analytical derivation

This work proposes a model-class-agnostic structural learning method to identify latent probabilistic structures directly from empirical probability tensors. Leveraging the vanishing binomials of toric models as algebraic signatures, the approach introduces minimal invariant constraints (MICs) as fundamental units that generalize classical notions of independence. By exploiting the correspondence between ideals and varieties, the method performs signature matching within an enumerable class of Kronecker-stack configuration matrices. This study represents the first systematic application of algebraic statistics to structure discovery in computational linguistics, demonstrating effectiveness on both synthetic and large-scale real-world linguistic data. The identified rank-one structures correspond to interpretable word sets, thereby establishing a novel pathway for applying algebraic statistics in this domain.

This work addresses the challenge of ensuring numerical stability, computational correctness, and physical consistency in high-stakes or scientific AI applications, where traditional post-training validation falls short. The authors propose embedding algebraic structural constraints directly into the model design phase to enable decidable correctness guarantees. Their key innovation lies in the first integration of Hindley-Milner type inference over finitely generated Abelian groups with a computable restriction of Solomonoff’s universal prior, yielding information-theoretically optimal hypotheses. This framework further combines dimensional type systems, program hypergraphs, graded Clifford algebraic inference, forward coeffect analysis, and exact posit accumulation to preserve model invariants while eliminating the cumulative computational overhead inherent in existing reliability approaches across deployment, inter-layer propagation, and inference stages.

This paper proposes a novel paradigm for machine learning grounded in abstract algebra, aiming to transcend the limitations of statistical modeling and numerical optimization by unifying data-driven tasks and formal reasoning problems—such as deriving Hamiltonian cycles from logical specifications. Methodologically, task objectives and data are encoded as axioms within an algebraic system; generalizable models are then constructed directly via subdirect decomposition, eliminating the need for validation-set-based hyperparameter tuning, enabling additive model composition, and guaranteeing asymptotic convergence to the intrinsic structure of the data. The core contribution is the first systematic adoption of subdirect decomposition as the foundational mechanism of learning, enabling purely symbolic, axiom-to-model derivation. Experiments demonstrate competitive performance against tuned multilayer perceptrons on MNIST, Fashion-MNIST, CIFAR-10, and medical image classification; moreover, the framework successfully solves specification-driven combinatorial problems without explicit search.

This work addresses the problem of modeling invariant functions over symmetric matrices (under conjugation by permutations) and point clouds (under rotations, reflections, and point permutations). Methodologically, inspired by Galois theory, we construct the first lightweight universal approximator framework that yields separating invariant features of dimensionality only $O(n^2)$ for symmetric matrices and further optimizes to $O(n)$ for point clouds—breaking the bottleneck of traditional high-dimensional invariant representations. Our theoretical foundation integrates invariant algebra and generating sets of rational function fields, coupled with DeepSets architecture and orbit-separation analysis under group actions. Experiments on molecular property regression and point cloud distance prediction empirically validate almost-everywhere orbit separation, enabling universal approximation of weighted graph functions. The proposed framework significantly enhances both expressive power and computational efficiency of invariant representations.

The “black-box” nature of large language models hinders rigorous performance verification. Method: We propose the first framework unifying mechanistic interpretability with formal performance verification: via weight-level mechanistic reverse-engineering, we decompose small-scale Transformer behavior on Max-of-K tasks into human-understandable algorithms and generate compact, machine-verifiable mathematical proofs (e.g., accuracy lower bounds). Contribution/Results: This establishes the first end-to-end closed loop from mechanistic understanding to formal proof. We discover that proof length positively correlates with both mechanistic insight depth and bound tightness, and identify “structural deficiency errors”—gaps between inferred mechanisms and true computational structure—as the key bottleneck limiting proof conciseness and fidelity. Validated across 151 random seeds and 4 values of K, our framework constructs 102 distinct strategies; empirical results confirm that shorter proofs reflect deeper mechanistic understanding, while higher-fidelity interpretations yield tighter performance bounds.

Current approaches to automatically verifying natural language mathematical proofs lack modularity and unambiguous structure, especially in advanced mathematics. This work proposes pseudo-formalization (PF), a representation that decomposes proofs into self-contained modules, each explicitly specifying premises, conclusions, and inference steps. We introduce a block verification (BV) algorithm that leverages large language models to independently validate each module. Our method uniquely combines the structural rigor of formal proofs with the expressive flexibility of natural language. Evaluated on both Olympiad-level and research-grade mathematical benchmarks, it substantially outperforms LLM-as-judge baselines, achieving marked improvements in both precision and recall for error detection. We also release ArxivMathGradingBench, an open-source benchmark for evaluating proof verification systems.

This study addresses the joint identification and counterfactual analysis in incomplete structural models featuring support and moment constraints. The authors embed counterfactuals directly into an augmented structural model, departing from the conventional “estimate-then-simulate” paradigm. By leveraging support function methods, they simultaneously achieve identification and inference, revealing a fundamental isomorphism between the two tasks. A key contribution is the formulation of irreducibility conditions that explicitly characterize all support implications. Under mild regularity assumptions, the support function approach preserves sharpness with respect to the moment closure—even in counterfactual settings where traditional sharpness fails. Moreover, for irreducible models, the identified set and the moment closure are statistically indistinguishable in finite samples.

This work addresses the limitations of current automatic formalization research, which predominantly focuses on well-supported mathematical domains and relies solely on kernel acceptance rate as a quality metric, thereby neglecting the practical needs of underrepresented areas such as numerical analysis and lacking comprehensive evaluation. For the first time, we employ a Lean 4 coding agent to formalize an entire textbook—*Numerical Methods for Ordinary Differential Equations*—from scratch and introduce a three-dimensional evaluation framework that jointly assesses semantic correctness, Mathlib reusability, and cross-file reusability. Through LLM-as-judge, semantic validation, and dependency analysis, we uncover pervasive issues in existing systems, including incomplete statements and weakened assumptions, demonstrating that kernel acceptance rate substantially overestimates formalization quality. Our approach establishes a reproducible, multidimensional auditing paradigm for trustworthy automated formalization.

In high-dimensional statistical inference, a computational-statistical gap often separates detection and recovery tasks, yet existing low-degree polynomial methods struggle to directly characterize computational lower bounds for recovery. This work proposes a general framework that leverages algorithmic continuity and cross-validation reductions to translate weak low-degree detection advantages into conditional computational lower bounds for recovery. The approach provides, for the first time, a model-agnostic and conceptually simple unification of the detection-recovery gap across multiple canonical models. Applying this framework to six problems—including planted matrix, dense subgraph detection, and the stochastic block model—it either reproduces known computational thresholds or offers new evidence for them, thereby demonstrating the framework’s broad applicability and effectiveness.

This work investigates the problem of efficiently recovering the parameters of a black-box linear model using factual, counterfactual, and robust counterfactual queries. It introduces the first general mathematical framework capable of characterizing the model’s decision regions without explicitly reconstructing its parameters, and systematically analyzes how different distance metrics affect query complexity. Theoretically, the study establishes that under differentiable distance functions, a single counterfactual query suffices to fully recover the model; in contrast, under polyhedral distances, the required number of queries grows linearly with the input dimension, doubling in the robust setting. These findings highlight the pivotal role of the choice of distance function in determining both the extractability and security of linear models.