Cofilling Shattering: A Syndrome-Support Hierarchy for Check Erasures

Let $A:\mathbb{F}_2^n\to\mathbb{F}_2^m$ be a binary linear map with fixed coordinate bases, let $C_A=\ker A$, and let $\lambda_A(y)$ be the minimum Hamming weight of a preimage of the syndrome $y$. We define $\operatorname{Shat}_{q,s}(A)$ as the least common check support of a $q$-dimensional syndrome subspace whose every nonzero element has coset-leader weight at least $s$. It therefore distinguishes release of $q$ independent syndromes from release of a subspace with no easy linear combination. Deleting check coordinates $F$ releases $\ker A_{\bar{F}}/\ker A$, canonically isomorphic to $(\operatorname{im} A)[F]$. Finiteness implies $R_q(C_A)\ge \mathsf{N}_2(q,s)$, where $\mathsf{N}_2(q,s)$ is the shortest length of a binary code of dimension $q$ and distance at least $s$; profile-Griesmer bounds independently control common check support. The hierarchy is coordinate-relabeling invariant but can change under a change of check basis. For the pair-repetition code $C_n=\{(x,x):x\in\mathbb{F}_2^n\}$, the standard realization $H_0=[I_n\ I_n]$ has $\operatorname{Shat}_{q,s}(H_0)=\mathsf{N}_2(q,s)$ whenever feasible. For every $q\ge 1$ and $s\ge 2$, with $n=\mathsf{N}_2(q,s)$, a row-equivalent realization of the same code has value $q$. For a simplicial coboundary map $A=\delta_k$, check erasure is top-face erasure and the released quotient is emergent cohomology. At $s=1$ the hierarchy reduces to generalized Hamming weights and is Tutte-determined; for $s\ge 2$, even identical labeled cut codes can have different values.