Dichotomy for the sharing floor

Prove or refute the conjecture that a linear-overlap layer of \(CCZ\) blocks attains the count \(6m+1-2s\) if and only if every connected component of its block-overlap graph satisfies \(s_C=m_C-1\).

Background

For linear-overlap layers, the isotropy floor is $6m+1-2s$, where $s$ counts shared incidences. The paper constructs optimal circuits when the overlaps can be routed through a degree-three forest.

Small cyclic overlap patterns show that the floor can fail when a component has $s_C\ge m_C$. The conjecture proposes that the componentwise condition $s_C=m_C-1$ exactly characterizes attainment, although the available routing construction does not establish the general sufficiency direction.