From Covering-Edge Descent Complexes to Functorial Presheaves and Coherence Defects

Covering-edge descent data on a finite poset is enough to build a degree-1 descent complex, but not enough to justify full presheaf or higher-categorical language. This paper analyzes the missing step: composition of restriction maps along comparable chains. The result is a two-path structural split. Path A assumes strict path-independence of saturated-chain composites. Under that condition, covering-edge restriction data extends uniquely to a functorial Banach-valued presheaf on the finite poset, giving access to the standard presheaf, Grothendieck-construction, and coherence-diagram interface. Path B does not assume strict functoriality. Instead, it measures the failure of composition as a coherence defect: the difference between a two-step restriction and the declared direct restriction. This defect acts as a higher closure or curvature obstruction. On the flat locus, where all composition defects vanish, the strict functorial interface is recovered. Away from that locus, the nonzero defect is not higher-categorical authority; it is the signed obstruction to strict functoriality. The paper is finite, structural, and categorical in scope. It makes no physical, dynamical, empirical, or infinite-time persistence claim. Its purpose is to state exactly when covering-edge descent data can be promoted to functorial presheaf language, and when failure of that promotion must instead be recorded as a typed coherence defect.