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We characterize uniform factor-balancedness in Arnoux-Rauzy words, generalizing the work of Espinoza, Popoli, and Stipulanti on ternary alphabets to finite alphabets of any size. In particular, we prove that an Arnoux-Rauzy word $\mathbf{x}$ is uniformly factor-balanced if and only if $\mathbf{x}$ has bounded weak partial quotients.
For every group $G$, of arbitrary cardinality, we construct a right-invariant metric with at most $32$ values whose isometries are exactly the permutations preserving every right-invariant metric on $G$. The proof combines subgroup-entry ranks and sign-variation colorings with a short-word rigidity theorem of Leemann and de la Salle. Their nonabelian orientation-rigidity theorem and direct regular-subgroup arguments yield the complete classification of natural groups in the right-translation sense: an abelian group $A$ is natural if and only if $2A=A$ or $2A=\{0\}$, and a nonabelian group is natural if and only if it is not generalized dicyclic. In particular, the additive group of every field is natural. The bound improves to $17$ for abelian groups and $5$ for Boolean groups, and the Boolean bound is sharp: $C_2^3$ admits no such metric with fewer than five values. Complementary constructions give one countable-valued hull metric realizing precisely the affine sign isometries simultaneously on all subgroups containing fixed coordinate markers, and signed-basis metrics with at most $p+5$ values over $\mathbb{F}_p$ for odd $p$. No other bound is claimed optimal, and no uncolored graphical regular representation is asserted.
Jev is a non-generative "System One" model that assigns probabilities to predefined answer options and cannot answer outside them. Its accuracy and calibration on medical question-answering and case-based diagnostic-reasoning tasks are unknown. We evaluated Jev 1.13 on four medical benchmarks: MetaMedQA, PubMedQA, DiagnosisArena-MCQ and the NEJM Case Challenges. GPT-6 Sol, with (medium) and without reasoning, was the reference. The primary outcome was top-1 accuracy; key secondary outcomes were calibration, selective prediction and recognition of unanswerable questions. All 8,469 requests returned a valid answer. Jev's accuracy was similar to that of GPT-6 Sol with medium reasoning on PubMedQA (78.4% vs 78.2%;), lower on MetaMedQA (74.8% vs 82.7%) and much lower on DiagnosisArena-MCQ (59.8% vs 82.4%;) and the NEJM cases (61.8% vs 82.4%). On MetaMedQA, Jev's probabilities were the best calibrated (expected calibration error 0.063 vs 0.146), and its answers with a probability of at least 0.9 (52.9% of questions) were 93.4% accurate, but GPT-6 Sol was as accurate when it accepted a similar proportion of questions. On DiagnosisArena-MCQ, Jev's probabilities discriminated poorly (AUROC 0.645 vs 0.768). Of the 162 questions whose correct answer was "I don't know or cannot answer", Jev chose that option for 10.5% (GPT-6 Sol, 8.6%). Median latency was 0.27-0.31 s; all 2,823 items cost USD 0.08. Jev was fast and inexpensive, and its accuracy was similar to that of a frontier LLM on research abstracts but lower on examination questions and much lower on complex diagnostic cases. Task-specific validation is required before clinical use.
Multi-tenant computing enables providers of commercial cloud quantum processors to rent disjoint sectors of a device to independent users. Average gate error, which cloud quantum computing providers report, does not determine how much one tenant learns about another. We propose an information-theoretic notion of information leakage across co-tenancy boundaries stemming from quantum state distinguishability. We measure this leakage on commercially-available 156-qubit (IBM Kingston) and 20-qubit (IQM Garnet) devices. Boundaries with identical benchmarked error can offer significantly different amount of information leakage because standard reported measures of error are blind to coherent-versus-stochastic nature of the error while the proposed notion of information leakage is not. A uniform Pauli randomisation implemented over the victim's whole register is used as a defence mechanism to reduce the information leakage to zero. The defence theoretically does not incur a fidelity cost, but the experiments show a non-trivial degradation caused by accumulation of errors. We provide a specific call-for-action to the providers of quantum cloud computing to report information leakage in addition to standard error rates in their device datasheet to enable users to compute privacy and security risks prior to engagement with the device.
