are worked out by pencils of conics: V(a,1,1,1,1) has line field Q(√Da, √Ea), abelian of exponent two for every a, while V(2,2,1,1,1) does not, its line field containing the splitting field of a cubic of discriminant −140 and Galois group S3. The two extremal cubics are identified with the Cayley nodal cubic and the Segre cubic over the base field, by exhibiting the map. The note ships with the program that recomputes every count it quotes and with the scripts behind every measured statement.