Papers

Algorithm-based fault tolerance detects corrupted matrix multiplications by comparing a checksum of the product against a checksum recomputed from the factors, flagging a fault when the residual exceeds a threshold. The classical threshold is tau = Ku, derived from a dot-product rounding bound that requires Ku <= 0.01. At bfloat16 with contraction length K = 4096, Ku is approximately 32, so the bound does not hold and the threshold is not defined. Prior work therefore evaluated checksum ABFT only at float32. We supply the missing calibration. This paper is a characterization of a detector under an output-level, post-GEMM perturbation model; it is not a characterization of faults in silicon, and no rate reported here estimates fault incidence in deployed hardware. We fit a Generalized Pareto distribution to 7.99e7 clean per-row residuals and extrapolate a threshold at a per-GEMM false-positive rate of 1e-6, obtaining 3.1445e-6, which sits between the two clean sample maxima we observed. The independence assumption behind the per-GEMM conversion is checked empirically rather than asserted, at a variance ratio of 0.985 against a binomial reference. Against this threshold the detector catches 304 of 400 single exponent-bit flips (76%), failing a bar of 90% that was registered before any data was collected, and catches 0 of 600 low-mantissa flips. Detectability is invariant in K across a 16x range even though the clean noise floor falls as K^-0.4999: signal and floor scale together, so bit position governs detectability and contraction length does not. This falsified the mechanism the study was designed around. We report four registered predictions that failed, including that one, and four retractions of our own earlier findings, two of which were caught by pre-publication audit rather than by prediction. Code, data, and timestamped pre-registration documents are released.

In a recent work [1] we demonstrated that the Fermi arcs in t-PtBi2 are indeed disconnected segments of the Fermi surface. We showed that the surface state band crosses EF between Weyl points and is has finite band gap at all other momenta until it merges with bulk bands. We did not find evidence of a superconducting gap in the spectra down to 3K. In recent Comment, Sergey Borisenko reanalyzed data from repository posted with our paper and concluded that a superconducting gap with i-wave symmetry is present in our data up to temperatures of 19K. We argue that this claim arises from incorrect identification of the location of Fermi arc and confusion between band gap and superconducting gap. This error was compounded by the fact that Author of the Comment analyzed data from interpolated images used for mapping the Fermi surface that are not suitable for extraction of energy gap on meV scale. This fact was previously communicated to the Author of the Comment during e-mail exchange. We maintain that our conclusion of absence of a superconducting gap above 3 K are valid and consistent with recent STM measurements performed on the same samples, which found surface superconductivity with Tc = 3K, below the temperature at which the data for our ARPES paper was measured.

Fast and accurate simulation of hemodynamic behavior within vascular networks is essential for numerous clinical applications. However, obtaining high-quality and computationally efficient flow measurements across complex vascular networks remains challenging. To address this, we first decompose the vascular network into a set of bifurcation units and then develop an operator network capable of mapping unit-specific parameters to the local solution fields of each bifurcation unit. By lumping the Windkessel-model outlet parameters and incorporating inlet boundary conditions from the solution of parent units, the flow and pressure fields can be rapidly approximated. Subsequently, operator-network-driven Schwarz waveform relaxation is applied across bifurcation units to correct discontinuities and improve numerical accuracy. On 7-segment and 55-segment arterial tree models, the proposed method achieves $13\times$ to $17\times$ wall-clock speedups over conventional 1D numerical simulation, with relative $L^2$ errors of 1% in both pressure and velocity. The resulting pulse wave velocity biomarkers agree with the conventional reference to within 1–2%, and the same trained operator generalizes to different tree-like 1D vascular network topologies.

In a binary alloy, spinodal decomposition has one composition direction in which the free energy can fall. A multicomponent alloy has more directions available, yet its elements usually separate together in a single, pseudobinary mode. Why does this happen, and what would allow two independent directions to become unstable? We examine these questions using binary interaction parameters and the composition-constrained curvature of the regular-solution free energy. Across equimolar BCC quaternaries, one unstable mode is the most common outcome at 500 K, while 15.2% have at least two. The tendency of unlike elements to mix helps explain the preference for one mode. We then show how particular arrangements of interactions, including a balanced repulsive triangle and an attractive-repulsive pair, guarantee two downhill directions. When those directions have similar strengths, simplified phase-field models produce a broad texture of locally different solid solutions rather than two chemical blocks. A phase-specific screen identifies five provisional alloys with nearly balanced modes, with Cu-In-Ir-Ti providing an example of this possibility. Thus, we establish a theoretical framework for engineering microstructures via spinodal decomposition derived from basic binary interaction tendencies.

