#01Nov 5, 2026
cs.LO
Typed Knowledge over Constructive Institutions
Hans-Martin Will
Type theories designed for mathematics or program verification need only that a proposition holds; proof irrelevance in conversion erases, at the kernel level, how a claim was established. A persistent knowledge store, by contrast, must retain that record: who established a claim, by which derivation or external tool, and under which assumptions. This volume presents the theory of Eigenius---a store of typed knowledge over constructive institutions---and develops its metatheory from that requirement. The kernel theory is a dependent type theory with cumulative universes, an impredicative universe of propositions, inductive families, and an intensional identity type. Proof irrelevance is an axiom with an explicit witness rather than a conversion rule, so proofs of a proposition are provably equal yet remain distinct terms. Definitional equality is joinability under an orthogonal rewriting system---confluent, injective on type formers, and decidable by a type-free test---while a separate subsumption judgment covers universe cumulativity, codomain-covariant function types, and declared class inclusions. We prove consistency via a set-theoretic model; develop a justification logic over the kernel's propositions to track evidence and support; organize environments as an institution so that store extensions are signature morphisms and external verdicts enter via comorphisms; formulate a typed knowledge-graph theory with records, classes, and refinements; and prove the correctness of conversion, normalization by evaluation, and bidirectional type checking relative to termination. A Lean 4 formalization accompanies the volume.
#02Nov 5, 2026
math.NA
Asymptotic-Preserving Tensor Neural Networks for 3D High-Field Kinetic Equations with Applications to Uncertainty Quantification
Shi Jin, Tian-ai Zhang
Kinetic equations in the high-field regime describe particle transport dominated by electric-field acceleration and collisional relaxation. These equations are widely used in simulations of plasmas and semiconductor devices. Typical models include the Vlasov--Poisson--Fokker--Planck (VPFP) and Boltzmann--Poisson systems. Their high dimensionality and multiscale nature pose two of the major challenges to accurate and efficient numerical solution. To address these challenges, we propose an asymptotic-preserving tensor neural network (APTNN) method. Specifically, to reduce high-dimensional computational costs and memory requirements, we adopt the canonical polyadic (CP) decomposition structure of TNNs to approximate their solutions. We show that the equation residuals retain this decomposition form, allowing high-dimensional loss integrals to be evaluated through combinations of one-dimensional integrals. The resulting computational and storage costs are reduced from exponential to polynomial growth with dimension. To improve the prediction accuracy in the high-field limit, we enhance the baseline AP loss by incorporating the first moment equation. The added loss term directly constrains the limiting flux closure to reduce its approximation error in the high-field regime. The enhanced AP loss can still be evaluated dimension-wise. Numerical results show that APTNN improves prediction accuracy in the high-field regime with significantly reduced computational costs and memory usage. For a 3D3V VPFP problem with multiple random inputs, training takes less than 30 minutes on a single GPU.
#03Nov 5, 2026
cs.LG
When Scores Conflict with Preferences: Pre-Alignment Diagnosis and Adaptive Correction of Constraint Drift in DPO
Ping Liu, Yan Yan
Direct Preference Optimization (DPO) is increasingly used in settings where pairwise preferences coexist with pointwise criteria such as response length, answer correctness, or safety scores. When the two disagree, standard DPO treats pairs whose preferred output scores worse on the pointwise criterion (conflict pairs) in the same way as the remaining (matching) pairs. We characterize a systematic failure mode, constraint drift: for sufficiently small updates, conflict pairs tend to increase constraint cost while matching pairs tend to repair it, and the net change depends on their balance. This balance yields a closed-form pre-alignment diagnostic, $ρ^{*}_{\mathrm{eq}}$, computable from dataset statistics, that indicates whether a dataset is self-correcting or requires intervention. Building on this diagnostic, we propose Conflict-aware DPO (CoDPO), which down-weights conflict pairs with a strength set from $ρ^{*}_{\mathrm{eq}}$ rather than tuned, and adds an adaptive margin shift that addresses residual drift during training. Empirically, drift grows monotonically with conflict exposure on the conflict-dominated domains, and the intervention decision given by $ρ^{*}_{\mathrm{eq}}$ matches the observed drift on every domain, including one that it identifies as self-correcting. On all four domains of the main comparison, CoDPO reduces drift while largely preserving the preference learning of vanilla DPO. Compared with hard filtering and recent constrained and multi-objective baselines, CoDPO also achieves a more favorable drift-preference trade-off.
#04Nov 5, 2026
cs.LG
Test-Time Self-Play: Scaling without a Verifier
Akira Yoshiyama, Akmal Ashirmatov, Jonas Hübotter and 2 more
Test-time scaling methods spend additional compute at deployment to improve a large language model's performance on a hard target problem. Most methods rely on a verifier that scores candidate solutions for correctness, using it to select among or refine the model's outputs. In many deployment regimes, however, a strong verifier on the target problem is unavailable. We introduce Test-Time Self-Play (TTSP), an approach to test-time scaling using asymmetric self-play without reliance on a verifier. On a difficulty-calibrated subset of LiveCodeBench v6, TTSP is the best-performing method at every verifier strength we evaluate, and the gap to the strongest baseline widens as the verifier weakens; in the no-verifier limit, TTSP yields a 35\% absolute improvement in mean@8. The gains hold across a range of verifier strengths, except when the base policy has near-zero coverage of the correct answer.
#05Nov 5, 2026
cs.CL
Encoding and Propagation of Grammatical Number in Large Language Models
Jacob Solinsky, Serguei Pakhomov
Sparse autoencoders (SAEs) have become an increasingly popular means of translating Large Language Model (LLM) hidden state vectors into a human-comprehensible list of features or symbols. As such, SAEs may provide insight into an LLM's knowledge representation and internal ``mental'' grammar - topics of pragmatic interest for mechanistic interpretation of AI behavior and of theoretical interest for the study of in silico representations of human language. Language models with fewer than 100M parameters, trained on small corpora, nonetheless exhibit strong grammatical competence and discourse coherence, despite producing content of limited reliability. We use SAEs that we trained on one such model (Pythia-70M) to conduct an in-depth examination of how a single grammatical phenomenon, number agreement, is represented and processed in an LLM. We demonstrate the mechanism and the methodology by which SAEs can be used to find and propagate ``plural noun'', ``plural matrix subject'', and ``generate a plural verb'' features which cause the LLM to produce plural verb forms such as ``are'', ``do'', or ``have'' instead of ``is'', ``does'', and ``has'' in response to prompts such as ``the men in the park''. We show that intervening to remove these features from the hidden states of plural subject noun tokens causes the model to produce singular verbs instead, and vice versa. We also show that corresponding ``singular noun'' and ``generate a singular verb'' features are not as effective for this task. This asymmetry suggests that Pythia-70m makes use of a privative representation of grammatical number in English. These findings replicated with larger LLMs Gemma-2-2b and Llama-3.1-8b yielded similar results. Our main contribution is a novel approach to the use of SAEs to understanding how LLMs encode human language that addresses a number of technical and methodological challenges.