On-policy distillation (OPD) is becoming an important component of large language model (LLM) post-training for transferring the reasoning capability of a strong teacher LLM to a weaker student LLM. OPD trains the student by minimizing the reverse KL divergence between the teacher and the student via rollouts generated by the student's policy. However, estimating the gradient of the reverse KL divergence in OPD remains a challenge. Using only the sampled token from the student-generated rollout is computationally cheap but provides limited distributional supervision, which will degrade accuracy. In addition, using the full vocabulary provides complete distributional supervision but is computationally expensive. Therefore, recent works propose Top-$k$ OPD (TK-OPD) that use selected top-$k$ tokens, which provides richer distributional supervision than sampled-token estimation at substantially lower computational cost than full-vocabulary estimation. Unfortunately, using only the selected top-$k$ tokens induces bias, leading to accuracy degradation, as the probability mass outside the selected top-$k$ tokens is discarded. To address the bias of TK-OPD, we propose Tail-Corrected Top-$k$ On-Policy Distillation (TT-OPD). It preserves the advantages of TK-OPD, including rich distributional supervision and low computational cost, while providing an unbiased estimator of the gradient of the reverse KL divergence. The key insight of TT-OPD is to use not only the selected top-$k$ tokens, but also the sampled token from the student-generated rollout, thereby recovering the discarded probability mass in expectation, avoiding the bias. Experimental results demonstrate that TT-OPD significantly outperforms other tested OPD variants.
Logical reasoning remains a major challenge for large language models (LLMs), particularly on structured problems that require precise constraint tracking, consistency preservation, and multi-step deduction. This challenge is especially acute for small-scale LLMs, which are more prone to producing inconsistent, redundant, or brittle reasoning trajectories. Existing approaches for improving logical reasoning largely optimize for final-answer correctness, providing only weak supervision over the intermediate reasoning process. In this work, we propose SPRING: (Solver-guided Process Rewards for Novel LogIcal ReasoNing Step Generation). SPRING uses SMT solver as a training-time verifier of intermediate reasoning steps to provide process-level supervision. It introduces the notion of a novel reasoning step, namely, a step that is logically valid, consistent with the evolving reasoning state, and not already implied by previously accepted non-contradictory deductions. Based on this solver-based assessment, it designs process rewards that encourage novel inferential progress while penalizing contradictory and uninformative reasoning steps. Evaluation across three logical reasoning benchmarks, ZebraLogic, AR-LSAT, and Knights and Knaves, and four LLMs shows that SPRING consistently outperforms base LLMs, outcome-only reward baselines, and Logic-LM. On ZebraLogic, SPRING improves puzzle accuracy by up to 49.71 and 15.43 points over the base LLM and strongest outcome-only baseline, respectively. On AR-LSAT, it improves overall accuracy by up to 64.93 and 12.14 points, respectively. On Knights and Knaves, SPRING achieves up to 93.14 puzzle accuracy and 96.05 person accuracy.
We introduce Triangular Resampling (TR), a post-training method for mitigating long-horizon error accumulation in motion diffusion models. Built on FloodDiffusion's triangular denoising schedule, TR addresses the mismatch between ground-truth-derived training windows and model-generated inference states. Replacing only completed motion history leaves this mismatch unresolved in partially denoised states within the active window. TR therefore extends rollout-based training to these states, using ground-truth clamping to limit excessive drift. For each replayed sample, TR draws one denoising threshold, shared across latent positions and replay updates, and replays multi-step triangular denoising without gradient tracking. After each update, states below the threshold are replaced with noise-matched ground truth, while those at or above it retain model predictions. The resulting latent window enters the standard training update. This rollout construction supports both supervised training (TR) and distribution matching (TR-DMD). On 120-second motion generation from HumanML3D test prompts, TR and TR-DMD achieve state-of-the-art FID AUC within their respective non-DMD and DMD comparison groups. Supervised TR reduces FID AUC by 40.9% and FID degradation slope by 55.3% relative to matched post-training without replay.
