Modular non-descent in algebraic quantum field theory and the selection of globally coherent vacuum data

In algebraic quantum field theory (AQFT), local von Neumann algebras associated with finite spacetime regions admit no canonical restriction, averaging, or coarse-graining maps. While a faithful normal state induces intrinsic modular structure on each local algebra via Tomita--Takesaki theory, this structure does not behave functorially under inclusions of regions. This raises a natural descent problem: whether one can choose coarse-graining maps between local algebras that are compatible with the state and compose consistently under refinement, or equivalently whether state-preserving conditional expectations exist between local algebras. In this paper I formulate the generic obstruction to such functorial descent in cohomological terms. The obstruction is associated not with modular flows on overlaps themselves, but with the incompatibility of chosen descent data modulo inner automorphisms. Relative modular theory provides a canonical diagnostic of this failure once descent maps are fixed, via Connes cocycles comparing induced states on a single algebra. I introduce a Laplacian-like coherence operator on patchwise reference data whose kernel characterises the maximally descent-compatible sector. Motivated by explicit models and by the split property, I argue that the obstruction is carried by ultraviolet-sensitive modular reference data localised near entangling surfaces, while infrared modular contributions remain compatible under refinement. Although partly motivated by questions surrounding vacuum energy, the results are formulated entirely within AQFT and modular theory and are intended as a contribution to the structural foundations of locality in quantum field theory.