Papers

We show that $m$-linear homogeneous rough singular integral operators $\LL_{\Omega}$ associated with $L^\infty$ functions $\Omega$ on $\mathbb{S}^{mn-1}$ with mean value zero may not be bounded from $L^1(\mathbb R^n)\times\cdots\times L^1(\mathbb R^n)$ to $L^{\frac 1 m,\nf}(\mathbb R^n)$ except possibly for the case $n=1$ and $m=2$.

Monocular videos of human manipulation provide abundant dexterous demonstrations, yet reconstructing hand-object interaction from a single view and transferring it to robot hands remain difficult, limiting their direct use for robot execution. Prior methods either require task-specific RL training, limiting scalability, or assume clean motion-capture trajectories and thus cannot operate directly on video. We present OmniHOI, a pipeline that turns an RGB video of hand-object interaction into an interaction-faithful trajectory on dexterous hands. The key idea is to enforce physical consistency using the evidence available at each stage: image evidence during reconstruction, contact geometry during retargeting, and dynamics during physics-in-the-loop refinement. Each stage optimizes the corresponding representation directly, correcting errors before they propagate downstream or must be absorbed by a learned policy. Across 150 motion-capture trajectories transferred to each of five dexterous hands with 6 to 22 DoF, we achieve 39-89% success, compared with at most 31% for prior transfer methods. On 60 monocular video clips, we achieve 53% success, compared with 28% for the best prior video-to-robot pipeline. Its trajectories also execute on a real bimanual robot across diverse tasks.

Knowledge distillation is an established technique for improving the capabilities of small, efficient student models by training them with the representations of larger, more capable teacher models. Much of the recent work in the distillation of large language models (LLMs) has focused on distilling abilities learned during post-training, such as instruction following, chain-of-thought reasoning, and tool usage. This has left a large research gap in general knowledge distillation for LLMs, which is essential for developing efficient and private systems suitable for deployment on edge devices. We take a first-principles approach, evaluating previous lessons from prior works and conducting new explorations to develop a distillation methodology suitable for modern LLMs. We present KDFP, a novel methodology for white-box general knowledge distillation in LLMs. We demonstrate that KDFP outperforms existing methods by 1.6% $-$ 4.9% across 9 benchmarks while increasing training efficiency by up to 99.1% through ephemeral parameter reduction.

An experimental campaign to measure double-differential cross sections (DDXS) for light-charged particles (LCPs) emitted from $^{\rm nat}$Fe was carried out in 2023 at the Neutrons For Science (NFS) facility of GANIL, France. The Medley experimental setup, specifically designed for light-ion measurements, was used for this campaign. The setup was installed in the white neutron beam line and irradiated with neutrons of energies up to 44 MeV. The Time-of-Flight (TOF) technique was used to obtain $^{\rm nat}$Fe(n,pX) double-differential cross sections for all neutron energies within the white-neutron beam's energy range. This work highlights important steps in the data-analysis procedure and reports the first results from this experimental campaign.

Differentiable renderers expose the derivative of any scalar objective with respect to every scene parameter, yet unlike the primal image, which decades of arbitrary output variables (AOVs) have taught us to decompose, inspect, and composite, these derivatives have no established representation for human inspection. We introduce the sensitivity AOV, a render output carrying the sensitivity of an objective to the scene parameters that influence it. A single reverse-mode pass populates a sensitivity buffer over the scene's parameter hierarchy, from which many views are read rather than re-differentiated, in direct analogy to deferred shading: image-space sensitivity at object and parameter-type granularity, projections onto a freely navigable scene from any inspection viewpoint, and, for spatially varying parameters, per-texel fields carried to the surface through texture coordinates. We separate the fixed camera that defines the objective from the free camera used to inspect the result, and position reverse-mode attribution against its forward-mode dual. Our aim is not a single algorithm but a scaffolding that establishes derivative outputs as first-class render products alongside the primal image.

