use genericity arguments

This study addresses statistical ill-posedness in distributional models—such as non-identifiability, singular Fisher information, and moment indeterminacy—arising from geometric degeneracies. It introduces transversality theory from differential topology to systematically characterize well-behaved models through the geometric properties of kernel-induced feature maps. Leveraging tools including Sard’s theorem, smooth mappings, jet spaces, and parametric transversality, the work unifies the explanation of six classical pathological phenomena, including representation collapse and the Behrens–Fisher problem, demonstrating that these issues are non-generic under typical conditions. The analysis provides a geometric perspective and theoretical foundation for robust statistical modeling, while presenting the underlying topological machinery in an accessible, pedagogically oriented manner.

This work investigates the distribution of critical points for degenerate quadratic optimization problems over algebraic varieties in overparameterized machine learning. When the number of training samples is smaller than the model’s degrees of freedom, the empirical loss function becomes rank-deficient; to address this, we introduce a projection morphism that reduces the degenerate problem to a non-degenerate one and systematically characterize the ramification locus as the dominant structural feature governing highly degenerate critical geometries. Innovatively extending Euclidean Distance Degree (EDD) theory to degenerate metric settings, we establish, for the first time, a rigorous correspondence between the ramification locus in algebraic geometry and critical points in overparameterized deep learning. This yields an explicit closed-form formula for counting critical points, validated empirically on canonical neural network architectures with high predictive accuracy—thereby substantially broadening the applicability of EDD theory to modern optimization and deep learning.

A fundamental information asymmetry exists between measurement-based mathematical representations and optimization-principle-based physical representations of experimental setups: certain partial orders require infinite information for physical realization, yet admit finite-information mathematical descriptions. Method: We introduce the “Debreu dimension”—a novel dimension concept unifying geometric intuition with order-theoretic rigor—thereby bridging the semantic gap between Dushnik–Miller dimension and classical geometric dimension. Under countability assumptions, we develop an explicit, geometry-guided constructive framework, integrating order theory, real-valued monotone function analysis, and set-theoretic tools. Contribution/Results: This work establishes a comprehensive dimensional classification scheme for preordered spaces, significantly enhancing both the geometric unification of partially ordered sets and the classification accuracy of real-valued monotone representations.

This paper addresses the lack of an information-theoretic characterization and a point-to-set principle for finite-state dimension (FS dimension). It introduces a precise quantification of the information content of real numbers under finite precision, integrating effective dimension theory, finite-state automaton complexity, and relativized algorithmic information theory. The work establishes, for the first time, a point-to-set principle for FS dimension and thereby defines a robust notion of relative normality. It then rigorously proves the equivalence between relative normality and FS dimension. The main contributions are: (1) an information-theoretic, exact definition of FS dimension; (2) a necessary and sufficient characterization linking relative normality to FS dimension; and (3) a foundational framework enabling future investigation of its equidistribution properties.

This work constructively establishes continuous and algebraic domain theory, overcoming reliance on the law of excluded middle, the axiom of choice, and universe resizing assumptions inherent in classical frameworks. Methodologically, it redefines continuity for dcpos within univalent type theory using categorical ideas of continuity, employs propositional truncation to ensure continuity is a decidable property, and rigorously constructs small-basis algebraic and continuous dcpos without size assumptions. Key contributions include: (i) the first formalization of core domain-theoretic concepts—such as algebraicity, continuity, and Scott-continuity—within a constructive, univalent setting free of choice and universe resizing; (ii) a characterization of small-basis dcpos as precisely those completely represented by small ideals; (iii) verification that the Scott $D_infty$ model forms an algebraic dcpo with small basis; and (iv) a fully constructive semantic foundation for the $lambda$-calculus. All results are formally verified in Agda.

This work addresses the limitations of traditional generalization analyses, which rely on the often unverifiable assumption of independent and identically distributed (i.i.d.) data and thus struggle to accurately characterize model performance on unseen data. The paper proposes a deterministic generalization analysis framework that dispenses with any prior probabilistic assumptions. By examining the sensitivity of optimization solutions to data perturbations, it decomposes the generalization error into geometric and probabilistic components, achieving their first-ever decoupling. The framework expresses generalization bounds via a variational principle, leveraging deterministic perturbation analysis and optimization sensitivity theory to capture the discrepancy between in-sample and out-of-sample performance. Error terms are evaluated through posterior statistical hypotheses, enabling the recovery of conventional high-probability or expected generalization guarantees—all without requiring distributional assumptions.

This work resolves the long-standing Jung–Tix problem in domain theory: whether there exists a subcategory of continuous domains that is cartesian closed and closed under (sub)probabilistic powerdomains. To this end, we introduce the category of finitely valuation-approximable domains (FVA), constructed via FS-approximations of the identity on finite posets. By leveraging coreflective embeddings and a finite separation saturation theorem, we prove that FVA is closed under function spaces, finite products, Scott-continuous retracts, and (sub)probabilistic powerdomains. This constitutes the first demonstration of a subcategory of continuous domains simultaneously enjoying cartesian closure and closure under probabilistic powerdomains, thereby affirmatively settling the generalized Jung–Tix problem and confirming the well-behaved nature of the valuation monad within this setting.

While existing machine learning models can process inputs of arbitrary dimensions—such as graphs or point clouds of varying scales—they lack rigorous theoretical guarantees of universality, as classical universal approximation theorems apply only to fixed-dimensional settings. This work introduces the first systematic definition and verification framework for universality across arbitrary input dimensions. By constructing an infinite-dimensional topological space encompassing all finite-dimensional inputs and their limits, and leveraging symmetry analysis together with the theory of compact families, we reveal fundamental limitations in the cross-dimensional universality of mainstream architectures and propose a concise, effective correction. Building upon this framework, we design a novel model that achieves provable universality over arbitrary dimensions, with formal guarantees rooted in function approximation theory.

This work addresses the limitations of existing Taylor expansion theory, which struggles to apply to classical web-based models of linear logic such as Köthe spaces and finiteness spaces, particularly when dealing with non-positive coefficients and partial summation structures. The paper introduces a general web-based semantic framework that accommodates partial summation and, for the first time, extends Taylor expansion theory to settings involving non-positive coefficients. This unified approach encompasses coherence spaces, probabilistic coherence spaces, finiteness spaces, and Köthe spaces. By integrating semantic tools from linear logic, differential λ-calculus, sequence space theory, and absolute convergence analysis of formal power series, the authors demonstrate that all major web-based models satisfy a generalized form of Taylor expansion, thereby broadening the mathematical foundations and applicability of differential program semantics.

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.