This paper develops the obstruction theory and derived geometric classification of categorified spectral objects, building upon the intrinsic geometric framework established in the companion paper. We prove that contextuality is not merely detected by geometry but is fundamentally a geometric phenomenon: it is the universal obstruction to classical geometric descent. We establish the Contextual Obstruction Theorem, which constructs a canonical obstruction class that vanishes if and only if the system admits a global classical semantic realization. We prove that this obstruction is universal among all contextual obstructions, and that its vanishing is equivalent to non-contextuality. We then establish the Contextuality-Singularity Correspondence, showing that contextuality is precisely the failure of smoothness of the categorified spectrum. We further prove that the spectral curvature tensor is naturally identified with the Atiyah-Kodaira-Spencer obstruction class, yielding the Curvature Criterion for Contextuality. We define the complete geometric spectral invariant as the tuple comprising all canonical geometric data, and prove the Completeness Theorem, establishing that geometry completely characterizes semantics. We develop a geometric stratification of the categorified spectrum by local geometric signatures, proving the Stratification Theorem and showing that contextuality is detected by the presence of strata with positive obstruction rank. Explicit computations for the Pauli system and the Mermin-Peres square demonstrate the full geometric encoding of contextual phenomena. Finally, we introduce the derived moduli functor of operator-semantic systems and formulate a Representability Conjecture asserting that this functor is represented by a derived stack. Conditional on this conjecture, we establish the Derived Spectral Moduli Principle, which unifies Hochschild cohomology, the tangent complex of the moduli stack, and the derived global sections of the tangent complex of the categorified spectrum. Thus this paper proves that contextuality is the universal obstruction to classical geometric descent, and that the derived geometry of the categorified spectrum provides a complete, functorial, and Morita-invariant classification of operator-semantic systems.
This paper develops the obstruction theory and derived geometric classification of categorified spectral objects, building upon the intrinsic geometric framework established in the companion paper. We prove that contextuality is not merely detected by geometry but is fundamentally a geometric phenomenon: it is the universal obstruction to classical geometric descent. We establish the Contextual Obstruction Theorem, which constructs a canonical obstruction class that vanishes if and only if the system admits a global classical semantic realization. We prove that this obstruction is universal among all contextual obstructions, and that its vanishing is equivalent to non-contextuality. We then establish the Contextuality-Singularity Correspondence, showing that contextuality is precisely the failure of smoothness of the categorified spectrum. We further prove that the spectral curvature tensor is naturally identified with the Atiyah-Kodaira-Spencer obstruction class, yielding the Curvature Criterion for Contextuality. We define the complete geometric spectral invariant as the tuple comprising all canonical geometric data, and prove the Completeness Theorem, establishing that geometry completely characterizes semantics. We develop a geometric stratification of the categorified spectrum by local geometric signatures, proving the Stratification Theorem and showing that contextuality is detected by the presence of strata with positive obstruction rank. Explicit computations for the Pauli system and the Mermin-Peres square demonstrate the full geometric encoding of contextual phenomena. Finally, we introduce the derived moduli functor of operator-semantic systems and formulate a Representability Conjecture asserting that this functor is represented by a derived stack. Conditional on this conjecture, we establish the Derived Spectral Moduli Principle, which unifies Hochschild cohomology, the tangent complex of the moduli stack, and the derived global sections of the tangent complex of the categorified spectrum. Thus this paper proves that contextuality is the universal obstruction to classical geometric descent, and that the derived geometry of the categorified spectrum provides a complete, functorial, and Morita-invariant classification of operator-semantic systems.
Classical Gelfand duality provides an equivalence between commutative C-star algebras and topological spaces, but fails to furnish a geometric object for noncommutative operator systems. To address this, we introduce a categorified notion of spectrum that captures the full operator-semantic structure. For an operator system A, we construct a spectral stack, denoted Spec(A), over the site of its commutative contexts. The construction proceeds in three stages. First, we encode the syntax of A via a colored operad called the synergy operad. Second, we aggregate local semantic data using a left Kan extension, providing an explicit coend formula. Third, we enforce descent via sheafification, yielding a stack that satisfies higher gluing conditions. We prove that Spec(A) satisfies a Yoneda-style universal property, making it the initial descent-complete semantic realization of A. This yields a contravariant functor from operator systems to spectral objects that admits a right adjoint, the global sections functor. Under semantic generation and descent completeness, the counit of this adjunction is an equivalence, establishing a reconstruction theorem: A is isomorphic to the endomorphisms of the structure sheaf on its spectrum. We further provide a recognition theorem characterizing the essential image of this functor and prove Morita invariance of the associated quasi-coherent sheaf categories. The construction naturally recovers classical Gelfand spectra and Bohrification as truncations of a Postnikov tower. Explicit computations for matrix algebras, Pauli systems, and the Mermin-Peres square yield a quantitative invariant of contextuality extracted from the inertia stack of Spec(A). Comparison theorems show that our framework subsumes Gelfand duality, Bohrification, and Tannaka reconstruction as special cases.
In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed as a compatible family of distributions, in order to define a cohomological obstruction for the family as a certain cohomology class. This class vanishes if the family has a global section. Thus the non-vanishing of the obstruction provides a sufficient (but not necessary) condition for the model to be contextual. We show that for a number of salient examples, including PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector configuration due to Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions, the obstruction does not vanish, thus yielding cohomological witnesses for contextuality.