Many treatments have effects that plausibly vary by latent classes. For example, the effect of personalized video recommendations may be stronger for users who seek a new discovery than for users seeking to rewatch something familiar. In the context of user journeys on digital platforms, we can conceptualize user moods as dynamic latent states that lead to distinct action distributions. Because mood is unobserved, we infer it from a user's pre-treatment actions using a hidden Markov model (HMM), and then leverage the latent state as an interpretable treatment effect modifier when evaluating experiment results. Under a treatment-effect sufficiency assumption, each observable conditional effect is a known mixture of the latent effects, so state-specific effects are identified when state posteriors are sufficiently heterogeneous. We propose an estimator that is Neyman-orthogonal to errors in the fitted HMM, enabling standard influence-function confidence intervals. In a real-world large-scale A/B test of a new recommendation model at Netflix, our latent-state analysis reveals that heavy-engagement users show significantly positive effects across all treatment variants, with limited evidence of benefit among less-active and rewatch-oriented users.

As AI models move toward clinical decision-making and personalized treatment, understanding what a model learns is important beyond predictive accuracy alone. We investigate whether personalized latent dynamics reveal clinically associated differences even when predictive fit is similar. A lightweight CNN–Transformer EEG foundation model pretrained on the Temple University EEG Corpus (TUEG) extracts segment-level representations. Using the Temple University Epilepsy Corpus (TUEP), representations are mapped to a shared latent-state space, and sparse multinomial logistic transition distributions (mLTD) are fit independently to each subject to obtain personalized transition-dependency graphs $W_n$. Analyses include $n{=}198$ subjects (99 epilepsy / 99 non-epilepsy). At $k{=}4$, epilepsy subjects exhibit substantially denser learned dependency structure ($p{=}1.1\times10^{-7}$), with the same pattern at $k{=}6$ (19.90 vs. 13.46; $p{=}5.2\times10^{-5}$). Graph-derived features provide moderate group discrimination under 5-fold subject-wise cross-validation (AUROC 0.68 at $k{=}4$; 0.65 at $k{=}6$). In contrast, held-out log-likelihood is nearly identical between groups at $k{=}4$ ($-0.992$ vs. $-0.991$; $p{=}0.95$), with similarly matched next-state prediction (AUROC 0.855 vs. 0.861; $p{=}0.54$). Thus, similar predictive fit does not imply similar learned dynamics: groups can be comparably predictable while differing substantially in the internal dynamical structure learned by personalized models. This distinction motivates evaluating learned structure alongside predictive performance in personalized clinical models.

We prove that the sine beta point process is tail trivial for every beta >0. The key ingredient in the proof is a certain uniform entropy bound, that makes use of the DLR equations of Dereudre et al. (2021) and the bounded two-point correlation density obtained by Qu and Valko (2025) and Assiotis and Najnudel (2026), that allows us to show that tail events are translation invariant. Tail triviality then follows either from the uniqueness theorem of Assiotis (2026) or from translation mixing, which we derive from the correlation estimates of Dumaz and Malvy (2026). As consequences, we obtain the following applications: (a) extremality of sine beta in the full convex class of fixed kernel Gibbs measures without stationarity and finite-energy conditions; (b) L^2-ergodicity of the associated infinite-particle reversible diffusion for every beta>0; and (c) the almost-sure collision/noncollision dichotomy with threshold beta=1 for the diffusion.

Accurate evaluation is central to selecting which LLM to deploy, yet testing a candidate on live traffic exposes real users to an unvetted model. Teams therefore evaluate candidates offline, on data produced by already-deployed models. This is off-policy evaluation (OPE), and it faces two distribution shifts: as a model is updated in post-training, its responses diverge from the logged ones (policy shift), and the reward definition under which it is judged changes with business requirements (reward shift). Classical OPE methods are ill-suited to this continual-deployment setting because they are defined per task and require fitting from scratch on every new logged dataset or reward definition. To address this, we propose PFN-OPE, a prior-data fitted network that amortizes OPE across a distribution of contextual-bandit tasks. We pretrain it once on tasks constructed from a pool of LLM responses scored by several reward functions, in which both shifts occur. At test-time it maps a logged dataset and one sampled target response per prompt to a value estimate in a single forward pass, with no per-task fitting. On HelpSteer2 and UltraFeedback with Qwen, Llama, and Gemma policies, PFN-OPE achieves 2.0 to 9.3 times lower error than the best baselines across all tested configurations in the reward-shifted settings.

