Universal Obstruction Cohomology of Categorified Spectral Objects:From Descent Filtrations to Derived Duality

This paper develops an intrinsic obstruction cohomology theory, denoted by Obs(A), for admissible operator-semantic systems A. Existing obstruction theories are typically formulated for topological spaces, schemes, or derived stacks. In contrast, the proposed theory is intrinsic to operator-semantic systems and is constructed directly from their categorified spectrum. It provides a universal cohomological framework for descent-compatible semantic obstructions without requiring an ambient geometric model. Building upon the intrinsic geometry of the categorified spectrum developed in the companion papers, including its tangent complex, curvature, and deformation theory, we prove that semantic realization naturally induces a canonical obstruction cohomology theory. The main contributions are fourfold. First, we construct the graded obstruction cohomology Obs(A) from a canonical descent filtration and establish a universal property: every representable descent-compatible obstruction invariant factors uniquely through Obs(A). Thus, obstruction cohomology serves as the universal source, or initial object, for such obstruction invariants. Second, we prove that obstruction cohomology satisfies an Eilenberg-Steenrod-type axiomatic framework on admissible operator-semantic systems. This framework includes long exact sequences, Mayer-Vietoris sequences, excision, and a conditional universal coefficient theorem. Third, we establish a Representation Theorem identifying obstruction cohomology, up to a degree shift by two, with the derived global sections of an intrinsic obstruction sheaf on the categorified spectrum. This representation yields a canonical duality between the tangent complex and obstruction cohomology, thereby connecting deformation theory and obstruction theory within a unified intrinsic geometric framework. Finally, we prove a Higher Contextuality Detection Theorem. Whereas existing approaches detect only a primary degree-two obstruction, the proposed theory detects contextuality at all cohomological degrees through a contextuality spectrum consisting of precisely those positive degrees in which obstruction cohomology is nonzero. Consequently, contextuality is governed by a graded obstruction cohomology rather than by a single obstruction class. These results establish a new intrinsic cohomological framework for operator-semantic systems and provide a foundation for studying higher-order semantic obstructions and contextuality beyond the reach of existing approaches.