Spiking Neural Networks (SNNs) offer an energy-efficient approach to processing event-camera data, yet out-of-distribution (OOD) detection remains challenging in this setting. Existing OOD detection methods often depend on model outputs or computational components that are unavailable in object detection SNNs or are poorly suited to low-compute deployment. To that effect, we show that the subthreshold membrane potential \(V_{\mathrm{mem}}(t)\) provides a useful internal signal for detecting distribution shifts. Simple per-channel statistics derived from these membrane dynamics enable OOD detection. To evaluate this approach, we introduce Gen1-C, an event-camera corruption benchmark developed upon the Prophesee Gen1 automotive detection dataset, containing six sensor-motivated histogram-level stress tests at five severity levels. We further propose the Multi-Descriptor Deviation (MDD), a corruption-blind method that operates on membrane-potential statistics. At the highest corruption severity, MDD achieves an AUROC of more than 0.88 on five of the six corruptions using only a bounded 64-frame observation window. Notably, the remaining corruption is also the one that has the smallest effect on the underlying detector. These results show that the temporal membrane-potential dynamics can provide an effective and low-cost signal for OOD detection in SNN-based event perception.

Resource limits are becoming an authorization boundary for AI agents that delegate work across concurrent and failure-prone workers. Parent-child allocation constraints, affine objects, and distributed escrow do not by themselves prevent overspend when replies are lost, effects complete after timeout, messages repeat, branches partition, or DAG joins alias one lineage. We formalize fault-tolerant budget conservation for distributed multi-agent delegation. Budgets are quantized resource vectors represented by exclusive escrow credits that move through a delegation DAG. Before dispatch, a branch converts credit into an operation reservation bound to lineage, epoch, normalized effect, maximum charge, receiver, and idempotency key. It persists a signed dispatch permit with quarantine; the gateway verifies that permit before first acceptance. Uncertain effects remain charged until authenticated settlement, a fenced authoritative no-effect proof, or permanent retirement. We prove ownership partition, ledger and effect conservation, descendant non-amplification, at-most-once settlement, late-completion safety, and partition confinement under explicit mediation, durability, authentication, normalization, and gateway assumptions. An indistinguishability result shows that partition-local availability requires exclusive preallocation. Bounded TLA+ checking, an independent JavaScript explorer, and crash-injected two-process SQLite experiments exercise the declared scope and detect timeout-refund and historical-certificate-validation mutants. The mechanism preserves the issued budget bound across the evaluated crash, retry, duplicate, partition, join, and late-completion schedules.

Backdoor attacks can be implanted in Large Language Models (LLMs) during training, causing unwanted behaviour when a trigger appears in the input. Existing backdoor defences for LLMs attempt to remove the backdoor but inadvertently shift the model's output distribution to benign prompts, which can result in degraded model performance and safety. We propose NEEDLE, a training-free method for targeted backdoor removal. Once a trigger has been identified, our method estimates a backdoor direction and a refusal subspace through activation vectors, then applies sequential weight orthogonalisation to suppress the backdoor while preventing changes in refusal-related representations. NEEDLE requires neither a clean reference model nor the original poisoned training data. Evaluation is conducted across multiple model families and attack types. NEEDLE achieves the lowest mean Attack Success Rate (ASR) among the evaluated defences, including 0% on challenging code injection attacks, while resulting in the lowest KL divergence and minimal changes in capability and safety.

Self-modifying AI agents can replace, fork, and roll back identity-bearing software while descendants remain executable. Per-successor authorization does not constrain the resulting population: siblings may duplicate quotas, combine permissions, survive ancestor cuts, or overlap predecessors during promotion. We define authorization succession, which conserves authority across the active frontier of a single-parent generation forest. Our external protocol binds each generation to a manifest, root, unique parent, complete lineage, and fresh population sequence. Separate invariants bound root-lifetime consumption and current population exposure. A staged reservation freezes predecessor residual authority during replacement, while a partitioning fork validates the complete child family. Each commit atomically fences the predecessor and activates successors. Ancestor cuts invalidate dependent descendants; rollback creates a fresh generation without restoring spent authority; and a new root requires an independent grant. Under complete mediation, authenticated records, sound effect abstraction, durable monotone state, and complete lineage accounting, we prove population-safe succession, fork conservation, revocation closure, atomic handoff, rollback non-reminting, and exclusion of self-certification. An executable evaluation covers 32 registered decisions through direct-call and mailbox mappings (64/64 replays; 28 allows, 36 denies). An independent checker accepts all 64 original traces and rejects 28/28 semantic mutants; 12/12 profile invariants, 16/16 crash cuts, and 32/32 contender schedules pass. Two external adapters reproduce all 32 decisions around measured OurArk and Darwin Godel Machine mutations, including fresh-process restart, atomic succession, and predecessor rejection. The results establish authorization succession for registered protected effects.