We describe Reservoir Light Replay Backpropagation (ResLRB), a method for reverse-mode differentiable light tracing whose memory is constant in path length and in the number of sensor connections per light path. Naive automatic differentiation of a light tracer records a computation graph that grows with the number of valid sensor connections along each path, since flux can be splatted from every scattering vertex; adjoint path replay removes the analogous dependence on depth for viewpoint path tracing, but its non-branching structure does not extend to the splatting case. We compress the adjoint graph during the primal pass by stochastically retaining a single representative sensor connection per light path via a streaming weighted reservoir. The adjoint pass reconstructs the path deterministically by replaying its pseudorandom number sequence and backpropagates only through the retained connection, yielding an unbiased gradient estimator whose peak memory is that of differentiating a single scattering event.

The diversity of robot embodiments and action spaces makes it challenging to build robot world models that generalize across different embodiments. We introduce the Latent Action-Conditioned Robot World Model (LAC-WM), which operates within a learned unified latent action space shared across diverse embodiments. This unified action space improves the world model's performance when adapted to previously unseen robot embodiments. We compare LAC-WM with an Explicit Action-Conditioned World Model (EAC-WM), which conditions on explicit motion labels. Our results show that explicit action conditioning leads to disjoint action representations across embodiments, limiting downstream performance when adapting to new robots. We evaluate both models on dexterous manipulation tasks and a modified LIBERO benchmark. LAC-WM improves downstream performance over EAC-WM by up to 46.7% on dexterous manipulation and 11.7% on LIBERO. Crucially, the unified latent action space allows LAC-WM's downstream performance to scale positively with the number of embodiments used during pretraining. In contrast, the disjoint action space in EAC-WM leads to decreased performance as the number of pretraining embodiments increases. These results highlight the importance of a unified action space for efficient cross-embodiment learning, addressing a key challenge in robotics. Project website: https://lacwm.github.io/

A large language model can only use the text that fits in its context window, and it recomputes its internal key-value (KV) state for a prompt every time the prompt is sent. We test a memory layer, the public package galahad-kv, that saves the KV state of each block of about 16,000 tokens to encrypted local NVMe disk and loads it back later, byte-exact, without recomputing it. We ran it on 50,000,000 tokens of real public text, served through vLLM on one NVIDIA H100, with Gemma 4 12B and Gemma 4 31B. Every block we probed was loaded back from the encrypted store with no recompute (100 of 100, at depths from 0 to 50M tokens) on both models. Loading a block was 2.8x to 4.3x faster than recomputing it and used 8.8x to 12.3x less GPU energy, and GPU memory stayed flat over the whole 50M-token stream. Asked about facts planted millions of tokens earlier, the 12B model gave the right answer 82 times out of 100 and the 31B model 98 times out of 100. Neither model made up an answer. The limits are as follows. This is reuse of stored state, not a wider attention window: one block is loaded at a time, and how well a question is answered depends on the model. Writing the memory is a one-time cost, and the store takes terabytes of local NVMe disk. We describe the test protocol, which is built to resist common ways of gaming long-context benchmarks, and give a single-GPU reproduction that uses public software and a free licence for the package.

JavaScript engines are pivotal to modern web browsers, enabling the execution of dynamic and interactive web applications. However, their complexity and widespread adoption make them prime targets for attackers exploiting vulnerabilities. While existing research has focused on detecting vulnerabilities of JavaScript engines, a significant gap remains in systematically understanding the characteristics of these vulnerabilities, including their symptoms, root causes, and exploitability. This paper bridges this gap by presenting the first comprehensive empirical study on vulnerabilities in JavaScript engines, investigating their characteristics and potential exploitation strategies. We construct a dataset comprising 241 vulnerabilities across four mainstream JavaScript engines from 2017 to 2024. Through in-depth analysis, we first develop taxonomies for symptoms and root causes. Building on this understanding, we investigate the exploitability of these vulnerabilities, identifying key prerequisites and extracting vulnerability trigger chains that demonstrate how logical errors propagate into memory safety violations. Additionally, we analyze the mitigation strategies to counter these exploits. Finally, we summarize key implications for various stakeholders, including developers and researchers, offering actionable insights to improve the security and resilience of JavaScript engines.

