Two-Cut Coherence of Quintic Forms: Lifting Separations and Second-Derivative Completeness

Abstract

For a homogeneous polynomial f of degree d, the degree-k restricted strength C_k(f) is the least number of products needed to write f with factor degrees k and d-k. We introduce a two-cut coherence parameter C_{k,l}(f): the least r such that f = sum_{i,j=1}^{r} p_i m_{ij} q_j with deg p_i = k, deg m_{ij} = l-k, and deg q_j = d-l. This requires two degree interfaces to be realized by a single common factorization. We show it equals the minimum common endpoint width of a three-block compressed transfer network, and equivalently the minimum, over all tensor lifts of f through commutative multiplication, of the larger of the two tensor-train endpoint ranks. In particular it lower-bounds homogeneous ABP width. Our main result is an extraction-completeness theorem for quintics at cuts (1,3). Let D(f) be the largest polynomial slice rank C_1 of a second directional derivative of f, and let t = C_3(f). Over an algebraically closed field of characteristic zero, ceil(D(f)/3) <= Cbar_{1,3}(f) <= C_{1,3}(f) <= t*D(f) + 2t^2, where Cbar denotes border complexity. Hence when C_3 is bounded, ordinary and border two-cut coherence are equivalent up to constants to a one-cut obstruction exposed by a second derivative. We also prove a border-stable lifting separation. For coprime nonzero cubics A and B, the quintic L = abA + cdB has ordinary and border local values C_1 = C_3 = 2, while ceil(max{C_1(A), C_1(B)}/3) <= Cbar_{1,3}(L) <= C_1(A) + C_1(B). Taking A to be a Fermat cubic in n variables, for which we show C_1 = ceil(n/2), gives an unbounded gap between separately optimal local interfaces and a common interface, even in border complexity. This refutes any universal bound of the form C_{k,l} <= C_k + C_l.