Software that hands branching decisions to a model needs a declared option and a probability it can threshold. Typed decision models, also called System One models, return such probabilities without generating text, while supervised classifiers and generative language models are the established alternatives. One harness sends eight decision-model checkpoints from six families, including the hosted model Jev, and four open generative models from three developers the same semantic requests, and scores trained and zero-shot classifiers on the same workflow, intent, emotion and social-science items. With task labels, a fine-tuned DeBERTa-v3-large has the highest observed accuracy on every labeled benchmark but one. Without labels, no decision model is significantly more accurate than Jev on workflows or intents, but Gemma-4-31B matches it on workflows and exceeds it on CLINC-150 at higher cost and latency. Stated probabilities of generative models become unreadable when replies miss the key format, whereas key likelihoods avoid this but can saturate. A guaranteed 5 percent risk leaves Jev 0.528 of the intent decisions, and an in-scope threshold still accepts 0.310 of out-of-scope requests. On typed-decisions, swapping yes and no flips 50.5 answers per hundred for Jev and at least 16.8 for every generative model tested, against at most 6.5 for four fine-tuned decision checkpoints. Exposure to a benchmark's training data explains the largest lead of an open checkpoint, which vanishes on rater-labeled emotions. An intent-trained first stage escalating to Gemma-4-31B reaches that model's accuracy at about Jev's price. The results yield condition-dependent design rules for automated decision gates.

We study the query complexity of optimization with exact scalar-value information. For globally $L$-smooth convex functions on $\mathbb R^d$ with a minimizer in a Euclidean ball of radius $R$, we prove the lower bound $\Omega(d\min\{d,\sqrt{LR^2/\varepsilon}\})$ for adaptive randomized algorithms in the stated accuracy range. Queries are unrestricted, and the bound contains no logarithmic factor. It holds for quadratics and matches the minimum of two elementary upper bounds on that class. Prescribed strong convexity introduces the additional cap $\sqrt{L/\mu}$. The proof compares the mean of a matrix with the mean of its inverse on arbitrary affine slices and bounds posterior moments after adaptive measurements. We extend these information bounds to saddle-point problems with independent block dimensions. For compact bilinear games, the coupling alone yields $\Omega(r\min\{r,\sqrt{CR_xR_y/\varepsilon}\})$ queries, where $r=\min\lbrace d_x,d_y\rbrace$. A strongly convex–strongly concave construction yields a condition-dependent lower bound even with known isotropic diagonal Hessians. Separate reductions establish scalar query lower bounds under mixed value and partial-gradient access, including for outer objective accuracy. A Euclidean-image radius yields smooth and nonsmooth extensions to independently specified norms and known penalties.

We show that, given a partially defined metric $d:R\rightarrow\mathbb{R}$ where $R\subset X^2$ is a closed equivalence relation on a compact metrizable space $X$, there is a metric $\hat{d}:X^2\rightarrow \mathbb{R}$ such that $d = \hat{d}|_{R}$

We show that there exists a finitely generated linear group whose non-abelian tensor square is not linear. On the other hand, the non-abelian tensor square of the braid group $B_n$, $n>3$, is linear.

We propose a BRST-invariant formulation of the Gribov–Zwanziger model in the Landau gauge, in which the Gribov scale is produced by the spontaneous breaking of a scalar quartet symmetry, rather than through the explicit breaking term of the conventional construction. We show that the resulting action preserves the ghost equation of the Landau gauge and reproduces the characteristic Gribov-type gauge propagator of the original Gribov scenario, a BRST-invariant mass term of the refined type being accommodated by the same mechanism. We establish the renormalizability of the model to all orders by means of algebraic renormalization techniques, no counterterm being required outside those already present in the classical action. We further show that the condensation of the quartet leaves unchanged the cohomology through which the observables are characterized, the operator realized in the displaced vacuum having the same cohomology as that of the symmetric phase, so that the mass scale is produced at no cost in the physical content of the theory, and the construction of genuine gauge-invariant observables in the Gribov–Zwanziger setting is thereby left open to the usual cohomological methods.

