Breaker Chains and Splitting of Interval Submodules in Zigzag Modules

Finite-dimensional zigzag modules decompose into interval modules. For an ordinary persistence module, every fully supported interval submodule of a restriction splits as a direct summand, whereas this need not hold for a zigzag module. We study this failure of splitting. A breaker pair is a local configuration of two overlapping interval summands with a backward arrow at the left boundary of the second interval and a forward arrow at the right boundary of the first. A single breaker pair gives a local nonsplitting interval submodule, but it need not control a prescribed larger restriction. We therefore introduce breaker chains. Our main result shows that a restriction $V[p,q]$ contains a nonsplitting submodule isomorphic to $I[p,q]$ if and only if its interval decomposition contains a breaker chain spanning $[p,q]$. Equivalently, the proper interval summands $J$ for which $\mathrm{Hom}(I[p,q],J)\mathrm{\ne }0$ have supports covering $[p,q]$. This cover formulation gives a constructive description of nonsplitting embeddings, a trichotomy for their splitting behavior, and an $O(m\log m)$ algorithm from an oriented barcode with $m$ intervals.