Range reduction plays a crucial role in the accuracy and performance of evaluating trigonometric functions, and is often the primary bottleneck for large floating-point inputs. While fast algorithms such as Cody--Waite work efficiently over narrow intervals, the Payne--Hanek algorithm remains the standard technique for accurate reduction across large floating-point inputs. However, many existing implementations of Payne--Hanek suffer from high latency due to heavy branching, conversion overheads, and the use of multi-word integer arithmetic, which hinders SIMD vectorization. In this paper, we analyze and present a branch-free variation of the Payne--Hanek algorithm using only floating-point arithmetic. Our method operates directly over large double-precision inputs ($|x| \ge 2^{16}$) and is well suited to hardware with FMA instructions. We formulate the precision constraints in terms of a truncation error budget, construct a compact lookup table indexed by the input exponent, and prove that the scaled reduced argument has absolute error below $2^{-110}$ and relative error below $2^{-60}$ for every input, including worst cases. The same routine can serve both as the complete range reduction of a single-stage implementation and as the fast path of a correctly rounded one, achieving higher throughput than existing implementations and lower latency than those returning a double-double reduced argument. The algorithm is currently implemented in the LLVM libc project.
Supervised fine-tuning can teach language models undesired behaviours alongside desired ones. Inoculation prompting (IP) aims to limit unwanted generalisation by requesting the undesired behaviour during training and removing the request at inference. However, undesired behaviour can still appear under unrelated prompts. IP can also hinder learning of the desired behaviour. We address these limitations in settings where both behaviours co-occur in most training examples, so filtering out examples with undesired behaviour leaves only a small clean subset. We introduce stratified inoculation prompting (SIP). SIP leverages a small clean subset to demonstrate that desired behaviour should persist without the undesired one across different contexts. SIP oversamples these clean examples under diverse non-eliciting prompts while inoculating the rest. SIP substantially reduces expression of undesired behaviour while preserving more of the desired behaviour than IP. These gains persist even when we extend IP to oversample the same clean subset at the same rate as SIP. Moreover, SIP yields lower emergent misalignment rates in all harmful-advice setups we tested. SIP can be further extended to limit the undesired behaviour even under prompts that explicitly request it. We introduce backdoor dilution, which weakens expression under the inoculation prompt, and password-locked inoculation, which concentrates elicitation on a designated password. Taken together, our findings show that changing the training contexts for a small clean subset can significantly improve selective generalisation.
We prove that each of the inequalities $P^{+}(p+1)>P^{+}(p-1)$ and $P^{+}(p-1)>P^{+}(p+1)$ holds for a positive proportion of primes. The argument combines a logarithmic balance identity with a uniform Selberg sieve over a thin cofactor strip. The same transfer mechanism applies to either shift and converts a one-sided large-prime-factor reservoir into an unconditional ordering result in both directions.
We determine the optimal query complexity of ground-state preparation to trace-distance error $\varepsilon$ when an energy threshold in the spectral gap is known. Let $U_H$ be an $α$-block-encoding of a Hamiltonian with unique ground state $|ψ_0\rangle$, and suppose $|\langleψ_0|U_I|0\rangle|\geγ$ for a state-preparation oracle $U_I$. The threshold lies at least $Δ/2$ above the ground-state energy and at least $Δ/2$ below every excited-state energy. We give two algorithms that prepare a state within trace distance $\varepsilon$ of the ground state. One uses $O((α/Δ)(γ^{-1}+\log(1/\varepsilon)))$ calls to $U_H$ in expectation; the other uses $O((α/(γΔ))\log(1/\varepsilon))$ calls to $U_H$ in the worst case. We prove a lower bound matching the expected query count; the corresponding worst-case lower bound follows from Somma and de Wolf [SdW26]. The respective bounds on calls to $U_I$ are $O(1/γ)$ in expectation and $O(γ^{-1}\log(1/\varepsilon))$ in the worst case. On $(N+1)$-dimensional systems, these $U_I$ bounds are also optimal when the expected or worst-case count of $U_H$ calls, respectively, is $o((α/Δ)\sqrt N)$. Both algorithms use a constant-accuracy spectral filter to construct a purifier, which we then sequentially compose during amplitude amplification to prepare a state with constant overlap with the ground state. The expected-query algorithm repeats the preparation followed by one high-accuracy spectral filter until success. The worst-case algorithm uses filters of increasing accuracy and limits the total number of queries.