As Large Multimodal Models (LMMs) transition toward natively unified architectures, evaluating their safety in synergistic harmful image-text generation tasks becomes a critical challenge. Unlike unimodal threats, synergistic risks emerge when text and image modalities are coordinated to produce harm that significantly exceeds their individual components. We introduce UnifiedAttack, a novel benchmark designed to evaluate LMM safety in collaborative scenarios by focusing on the harmfulness gain achieved through cross-modal synergy. The benchmark incorporates samples filtered for their multimodal potential alongside a novel subset of synthesized disinformation queries. To verify identified vulnerabilities, we propose a synergistic hijacking framework featuring In-Context Reskinning (ICR) and Cognitive Planning Injection (CPI). ICR utilizes few-shot learning to wrap adversarial intent in benign virtual shells to desensitize safety filters, while CPI hijacks the reasoning path by enforcing a plan-then-execute paradigm. By compelling the system to commit to a neutral logical plan, we exploit its internal drive for consistency to induce the synchronized generation of harmful multimodal content. Extensive evaluations on state-of-the-art architectures demonstrate that UnifiedAttack consistently bypasses modern alignment. Our findings reveal that the structural helpfulness and logical coherence of unified models can be systematically weaponized, highlighting the urgent need for logic-aware defenses in synergistic generation tasks. Code is available at https://github.com/bingjunluo/UnifiedAttack .

We derive a short-maturity expansion for up-and-out put barrier option prices under continuous stochastic volatility when the strike and the barrier approach the spot at the diffusive scale. Assuming joint weak convergence of the normalized terminal return, the relative volatility fluctuation, and the running maximum, together with uniform integrability, we show that the leading term is the time-inhomogeneous Black-Scholes barrier price fitted to the forward variance curve. The first model-dependent correction is of order $\theta^{H+1/2}$, where $\theta^H$, $H \in (0,1/2]$, is the order of the relative volatility fluctuation, and is represented explicitly through killed Brownian transition densities. For regular volatility models, where $H= 1/2$, the coefficient is determined by the short-maturity at-the-money implied-volatility skew. For rough volatility models with $H < 1/2$, the coefficient reduces to a one-dimensional integral. Numerical experiments show that the correction materially improves the Black-Scholes approximation.

We establish the sharp asymptotic and symmetry theory for the critical quasilinear Hartree equation \[ -\Delta_pu=\big(|x|^{-\sigma}*u^{p^*_{\sigma}}\big)u^{p^*_{\sigma}-1} \quad\text{in }\mathbb R^N, \qquad p^*_{\sigma}=\frac{p(2N-\sigma)}{2(N-p)}, \] in the complementary range $2p<\sigma<N$, beyond the quasilinear theory previously available for $\sigma\le2p$ [Math. Ann., 391(2):2653–2708, 2025]. This range is structurally different: $p^*_{\sigma}<p$, so the effective potential obtained by factoring out $u^{p-1}$ becomes singular as $u\to0$. Nevertheless, whenever $p^*_{\sigma}>1$, every nonnegative finite-energy weak solution is automatically locally bounded. Our proof provides a quasilinear regularity mechanism adapted to this complementary range, where the potential-theoretic route available for $\sigma\le2p$ breaks down. For positive finite-energy solutions with $p^*_{\sigma}>1$ we moreover uncover a sharp transition governed by the sign of $\sigma(p-2)+2p$. If this quantity is positive, the classical $p$-harmonic profile persists. At the threshold, \[ u(x)\simeq |x|^{-\gamma_h}(\log|x|)^{1/(p-p^*_{\sigma})}, \qquad \gamma_h=\frac{N-p}{p-1}, \] with the matching two-sided gradient rate. Below the threshold, \[ u(x)=A_*|x|^{-\gamma_*}+o(|x|^{-\gamma_*}), \qquad \gamma_*=\frac{2(N-p)(\sigma-p)}{p(\sigma-2p)}<\gamma_h, \] and the rescaled gradients converge uniformly to the corresponding radial profile. In all three regimes the solutions are radially symmetric about some point and strictly decreasing in the radial variable.

