obstruction theory

This work addresses the failure of local consistency assumptions in existing continuous generative causal models when confronted with nontrivial homological structures—such as structural conflicts or latent confounding. The authors formalize structural causal models as cellular sheaves over Wasserstein space and introduce a computable causal sheaf Laplacian by integrating entropy-regularized optimal transport with cohomological theory. They further propose an O(1) memory-efficient reverse-mode gradient algorithm. By unifying cellular sheaf theory, nonlinear Fokker–Planck dynamics, and Sinkhorn optimization, this study establishes the first sheaf-theoretic foundation for generative causal modeling. Applied to counterfactual inference in single-cell RNA sequencing, the proposed topological causal scoring significantly enhances sensitivity to complex causal structures, revealing that thermodynamic noise can traverse topological barriers via “entropic tunneling.”

This work establishes a rigorous semantic foundation for cryptographic constructions such as Σ-protocols by leveraging Grothendieck topologies and sheaf theory. Drawing on tools from topos theory—including internal logic, subobject classifiers, and descent theory—it systematically develops a theoretical pathway from torsors to sheaf topoi, unifying local consistency and global realizability within a categorical framework. By integrating structural elements such as the Yoneda lemma, Kan extensions, and intuitionistic logic, the paper offers the first sheaf-theoretic formalization of simulatability and local–global reasoning in Σ-protocols, thereby providing a novel and mathematically precise semantic basis for cryptographic protocols.

Existing program analysis techniques lack a unified mathematical framework capable of simultaneously addressing type checking, vulnerability detection, and behavioral equivalence verification. This work proposes the first unified model based on Čech cohomology, representing program semantics as a presheaf over the category of program sites. In this framework, the zeroth cohomology group \(H^0\) captures globally consistent typing, while the first cohomology group \(H^1\) characterizes errors or failures of equivalence arising from local inconsistencies. By integrating tools from algebraic geometry to model Python semantics and leveraging the Mayer–Vietoris sequence, the approach enables incremental obstruction analysis and minimal repair counting. The resulting tool, Deppy, achieves 100% vulnerability recall, 99% zero-false-positive equivalence accuracy, and 98% specification conformance accuracy on a benchmark suite of 375 programs, substantially outperforming state-of-the-art type checkers such as mypy and pyright.

In dependent type theory, controlling the granularity of definition unfolding has long posed a dilemma: global unfolding renders proofs brittle, while manual annotation sacrifices automation and robustness. This paper introduces a novel localized unfolding mechanism grounded semantically in extension types, whereby definitions remain inert by default and admit on-demand, local, context-sensitive selective unfolding. Departing from traditional global toggle paradigms, we design a core calculus within homotopy type theory and implement a formally verified prototype in the cooltt proof assistant. Theoretically, we establish a normalization theorem ensuring computational soundness. Empirically, our approach significantly improves proof stability, effectively curbs target-term size explosion, and reconciles expressive reasoning power with maintainability—demonstrating both foundational advancement and practical viability.

This paper addresses the lack of a unified framework for rewriting theory in strict higher-dimensional categories. Methodologically, it reconstructs algebraic rewriting theory around the central notion of *polygraphs*, embedding them systematically into the setting of strict $n$-categories and homotopical algebra, thereby developing a comprehensive $n$-polygraph theory; notably, it proves—within the folk model structure—that $n$-polygraphs are cofibrant objects, the first such result. The main contributions are threefold: (i) a deep unification of polygraphs with strict higher-dimensional categories and homotopical algebra; (ii) a systematic generalization of low-dimensional confluence results to arbitrary dimension; and (iii) a universal criterion and algorithmic procedure for verifying coherence of higher-dimensional algebraic structures. These advances provide novel structural tools for higher algebra, type theory, and formal verification.

This work addresses the challenge of determining whether a scientific theory can be transferred to a new context or requires an extension of its representational language. It proposes a diagnostic framework grounded in finite sheaf theory, formalizing “obstructions” as computable metrics—such as residual fitting error, overlap incompatibility, and constraint violation—through the local–global structure of source, overlap, target, and validation diagrams. The approach introduces constellation kernels as probes for representational similarity and, for the first time, translates the sheaf-theoretic notion of obstruction into an actionable diagnostic tool for AI-based scientific agents. This enables clear discrimination between internal deformations of a theory and cases necessitating genuine language expansion. Evaluated on the transition-card benchmark, the method accurately ranks transfer obstructions, identifies minimal-obstruction candidates, and cleanly separates the two types of theoretical change.

This study addresses the local-to-global consistency problem for finite closure systems over overlapping domains: given a collection of local closure operators, can they be coherently extended to a global conservative closure system? To this end, the paper introduces the “atlas-induced closure” construction and establishes a semantic correspondence between indexed truth spaces and closure systems, revealing an intrinsic connection between regional inclusion relations and closure derivations. The central contribution is a finite, directly computable criterion—termed “local visibility obstructions”—that precisely characterizes the necessary and sufficient conditions for the existence of a global conservative realization. In the absence of such obstructions, the atlas-induced closure yields the unique global conservative extension, and an efficient algorithm is provided for detecting these obstructions.

This work formalizes causal reasoning within the framework of toposes, addressing the rigorous verification of interventions, mechanism grafting, and an intuitionistic do-calculus. Leveraging Cubical Agda, it models causal worlds as presheaf categories (1-toposes), defines interventions via characteristic maps of subobject classifiers, and conducts reasoning in the internal intuitionistic language. The main contributions include the first machine-verified core of a topos-theoretic causal model; a correction of the missing Lawvere–Tierney axiom by introducing the double-negation topology; the identification of a novel phenomenon termed the “contextuality barrier”; a proof that interventions and Pearl’s rules are j-stable under arbitrary topologies; and the establishment of an equivalence between counterfactual transportability and j-stability.

This work establishes a fundamental theoretical barrier to the verification of AGI alignment, demonstrating that correctness in alignment is structurally unverifiable. By integrating formal methods from mathematical logic, computability theory, and descriptive complexity, the paper introduces the “Alignment Unverifiability Theorem” and the “Finite-Structure Unverifiability Theorem,” thereby articulating—for the first time—a trilemma among soundness, completeness, and tractability. This trilemma arises intrinsically from limits imposed by descriptive complexity. The study further reveals that prevailing AI safety mechanisms are not ad hoc fixes but structural compromises that deliberately sacrifice logical expressiveness to obtain decidable fragments of safety. Consequently, the work delineates the theoretical boundaries and feasibility limits inherent to aligning artificial general intelligence.

This work addresses the challenge of efficiently maintaining first-layer sheaf cohomology under dynamic edits to one-dimensional cellular complexes, where traditional methods incur $O(n^3)$ recomputation time per edit and thus struggle with large-scale streaming updates. Under a local geometric boundedness assumption, the authors propose a lazy-update mechanism that combines cellular sheaf theory, local block decomposition, and the Mayer–Vietoris sequence to defer local feature resolution and global assembly until necessary. This approach achieves, for the first time, $O(1)$ edit latency—constant with respect to the total complex size $n$—for sheaf cohomology in dynamic settings. Theoretical analysis shows that such locality is unattainable for nontrivial sheaves without partitioning. Experiments on a Barabási–Albert graph with $5 \times 10^6$ vertices and $1.7 \times 10^7$ edits confirm the method’s efficacy: median edit latency is only 35 microseconds, with results exactly matching those of batch recomputation.