Of the approximately 4,500 Oracle Bone Inscription (OBI) characters discovered from the Shang dynasty, only about 1,600 have been deciphered. Many computational approaches compare OBI with glyphs from one historical period at a time. However, during the evolution of Chinese characters, significant structural or semantic changes often occur in uncertain dynasties. A single-period reference may be insufficient when relevant forms change substantially between observed eras. Therefore, we propose the \textbf{Manifold-based Script Evolution Framework (MSEF)}, a framework that models the evolution series (OBI, Bronze, Seal, Clerical, Regular) of Chinese characters as the continual evolution of a manifold space. MSEF represents each character as an era-specific manifold point and learns continuous inter-era transition rules via Neural Ordinary Differential Equations. Both manifold space and transition dynamics can be trained end-to-end through character evolution pairs across any two eras.
We study the classical multi-item monopoly setting with a single additive buyer and $m$ heterogeneous items whose values are independent but not necessarily identically distributed. Optimal truthful auctions may be randomized and complicated. We analyze the approximation ratios of three simple deterministic auctions: selling all items separately, selling them as a single grand bundle, and choosing the better of the two. Our technical cornerstone is a nonlinear mathematical programming formulation of the worst-case approximation ratio of selling separately, in discrete auctions where values lie in the grid $\{0,1/K,2/K,\dots ,1\}$. For two iid items, we construct novel tight Lagrangian dual certificates that determine this ratio exactly for any discretization parameter $K$. Taking $K\to\infty$, we obtain the tight bound $1+W(1/e)\approx 1.278$ in the continuous-valued setting, where $W$ denotes the Lambert-W function, closing the $[1.278,1.368]$ gap from the work of Hart and Nisan [EC'12, JET 2017]. For $m\geq2$ independent items, a different dual construction gives an upper bound on the approximation ratio of selling separately in terms of basic statistics of the item values. Combining this bound with new inequalities relating optimal revenue (REV), separate-selling revenue (SREV), and grand-bundle revenue (BREV), we derive improved guarantees for all three auctions. Most notably, we prove \[REV\leq 3 \max\{SREV,BREV\},\] improving upon the long-standing $5.2$ factor of Ma and Simchi-Levi [arXiv 2015, AISTATS'21] and the $6$ factor of Babaioff, Immorlica, Lucier and Weinberg [FOCS'14, JACM 2020]. For iid items, we also prove $REV\leq 4.18 BREV$.
Distributional training provides collective supervision for one-step visual generation by matching real and generated features in frozen representation spaces. We introduce a unified theoretical framework that separates distribution modeling from matching discrepancy and connects global objectives to pointwise feature updates through Wasserstein gradient flow. Under this framework, FD-Loss and Gaussian-kernel Drifting are recovered through Gaussian optimal transport and kernel-density-based KL matching, respectively. The framework motivates MGFlow, which models feature distributions with Gaussian mixtures at an adjustable granularity between global moments and sample-based representations. MGFlow supports both optimal transport and score-based matching, and couples mass-constrained sample assignment with paired component updates to address mode collapse that mixture expressivity alone does not resolve. On ImageNet $256\times256$, MGFlow substantially surpasses the FD-Loss baseline, achieving state-of-the-art results with 1.45 $\mathrm{FDr}^6$ on pMF-H and 1.64 on JiT-H. For text-to-image generation, MGFlow post-trains FLUX.2 [klein] 4B into a one-step generator that outperforms the original four-step model on both GenEval and PickScore.