How students interact with artificial intelligence (AI) systems in educational settings may determine whether that interaction supports or displaces critical thinking. This paper introduces two contributions. The first is the Three Paths of Student-AI Interaction, a typological framework identifying three qualitatively distinct modes of student-AI engagement: Passive Review, Direct Question, and Strategic Dialogue. The second is the Next Level Teaching Blueprint (NLTB), a three-stage instructional design system intended to make Strategic Dialogue more likely. Qualitative content analysis of 50 randomly sampled student-AI interaction messages from an undergraduate research methods course was used to examine the typology. Two human coders achieved 68% path-level agreement ($\kappa$ = .48), with 80% agreement on Strategic Dialogue identification specifically. GPT-5, used as a third coder, produced a similar overall distribution and introduced a coding category absent from the human scheme. Path 1 (Passive Review) accounted for 46% of exchanges in the primary researcher's classifications, Path 2 (Direct Question) for 18%, and Path 3 (Strategic Dialogue) for 36%. A second, descriptively examined dataset contained predominantly Strategic Dialogue content, offering a preliminary indication that instructional framing may influence which path students take. Together, the Three Paths framework and the NLTB contribute a language for describing student-AI interaction and a design approach for supporting higher-order engagement.

Let $N\ge 2$ and $1<p<N$. Let $H$ be a possibly non-reversible Minkowski norm on $\mathbb{R}^N$, with $H\in C^2(\mathbb{R}^N\setminus\{0\})$ and $D^2(H^2/2)$ uniformly positive definite away from the origin. We classify all nonnegative local weak solutions $u\in W^{1,p}_{\rm loc}(\mathbb{R}^N)$ of \[ -\operatorname{div}\big(H(\nabla u)^{p-1}\nabla H(\nabla u)\big)=u^{p^*-1},\qquad p^*=\frac{Np}{N-p}, \] without any finite-energy, integrability, or asymptotic assumption. Every nonzero solution is an anisotropic Sobolev bubble. This settles, for the full range $1<p<N$, the anisotropic analogue of Problem A of Antonini–Ciraolo–Gatti, whose Euclidean case was settled by Zhang, and removes the restriction on $p$ in the non-energy anisotropic classification of Chen–Wu–Yan–Yang. We also prove a scale-invariant Schoen-type Harnack inequality for local solutions. The proof follows the strategy of Zhang for the Euclidean $p$-Laplacian, combining rigidity of bounded blow-up profiles and a first-contact argument with the integral bootstrap of Ciraolo and Gatti. The anisotropic analysis requires two additional ingredients. First, we establish a Bochner identity for the conjugated stress tensor in distributional form on the noncritical set, using only $C^2$ regularity of $H$. Second, we prove a two-term expansion at isolated singularities of Finsler $p$-harmonic functions, including the gradient control needed by the Pohozaev invariant. For non-reversible anisotropies the argument is formulated with the reflected norm. A Morrey-type estimate proved directly for the relevant Leray–Lions operator replaces the potential-theoretic input used in the Euclidean argument.

The quantum Hall effect, topological insulators and quantum capacitance are usually treated as distinct topics, yet they share a common thread. Bulk topology determines a protected boundary state, and quantum capacitance is one of the few electrostatic methods able to detect the density of states. This paper examines that connection. The integer quantum Hall effect showed that quantised boundary transport has a topological origin given by the Chern number, later extended to zero field in Z2 topological insulators with helical edge and Dirac surface states. Quantum capacitance, developed for two-dimensional electron gases, has since become a key technique for mapping the Landau level density of states underlying the quantum Hall effect and the linear Dirac density of states of topological surface states. This picture is complicated by self-consistent screening theory, which shows that the quantum Hall bulk splits into compressible and incompressible regions that shift with field. Local quantum capacitance measurements confirm that the bulk is not uniformly insulating across a quantised plateau, even though the plateau stays quantised. The paper argues that quantum capacitance is not a secondary feature of these phenomena but one of the clearest windows into the boundary density of states that topology imposes, and that it reveals the local, field-dependent nature of the insulating bulk assumption underlying the analogy between the quantum Hall effect and topological insulators.

We give an exposition of some of the ideas leading up to Jacob Tsimerman's groundbreaking work developing the theory of o-minimal definability in arithmetic and algebraic geometry.

