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Automated computational science could accelerate discovery by selecting, adapting and developing numerical methods. This prospect requires mathematical knowledge that AI systems can interpret and reuse with explicit guarantees. Here we present FARMS, a faithful and efficient foundation that transforms computational mathematics into AI-ready computing knowledge. FARMS connects formal statements to source context, algorithmic meaning and quantitative guarantees through source preparation, reusable representations and semantic review. Across 171 targets in randomized numerical linear algebra and numerical partial differential equations, 157 were accepted by the pipeline, with mean estimated costs of US \$0.85-2.29 per target across sources. Lean-checked implications from explicit source premises connect selected statements to computational skills, illustrated by a heat-flow solve and wave-response compression. FARMS therefore supports an affordable, reusable knowledge foundation for trustworthy automated computational science.

Lean's rich expressiveness supports higher-order functions, inductive datatypes, and user-defined abstractions, making it challenging to translate proof obligations into first-order Satisfiability Modulo Theory (SMT) queries and reconstruct kernel checked proofs. We present crush, an extensible SMT-backed proof automation framework for Lean. To bridge the gap between Lean's higher-order constructs and the solver's first-order logic, crush uses defunctionalization to encode function values and their applications. We formalize this translation for a supported higher-order fragment and prove its soundness. Moreover, crush provides metaprogramming interfaces for extending both Lean-to-SMT translation and the reconstruction of Lean proofs from solver certificates without modifying the translation pipeline. We implement crush as both a tactic and a backend for other third-party automation systems. We show that across four such use cases that target verification and general proof automation, crush discharges more obligations than the competing backends and reconstructs more kernel-checked proofs than the baselines.

AI has made remarkable progress in mathematics, solving challenging problems with machine-checkable proofs in Lean, yet typically relying on predefined mathematical definitions and formal specifications. This raises a natural question: can it reason rigorously about an unknown world? We introduce the Algebraic World Model (AWM), a framework that turns interaction with unfamiliar environments into explicit mathematical models supporting formal reasoning. AWM explores the environment and develops executable Python hypotheses, continually testing and revising them against observations. It then structures these hypotheses into typed objects and shared rules, simplifies their structure through category-theory-guided transformations, and formalizes the resulting model in Lean 4 to prove reachability and impossibility. AWM can formally prove that a goal is unreachable, even beyond explored trajectories. On two impossible game levels, it constructs closed, kernel-checked Lean proofs excluding every legal winning sequence in all six runs, while Claude Opus 5.5 and GPT-6 Astra, equipped with coding tools, produce none in twelve baseline runs. Four of AWM's six initial bounds are correct, compared with one for Opus 5.5 and none for GPT-6 Astra; given only feedback that a cheaper win exists, AWM revises its rules and constructs new proofs, reaching six of six correct bounds. Moreover, AWM produces interpretable mathematical models whose rules can be inspected and tested step by step. Across all 25 public ARC-AGI-3 games, these models enable Lean to replay the winning trajectories of all 183 levels and generalize beyond observed play, predicting unseen transitions and detecting changes in game mechanics. With Claude Opus 5, AWM clears all levels with a score of 99.9, while its certified winning trajectories use only 43% of the human baseline's actions.

Recently, Guilmant, Lefaucheux, Ouaknine, and Worrell [2] proved decidability of the termination problem for linear constraint loops over two real-valued variables, thus settling a wellknown open problem. Their algorithm is based on a theoretical result that such a loop does not terminate if and only if it has a witness of an infinite run of a specific shape. By analysing the structure of the witness carefully, we show that termination such loops can be decided in polynomial time, improving on the polynomial space bound argued by Guilmant et al. We also present an efficient implementation of the polynomial-time algorithm based on the satisfiability modulo theories (SMT) solver Z3.

Logic-based workflow patterns capture recurring behavioural structures together with reusable temporal specifications. Their composition raises distinct problems as workflow structures become richer, including instantiation, nesting, conditional execution, repeated occurrences, and traceability of generated formulae. This paper proposes a hierarchy of Pattern Composition ($ΠC$) procedures, from $ΠC_0$ to $ΠC_4$, that organizes these requirements as progressively richer compositional capabilities. The hierarchy separates compositional complexity from the expressive power of the temporal-logic fragment, represented by $TL_j$, yielding a framework $ΠC_i[TL_j]$ for developing logical workflow specifications. We develop activation-aware Pattern Composition ($ΠC_2$), which distinguishes the structural presence of a nested pattern from its execution in a particular workflow context. We formalise its recursive procedure, analyse its properties, and define an evaluation protocol for conditional activation, conservativity under mandatory composition, and specification growth. The hierarchy also characterizes selective occurrence qualification for iterative patterns ($ΠC_3$) and provenance for tracing generated formulae to their sources ($ΠC_4$). Finally, we outline $ΠC_{\mathrm{Assist}}$, an orthogonal mode in which AI proposes and refines pattern expressions subject to the hierarchy's compositional checks.

We devise two complementary characterizations of hereditary history-preserving bisimilarity (HHPB): a denotational one, over stable configuration structures, and an operational one, in a reversible process calculus. Our characterizations rely on forward-reverse bisimilarity (FRB) augmented with backward ready multiset equality, which is able to distinguish between autoconcurrency and autocausation. With respect to previous work, we thus shift the emphasis from uniquely identifying events, as done in other characterizations, to counting occurrences of identically labeled events associated with incoming transitions. This yields a more lightweight behavioral equivalence than HHPB: our bisimulations are still binary relations instead of ternary ones and do not need to incrementally build any labeling- and causality-preserving bijection for occurred events. We show that our characterizations are valid in the absence of non-local autoconflicts. We then study the logical foundations of these characterizations by relating event identifier logic, which captures HHPB, and backward ready multiset logic, developed for our extension of FRB.

We present Vampire-Hammer, a hammer for the Lean interactive theorem prover that discharges goals using the Vampire automated theorem prover and reconstructs proofs checked by the Lean kernel. Vampire-Hammer uses the Lean-auto framework to process goals and premises before dispatching them to Vampire for proof search. Doing so, we rely on standard TPTP problem formats and annotated TSTP proofs that support proof reconstruction. With a 10-second budget on 196,083 eligible Mathlib theorems, Vampire-Hammer solves 21,158 problems not solved by other evaluated tools within the same budget.