With the constant advancements in AI, one possibility is to model agents after humans and, in turn, use these agents to carry out synthetic interactions. Such models could be used to predict interactions between their real counterparts, or potentially interactions at larger scales. In this paper, we test a more controlled version of this question through chess. We use 8 elite chess players, seal their direct pairwise games, learn each player independently using different methods, and then compose the resulting models on the withheld dyads. To evaluate the generated interactions, we use two measurements: opening-family total variation distance and win-draw-loss (WDL) total variation distance. M1 primarily improves WDL fidelity while producing smaller opening-family improvements, whereas M2 produces much larger opening-family improvements while having little effect on WDL-TV. For opening-family behaviour under M2, the correct assignment of the eight learned player identities also gives the closest match among all $8! = 40{,}320$ possible assignments. These results show that at least some properties of previously unseen interactions can be recovered from independently learned individuals. The partial recovery observed here may reflect limitations of the current individual modelling methods rather than a fundamental limit on compositional interaction recovery. An additional post-hoc method that combines the two mechanisms improves both measurements, suggesting that recovery across these behavioural properties is not necessarily mutually exclusive.
We establish the optimal query complexity of smooth convex quadratic optimization from exact function values. For dimension $d$, smoothness $L$, and minimizer radius $R$, the sharp rate at small relative error is $Θ(d\min\{d,\sqrt{LR^2/\varepsilon}\})$. The lower bound has no logarithmic loss and holds even for adaptive randomized algorithms with unrestricted query locations, precision, and computation. It also matches upper bounds for general smooth convex objectives at moderate accuracy. We obtain, to our knowledge, the first lower bounds matching known smooth and nonsmooth rates, up to logarithmic factors, under Euclidean regularity and $\ell_p$ localization or feasibility, $1\le p<2$. These bounds determine the optimal polynomial dimension dependence in these geometries. A geometric framework covers arbitrary norms and known regularizers; exploiting regularizer sublevels can strengthen lower bounds by an unbounded factor. For convex-concave saddle-point problems with unequal block dimensions, we isolate coupling costs even with known, perfectly conditioned block Hessians. Separate reductions show that scalar evaluations remain necessary despite unlimited access to one block's partial gradients.
Representations continually change as a network learns new tasks. We ask whether early representational changes naturally form a geometric structure that continues to shape later learning. We identify a low-dimensional subspace of early representation drift, which we call a scaffold, and test whether it is reused across subsequent tasks. Across four pretrained visual encoders and two datasets, later representational changes consistently favor this early-defined subspace over matched random alternatives. This reuse is history-dependent: when networks experience different early tasks but identical later training inputs, each network preferentially reuses the scaffold induced by its own learning history. The same preference appears in individual optimizer updates, even though the network's dominant local response directions shift away from the original scaffold. Finally, constraining motion within the scaffold slows new-task acquisition more than matched random constraints, while effects on old-task retention are less consistent. In summary, these results suggest that early experience leaves a persistent geometric imprint on how neural networks adapt to future tasks.Code is available at https://github.com/YuantaoDeng/latent-scaffold.
The shallow embedding of higher-order modal logic in classical higher-order logic, used in Benzmüller and Scott's Notes on Gödel's and Scott's variants of the ontological argument (2025), reaches beyond the modal object language of the arguments: its property quantifiers range over terms that may also express nominals and satisfaction operators of hybrid logic, and a proof using one proves a theorem of the embedding that need not be one of the modal logic. That the framework affords this is not new, and whether a result is one of the modal logic can be settled in two ways: by replaying it in an explicit proof calculus, done by hand for chosen theorems, or by analysing the proofs the embedding itself produces, done here mechanically, for every result at once. Every statement the Notes prove has a proof inside the object language: 294 written out by hand and machine-checked, none using a nominal. The proofs the Notes themselves give instantiate no nominal either; what the detector flags there are terms a prover substituted. The three questions the Notes leave open are settled too, without nominals, but the conjunction axiom has to be emended: generalised in the Notes to Gödel's "any number of summands", it covers the conjunction of no properties, and of one; the empty one alone settles all three, and the two together yield what a separate axiom of Gödel's is for. This article restricts the conjunction axiom to at least two different conjuncts, the reading Gödel's footnote suggests, and the questions are settled again, by proofs that turn on the argument rather than a degenerate instance. The restriction holds of the object language only: with a nominal the axioms make the accessibility relation the identity and the readings coincide. Every theorem is verified in Isabelle/HOL and independently in Lean 4; the countermodels are Nitpick's, certified by the build.