Evaluating the drag coefficient Cd for a sphere moving in a fluid at finite Reynolds number Re has become a classical research subject in fluid dynamics, with publications appearing continuously by numerous authors over the years due to the unavailability of closed-form mathematical solutions. To date, mathematical formulas for calculating Cd as a function of Re can only be obtained by correlation curve fitting to experimental data. The consequence is the coexistence of various formulas with different mathematical fitting functions and an increasing number of fitting parameters. The present study focuses on examining the accuracy of Klyachko's 1934 formula for Cd with two common fractions as fitting parameters, which could enable closed form integration of a simplified equation governing rectilinear motion of aerosol spheres. The computed values of root-mean-square relative error from recently developed formulas show reasonable accuracy of the Klyachko formula for Re less than 1000 and its extension to Re less than 2e5 as a piecewise continuous function, with the possibility of significant improvement through a slight modification. Hence, a few demonstrative examples are presented using the extended Klyachko formula in the study of rectilinear acceleration, the calculation of the terminal velocity of aerosol spheres, and so forth, to shed fresh light on how this classical problem can be studied.

Multimodal on-policy distillation (OPD) aims to improve visual grounding while preserving the strong reasoning capabilities of language models. Recent multi-teacher approaches combine LLM and VLM teachers to provide complementary supervision. However, directly using a VLM's full predictive distribution entangles its visual grounding signal with its own language prior, preventing the grounding information from being transferred independently. Conversely, increasing the strength of visual supervision can improve perception but may overemphasize visual evidence and degrade language reasoning. To address this trade-off, we introduce LEGO-OPD, which selectively composes factors from a Language Expert and a Grounding expert into One teacher distribution for multimodal OPD. Under a generalized Bayesian formulation, the language expert provides a prior over candidate tokens, while the grounding expert contributes a visual likelihood that updates this prior, rather than transferring its complete predictive distribution. This factorized composition allows language reasoning and visual grounding to be controlled independently. We further introduce adaptive calibration to determine how strongly the visual likelihood should update the language prior at each decoding prefix. Specifically, LEGO-OPD uses the grounding expert's image-induced prediction shift as a prefix-dependent reference, preventing both insufficient and excessive visual supervision. Experiments with Qwen3 models show that LEGO-OPD consistently outperforms the evaluated single- and multi-teacher OPD baselines on both multimodal and text-only reasoning tasks. Moreover, it improves the initial student's visual perception while preserving text-only reasoning.

Distilling the reasoning capabilities of large language models (LLMs) into smaller students is a central challenge for efficient deployment. Current approaches face a fundamental tension: optimizing purely for verifiable task rewards (e.g., via GRPO) leads to reward hacking, where students arrive at correct final answers through flawed intermediate logic, while regularizing with soft divergence penalties against a teacher (e.g., KL-based distillation) dilutes task performance and, critically, allows the student to compensate for severe logical violations at one step with high teacher agreement at others. We argue that this averaging is fundamentally misaligned with the nature of reasoning: a chain-of-thought is only as valid as its weakest link. Motivated by this observation, we formulate reasoning distillation as a constrained reinforcement learning problem in which the task reward is maximized subject to a worst-case constraint on the teacher log-likelihood along every prefix of the trajectory. To avoid the prohibitive cost of dual Lagrangian solvers and the test-time teacher dependence of state-augmented methods such as Saute, we derive an unaugmented constrained MDP whose reward transformation preserves the hard-constraint semantics, admits a low-variance policy gradient decomposition into single-step and long-term terms, and provably satisfies the worst-case constraint almost surely in the penalty limit. Through extensive experiments on mathematical reasoning and code generation tasks, we demonstrate that our method significantly expands the accuracy-fidelity Pareto front. By matching the high Final Answer Correctness of pure RL and drastically reducing teacher constraint violations, we ultimately achieve the highest rigorous Reasoning Success Rate across all evaluated settings.

