五次型的双割相干性：提升分离与二阶导数完备性
Two-Cut Coherence of Quintic Forms: Lifting Separations and Second-Derivative Completeness
AI总结：
该研究定义五次型双割相干性参数，证明其与压缩传输网络等的等价性，推导割点(1,3)处的完备性定理，构造实例否定C_{k,l}≤C_k+C_l的通用边界。
AI中文摘要：
对于次数为d的齐次多项式f，次数k受限强度C_k(f)是将f写成因子次数为k和d-k的乘积所需的最少项数。我们引入双割相干性参数C_{k,l}(f)：即满足f = ∑_{i,j=1}^{r} p_i m_{ij} q_j的最小r，其中deg p_i = k、deg m_{ij} = l-k、deg q_j = d-l，这要求两个次数界面由单一公共因式分解实现。我们证明该参数等于三区块压缩传输网络的最小公共端点宽度，等价于通过交换乘法对f进行所有张量提升中，两个张量列车端点秩的较大值的最小值，尤其它是齐次ABP宽度的下界。我们的主要结果是割点(1,3)处五次型的提取完备性定理：设D(f)为f的二阶方向导数的最大多项式切片秩C_1，t = C_3(f)；在特征为零的代数闭域上，ceil(D(f)/3) ≤ C̄_{1,3}(f) ≤ C_{1,3}(f) ≤ t·D(f) + 2t²，其中C̄表示边界复杂度。因此当C_3有界时，普通和边界双割相干性与二阶导数暴露的单割障碍在常数范围内等价。我们还证明了边界稳定的提升分离：对于互素非零三次式A和B，五次式L = abA + cdB的普通和边界局部值满足C_1 = C_3 = 2，且ceil(max{C_1(A), C_1(B)}/3) ≤ C̄_{1,3}(L) ≤ C_1(A) + C_1(B)。取A为n变量的费马三次式（我们证明其C_1 = ceil(n/2)），可得在最优单独局部界面与公共界面之间存在无界间隙，即使在边界复杂度中也成立，这否定了C_{k,l} ≤ C_k + C_l这类通用边界。
英文摘要：
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.