Quantum measurements need not reveal preexisting values: the outcome assigned to an observable can depend on which compatible observables are measured with it. This contextuality underlies the Peres-Mermin proof of the Kochen-Specker theorem and, in quantum computation with magic states, tracks the even/odd parity of the local dimension separating the classically simulable from the genuinely quantum. ATTRIBUTION: the even/odd value classification of state-independent qudit contextuality (the parity dichotomy and the fixed obstruction value d/2) is due to Abramsky, Cercelescu and Constantin, FSCD 2024; this paper supplies its geometric and cohomological ACCOUNT, not a rediscovery of it. Organizing all measurement contexts of M_n(C) into the stack [X/PU(n)] of maximal abelian subalgebras, we show the lifting class of U(1) -> U(n) -> PU(n) is vacuous on the action groupoid U(n) x X but, on PU(n) x X, is a canonical degree-2 class of order exactly n. Its real shadow, through the signed-permutation group B_n, is a Z_2 extension splitting if and only if n is odd, with "-I is a rotation square iff n is even" as the mechanism; at n=4 it restricts to the real two-qubit Pauli class, whose pairing with the Peres-Mermin word is -1, exactly the obstruction to a noncontextual value assignment. We compute H^2 = Z/n (+) Z_2 for n >= 4 and identify the canonical class as a generator of the Z/n summand. STATUS AND SCOPE. Every step is tagged Complete, Sketch (reduced to a named standard theorem whose hypotheses are verified), or Gap, and a status table closes the paper. The finite and algebraic core is machine-checked by scripts in the public suite. The geometric layer rests on standard but non-elementary facts about differentiable-stack cohomology and the Lyndon-Hochschild-Serre spectral sequence; that bookkeeping is reduced to named standard theorems and AWAITS SPECIALIST REVIEW. This is a preprint, not a peer-reviewed article, and should be read at that strength. The finite Weyl value spectrum is developed in a companion paper. Code, data and verification suite: https://github.com/manuflog/contextuality-obstructions Note on methods: large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance, and the author takes full responsibility for the results, including any errors.
Quantum entanglement is a basic resource of quantum theory, but its usual definition assumes a fixed decomposition into subsystems. We develop an algebro--geometric framework to address quantum entanglement. For the Severi--Brauer scheme SB(A)→X of an Azumaya algebra and a factorization type d, we prove that a global locus of product states exists exactly when the associated torsor under PGLn reduces to the stabilizer Gd of the Segre variety. Thus entanglement in families is measured by a geometric obstruction to global subsystem structure.
Beyond this reduction criterion, we identify the cohomology that carries computable entangling obstructions. After determinant and incidence data L⊂R⊂K are chosen, the relevant classes lie in the mapping cone relative cohomology E∙(BL→BR). Their absolute images on X, modulo classes induced from BK, give determinant entangling obstructions beyond the Brauer class. We construct the moduli of subsystem structures as the quotient P/Gd and realize it as a locally closed locus in the relative Hilbert scheme, with a compactification by degenerations of product state loci. In the bipartite case, a chosen subsystem structure globalizes the Schmidt rank stratification to a flat filtration with base change compatible resolutions and fiberwise constant numerical invariants. Examples show that reducibility can depend on the underlying torsor, not only on the Brauer class, with an interpretation by entangling monodromy.
Recent work by Abramsky and Brandenburger used sheaf theory to give a mathematical formulation of non-locality and contextuality. By adopting this viewpoint, it has been possible to define cohomological obstructions to the existence of global sections. In the present work, we illustrate new insights into different aspects of this theory. We shed light on the power of detection of the cohomological obstruction by showing that it is not a complete invariant for strong contextuality even under symmetry and connectedness restrictions on the measurement cover, disproving a previous conjecture. We generalise obstructions to higher cohomology groups and show that they give rise to a refinement of the notion of cohomological contextuality: different "levels" of contextuality are organised in a hierarchy of logical implications. Finally, we present an alternative description of the first cohomology group in terms of torsors, resulting in a new interpretation of the cohomological obstructions.
Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting observables, which are not assumed to be jointly measurable, cannot be consistently ascribed deterministic values even if one enriches the description of a quantum state. Here, we consider the geometrically dual objects of noncommutative operator algebras of observables as being generalisations of classical (deterministic) state spaces to the quantum setting and argue that these generalised state spaces represent the objects of study of noncommutative operator geometry. By adapting the spectral presheaf of Hamilton–Isham–Butterfield, a formulation of quantum state space that collates contextual data, we reconstruct tools of noncommutative geometry in an explicitly geometric fashion. In this way, we bridge the foundations of quantum mechanics with the foundations of noncommutative geometry à la Connes et al. To each unital C*- algebra A we associate a geometric object—a diagram of topological spaces collating quotient spaces of the noncommutative space underlying A —that performs the role of a generalised Gel'fand spectrum. We show how any functor F from compact Hausdorff spaces to a suitable target category C can be applied directly to these geometric objects to automatically yield an extension F~ acting on all unital C*-algebras. This procedure is used to give a novel formulation of the operator K0-functor via a finitary variant K~f of the extension K~ of the topological K-functor. We then delineate a C*-algebraic conjecture that the extension of the functor that assigns to a topological space its lattice of open sets assigns to a unital C*-algebra the Zariski topological lattice of its primitive ideal spectrum, i.e. its lattice of closed two-sided ideals. We prove the von Neumann algebraic analogue of this conjecture.