When selecting mathematical training data for LLMs, a natural organizing principle is topic: probability examples for probability targets. An alternative is reasoning approach: worked solutions that share a solution method with the target, even when the mathematical domain differs. We ask which relation produces greater transfer after fine-tuning. We evaluate two counterbalanced $2\times2$ designs: probability and combinatorics crossed with invariant reasoning and double counting (2,000 problems), and number theory and geometry crossed with complement and pigeonhole reasoning (800 problems). In each design, every cell serves as the held-out target in turn: same-approach (SA) sources share the target's method but change the topic, while same-topic (ST) sources share the topic but change the method. Every source appears once in each role, so additive source-quality effects cancel from the equally weighted aggregate contrast. Across five base models and three training seeds per design, SA outperforms ST in all 40 seed-pooled model–target comparisons. Model-level advantages range from 8.2 to 16.2 percentage points in the primary design (mean: 10.8) and from 12.0 to 16.0 in the second design (mean: 14.3); all ten model-level 95% confidence intervals exclude zero. In both designs, ST sources are more similar to targets under embedding and lexical measures, so the SA advantage runs opposite to the measured ordering of statement-level resemblance. These findings identify reasoning approach as a more effective matching criterion than topic for mathematical transfer across the evaluated topic–approach combinations.

Long-term object search requires learning where objects usually appear from repeated but uneven observations of a changing environment. We formulate retrospective open-vocabulary memory as probabilistic inference from censored observations, where the key idea is to reason with evidence per opportunity: a detection or non-detection should influence belief only in proportion to the robot's opportunity to observe the corresponding location. We introduce ECROM, which uses this principle to estimate long-term prevalence for concepts specified only at query time and converts the resulting belief directly into an active-search prior. To evaluate this problem, we introduce a controlled long-term benchmark in ten HM3D homes that independently varies object placement and observation opportunity across repeated traversals. ECROM improves support-level AP on held-out queries by 4.5 points and search SPL by 4.2 points over the strongest competing memory in each metric. The benchmark, dataset, and code will be open-sourced. Project page: https://jiaming.im/ecrom/

Structured state-space sequence models (SSMs) achieve linear-time inference and logarithmic-time parallel training via associative scans, yet diagonal transitions generate solvable semigroups unable to track non-solvable group words or 3D rigid-body manifolds. We present Non-Commutative Lie-Algebraic State-Space Models (Lie-SSMs), parameterizing recurrent updates on Lie algebra tangent spaces via the matrix exponential. First, using an o-minimal cylindrical flag collar, we establish an unconditional single-layer finite-division criterion for contracting $SO(3)$ and $SU(2)$ heads, proving odd-prime Sylow abelianity, odd-$p$-core centrality, and exponent-30 commutator obstructions excluding $S_5$ in $SO(3)^m, SU(2)^m,$ and $SO(4)^m$ while identifying the sharp threshold $d^*(S_5)=d^*(A_6)=5$ in $SO(5)$. Second, Depth-2 semidirect and Schreier $2$-cocycle cascades resolve $S_5$, $2I\cong SL_2(F_5)$, and $\mathrm{Heis}(F_p)$ in $O(\log T)$ scan depth, whereas unnormalized contracting coupling collapses exponentially without Directional RMSNorm. Third, we characterize the $\Theta(\varepsilon^{c-1})$ Chen–Magnus–Hall–Witt commutator saddle and $1/|[G,G]|$ coset-centroid dilution, proving generic first-order symmetry breaking at $\varepsilon=0$ and teacher-forced local Polyak–Łojasiewicz convergence under dense prefix supervision and factored readouts ($99.90\%/99.40\%$ on $S_5$ at $T=128$; $100.00\%$ on $D_{16}$). Across $SE_2(3)$ Left-Jacobian odometry, Hillis–Steele/Blelloch GPU scans ($48.24\times$ forward, $20.27\times/10.28\times$ full-layer speedup at $T=2048$), and three 3D IMU benchmarks (UCI HAR, UCI HAPT-PT, and 17-subject REALDISP), Lie-$SO(3)$ models reach $95.69\%\pm 0.14\%$ on HAR, $88.44\%\pm 0.83\%$ episode logit-vote ($83.97\%\pm 1.20\%$ C3 window in 30-subject 5-fold CV; $87.16\%\pm 0.64\%$ 9-subject split) on HAPT-PT, and $84.23\%\pm 1.00\%$ on REALDISP.

