Paper069_The_Nullspace_Dichotomy_V4

ABSTRACT. Let ℱ be a k-sunflower-free family of r-sets and Mc its degree-c lifted incidence matrix. We prove that the Sunflower Conjecture reduces to one matrix rigidity statement. The Extraction Algorithm converts any minimal-support null vector of Mc⊤ into a k-sunflower, conditional on the Signed Growth Lemma. The Signed Growth Lemma is the following purely linear-algebraic statement: a vector α in the kernel of the global star operator Mc⊤ cannot have all local signed petal sums simultaneously small under the petal matrix P(t), unless α is trivial. Two matrices. One vector. One kernel constraint. One lower bound. The unsigned version of this statement is a pigeonhole on column sums and is exactly the Erdős–Rado 1960 proof recovering the factorial bound. The signed version is strictly stronger by an exponential factor. The difference is not analysis. It is the difference between probing a 0-1 matrix with an arbitrary positive vector and probing it with a vector constrained to lie in a specific kernel. The Sunflower Conjecture is the claim that this kernel constraint is rigid enough to prevent simultaneous near-annihilation.