formalize fixed-point theorems

This work addresses the constructive problem of fixed-point iteration for continuous self-maps on complete lattices by introducing a purely algebraic reasoning framework, termed AIC, which eschews traditional index-based analytic methods in favor of equational logic to express fixed-point iterations as algebraic identities. The framework not only provides an algebraic reconstruction of classical results such as the Tarski–Kantorovich principle and its k-induction generalizations but also yields a novel fixed-point theorem based on limsup and liminf constructions. Furthermore, it reveals the incompleteness of finite axiom systems for this setting and demonstrates the necessity of infinitary axioms. All theoretical developments are formalized in Isabelle/HOL and fully verified automatically via Sledgehammer, thereby establishing precise completeness boundaries for the AIC system.

This work presents the first unified formalization in Lean 4 that seamlessly connects Scarf’s theorem, Brouwer’s fixed-point theorem, and the existence of mixed Nash equilibria in finite games within a single combinatorial proof framework. By leveraging an indexed-order formulation of Scarf’s theorem, room–door structures, parity arguments, and explicit embedding–projection constructions—combined with compactness and continuity reasoning—the study establishes a rigorous derivation from triangulated simplices to product spaces of simplices. The project not only delivers fully formalized combinatorial proofs of these three foundational results but also introduces BrouwerBench, a benchmark comprising 80 tasks designed to evaluate formal proof systems’ capacity to understand and reason about deep mathematical structures.

This study investigates the computational expressivity of the intuitionistic proof system μLJ—featuring a least fixed-point operator—and its cyclic extension CμLJ, aiming to precisely characterize the class of first-order total functions they represent. Methodologically, it integrates sequent calculus, realizability semantics, computability modeling, reverse mathematics, and ordinal analysis. The main contributions are: (i) the first rigorous proof that, under the “proofs-as-programs” paradigm, both μLJ and CμLJ exactly capture the first-order total functions provably total in the second-order arithmetic subsystem Π²₁-CA₀; (ii) the establishment of computational equivalence between cyclic and standard fixed-point proof systems; (iii) the development of a novel computability semantics tailored to cyclic proofs; and (iv) a reverse-mathematical foundation for the Knaster–Tarski fixed-point theorem. These results resolve a long-standing open problem on compactness boundaries, providing the first exact upper and lower bounds.

This work addresses the challenge of preserving and verifying structural correctness—such as dimensional consistency, stratification, escape properties, and numeric representations—during compilation of ML-family languages. To this end, it introduces an internal scaffolding mechanism based on fixed-point combinators, integrating closed negative types and fractional types to encode program semantic structure into MLIR intermediate representations at the middle end of compilation. Leveraging categorical constructions, the approach enables accompanying verification without requiring developers to explicitly engage with category theory, thereby ensuring structural integrity throughout the entire compilation pipeline. By exploiting MLIR’s dialect system, attribute infrastructure, and static single-assignment form, proof artifacts remain amenable to continuous toolchain validation as code is progressively lowered, achieving, for the first time, end-to-end verifiable preservation of structural properties.

This paper establishes a rigorous categorical semantics for e-graphs (equivalence graphs) within the framework of monadic categories, supporting double-pushout (DPO) rewriting. Method: The authors generalize e-graphs to monadic categories by introducing *equivalence hypergraphs* (e-hypergraphs)—a compositional structure whose vertices are algebras over a monad and whose hyperedges encode algebraic operations, thereby internalizing structural equations up to isomorphism. The approach integrates category theory, semilattice-enriched categories, and hypergraph-based combinatorial modeling to yield a sound and complete semantic framework. Contribution/Results: The resulting framework provides an algebraic and monadic foundation for equivalence reasoning in e-graph–based program optimization, and extends the formal applicability of e-graphs to SMT solving and algebraic optimization—enabling principled, categorical treatment of equational rewriting beyond traditional graph-based methods.

This work addresses the challenge in semantic modeling of scoped computational paths, where algebraic structure and topological coherence are difficult to reconcile due to incompatibility between product and quotient topologies. By constructing a topological semantics for scoped rewriting systems, it endows rewrite steps with continuous geometric realizations and characterizes name-carrying rewrites via endpoint-fixing homotopies. The paper introduces an unconditional construction of a final composable topology that explicitly resolves—rather than obscures—the product-quotient inconsistency. It establishes quadruple equivalence criteria and sufficient conditions for compact Hausdorffness, and proves, via continuous sections in a universal representation, that the coherent path quotient is homeomorphic to the standard fundamental groupoid. Formal verification in Lean 4.24.0 covers finitely generated cases such as the circle and torus, yielding normal forms and ℤ, ℤ² classifications via winding numbers, and confirming that the realization map is a continuous groupoid homomorphism and faithful under geometric completeness.

This study systematically evaluates the rigorous proof reasoning and explicit construction capabilities of large language models on Olympiad-level combinatorics problems. To this end, we introduce a benchmark comprising 100 expert-annotated competition problems, categorizing tasks into analytical (proof-oriented) and constructive (implementation-oriented) types. We propose a unified evaluation protocol that integrates rubric-guided proof assessment with deterministic verification of constructions, enhanced by a Best@4 multi-solution sampling strategy. Experimental results show that the strongest model achieves an average score of 65.4% overall (75.3% under Best@4), with markedly divergent performance across the two task types, revealing current limitations in creative mathematical reasoning—particularly on existence and construction problems. This work presents the first fine-grained distinction and joint evaluation of these capabilities, offering a new benchmark and diagnostic framework for mathematical reasoning research.

This work addresses the challenge of satisfiability in the existential theory of the reals (∃ℝ), particularly the realizability of configurations in discrete geometry, by introducing a novel paradigm termed Satisfiability Modulo Realizability. The approach encodes geometric problems as SAT instances over abstract order types and integrates diversity-driven sampling, partial realizability feedback, and a new flip-based heuristic to efficiently guide the solver away from non-realizable regions during search. Leveraging this framework, the study resolves a long-standing open problem in discrete geometry by proving that the largest point set containing no empty convex hexagon or convex heptagon has size 23, thereby substantially advancing the computational frontier in this domain.

This study investigates the provability complexity of infinitary, well-founded proof systems in linear logic extended with least and greatest fixed points. By introducing transfinite branching rules to construct infinitary proofs, and combining cut elimination with focalization strategies, the authors employ a refined rank measure on formulas to tightly control proof tree heights. This approach establishes an exact correspondence between provability in the system and levels of the hyperarithmetical hierarchy. The main contribution is the first complete characterization showing that the provability strength of this fixed-point linear logic precisely coincides with the $\omega^{\alpha^\omega}$-th level of the hyperarithmetical hierarchy, where $\alpha$ is a computable ordinal, thereby pinpointing its proof-theoretic complexity within the higher-order arithmetical hierarchy.

This work proposes a systematic approach to constructing a category of combinatorial manifolds satisfying prescribed local properties and applies it to directed topology and automata theory. By forming a coreflective subcategory within the category of relational presheaves and endowing it with a model structure via a unique factorization system, the cofibrant objects are precisely the combinatorial manifolds. This framework introduces manifold-theoretic ideas into automata theory for the first time, yielding a categorical abstraction and proof of Kleene’s theorem. Furthermore, it establishes the category of Euclidean precubical sets as a coreflective subcategory of relational precubical sets and reveals its combinatorial correspondence with the blowup construction in directed topology.