Structured tool calls often fail after only a small number of fields violate a schema or an execution contract. Regenerating the complete object enlarges the action surface and makes repeated repair difficult to audit. We introduce ContractRL, a contract-constrained sequential repair protocol that models verifier-guided JSON repair as a bounded decision process. At each step the policy observes the candidate, typed verifier feedback, JSON Pointer, immutable repair history, and remaining budget; a contract-derived action mask filters malformed or prohibited RFC-6902 operations before a deterministic validator performs the transition. We specify a contract-constrained group-relative objective for patch, retry, and abstention decisions while keeping canonical targets and semantic labels outside the online state until trace freeze. Under identical verifier information, ContractRL attains 0.9362 semantic success with 34.4 generated tokens, compared with 0.9076 and 44.9 tokens for Patch-SFT and 0.9148 and 137.2 tokens for full regeneration over 192 cases per seed and five seeds. Policy optimization improves semantic success from 0.9186 for supervised ContractRL to 0.9375. A separate three-seed paired evaluation against Patch-SFT yields a semantic difference of +0.0396 (95% CI $[+0.0137,+0.0662], p=0.0039$). Feedback, action-mask, budget, and schema-shift analyses connect these gains to localized correction, while adversarial and multi-turn evaluations characterize the remaining failure modes.

Tool-using agents can expose citations and execution logs while leaving a critical association unaudited: whether the claim shown to a user is the claim emitted by the committed execution and supported by the cited source. A valid citation and a valid trace can therefore remain individually well formed while being transplanted across claims, actions, runs, or source versions. We introduce a claim-anchored execution contract that jointly binds the emitted claim, its exact source span, the ordered execution prefix that produced it, and the source version and access state observed by that execution. Each receipt contains an emission anchor that deterministically locates the claim inside a committed answer or claim-bearing action, together with source identifiers, offsets, hashes, quotes, and a domain-separated execution commitment. A deterministic integrity verifier reconstructs these bindings before semantic or task labels are joined. We separate this integrity plane from a pluggable support plane, so structural validity is not used as a proxy for entailment. The contract exposes seven independently testable properties: claim-emission binding, source binding, ordered-execution binding, oracle separation, persisted-object replay, execution-rerun consistency, and version/access binding. Across 1,280 cross-object attacks, the joint contract detects 1,275 substitutions (0.9961). Removing a targeted property reduces its attack-detection rate to 0.0156-0.0625. On an independently adjudicated 384-pair split, the conflict-aware support guard reaches F1 0.8865 and false acceptance 0.0729; on unseen failure families, these rates are 0.8679 and 0.0938.

We present fully two-dimensional self-consistent calculations of the electron density, local filling factor and current distribution in a six-terminal GaAs/AlGaAs Hall bar of width W = 3 micrometers with two additional square contacts, A and B, placed on the channel axis and connected to the outside by air bridges, following the interior-contact experiment of Kendirlik et al. [Nat. Commun. 8, 14082 (2017)]. Within the screening theory of the integer quantized Hall effect, the Thomas-Fermi-Poisson equations are solved on the full device plane, including the probe arms and the depletion regions around the interior contacts, and the current is obtained from a local Ohm's law with a filling-factor dependent conductivity tensor, for source-drain and interior-contact excitation separately. Across the nu = 2 plateau the interior contacts pass through three regimes as the field is lowered: galvanically connected to the current path in a compressible bulk, floating in the Hall potential of a fully incompressible bulk, and isolated from both the bulk and the edge strips by closed incompressible rings encircling each contact. In the floating and isolated regimes the two-terminal resistance between the interior contacts rises by orders of magnitude. The isolation interval lies within the Hall plateau and ends before its low-field edge; its high-field end is sharply set by the disappearance of the incompressible bulk, while its low-field end shrinks with temperature and Landau-level broadening. The resistance between an interior contact and a perimeter probe approaches one half of the quantized Hall resistance across the plateau. The topology of the incompressible regions and the boundaries of the isolation interval are insensitive to the regularization of the local conductivity, to the grid and to moderate disorder, whereas the magnitude of the isolation resistance is not.

Continuous policy optimization may spread an update across many states, even when deployment permits only a few complete state-level changes. We study how much discounted return can be retained when continuous row mixtures are rounded to sparse binary policies in finite MDPs. Policy-dependent visitation couples the row edits, while long horizons make global curvature bounds conservative. From $2d+2$ Bellman solves, we derive reusable envelopes that support uniform and candidate-specific guarantees before rounding. A rank-two rational representation of each exchange further permits weighted curvature integration along the realized trajectory. We prove that linear dimension dependence is unavoidable when the budget scales, and that exact global curvature thresholding is hard. Candidate-specific bounds raise pre-rounding certification coverage from $48.2\%$ to $74.1\%$ on the structured suite. At $\gamma=0.95$, local integration lowers the median bound-to-loss ratio from $402.3$ to $2.08$ on coupled instances.