Optimal dynamic treatment regimes (ODTRs) are increasingly used to generate sequences of personalized recommendations that maximize a final outcome of interest. Although ODTRs are widely used in biostatistics and precision medicine, their applications in education remain limited, especially for longitudinal decision-making. This study translates ODTR methods for single- and multi-stage designs into individualized high school math course recommendations, providing step-by-step guidelines and demonstrations with data from the High School Longitudinal Study of 2009 (HSLS:09). We use Targeted Maximum Likelihood Estimation with ensemble machine learning to estimate ODTRs that maximize students' math achievement and college enrollment, imposing feasibility constraints based on domain knowledge and propensity-score thresholds to ensure recommendations are practically implementable. Results show that unconstrained regimes yield higher estimated values but infeasible recommendations, while feasible regimes produce realistic pathways with stronger empirical and expert-informed support. Finally, we discuss practical considerations for designing ODTRs in education.

We formulate diffractive ring focusing of paraxial beams carrying orbital angular momentum (OAM) through a finite circular aperture as a variational optimization problem. For a fixed topological charge, we determine the real-valued radial pupil that maximizes the intensity at a prescribed ring radius and propagation distance under a normalization constraint on the input field. The resulting problem admits an exact analytical solution: the optimal pupil combines the familiar Fresnel phase structure with an OAM-dependent Bessel radial modulation, and the associated eigenvalue yields a rigorous upper bound on the achievable ring intensity. We further analyze the lens-assisted configuration and derive a simple bound that allows direct comparison between diffractive and refractive vortex focusing. The theory also provides a quantitative framework for evaluating the performance of practical mask designs, including binary Fresnel-Bessel zone plates and amplitude-weighted pupils. We have experimentally tested our predictions in the optical domain by implementing the corresponding vortex beam pupils and comparing their propagation and focusing performance with the optimal solution. Our results establish the vortex-beam analogue of optimal amplitude-only focusing and identify the fundamental limits governing diffractive concentration of structured waves

We consider the SM extended by a 4th generation of vector-like leptons. This model generates strongly correlated contributions to the lepton masses, Higgs couplings and dipole moments. At leading order the Higgs coupling restricts the magnetic and electric dipole moments to an ellipse. Here we compute the one-loop corrections to the Higgs coupling and two-loop corrections to the dipole moments to obtain the correlation at next-to-leading order using the framework of SMEFT. In addition we resum the leading logarithmic corrections from SM couplings through the SMEFT renormalisation group equations. With these results we compute the RG improved correlation at NLO and discuss the implications for the muon, electron and tau correlations. Specifically, for the muon we find that the corrections contract the ellipse such that a larger part of parameter space becomes compatible with the updated SM value of the muon magnetic moment.

We obtain a near-minimax rate on covariance estimation of adversarially corrupted Gaussian data in Frobenius norm when the covariance estimation is assumed to lie in a star-shaped set $K \subseteq \mathbb{R}^{n \times n}$. Assuming a known upper bound of $\varepsilon < 1/32$ on the fraction of the $N$ observations of data $X$ are arbitrarily corrupted by an omniscient adversary, we obtain the near-minimax rate (up to constants) of \begin{align*} \left(\varepsilon^2 \vee \delta^{\star2} \right)\wedge d^2 \lesssim \inf_{\hat\Sigma} \sup_{\Sigma \in K} \sup_{\mathrm{C}} \mathbb{E} \| \hat\Sigma(\mathrm{C}(X)) - \Sigma \|_F^2 \lesssim \left(\varepsilon^2\log^2(1/\varepsilon) \vee \delta^{\star2} \right)\wedge d^2 \end{align*} where $d$ is the diameter of the set $K$, $N \gtrsim \sup_{\delta > 0} \log \mathrm{M}^{\operatorname{loc}}(\delta, 2c)$, and \begin{align*} \delta^*=\sup\left\{ \delta\geq 0: \frac{N}{\Upsilon^2} \delta^2\leq \log \mathrm{M}_K^{\operatorname{loc}}(\delta,c) \right\} \end{align*} for an arbitrary large constant $c$ and the local metric entropy of $K$, $\mathrm{M}^{\operatorname{loc}}_K(\delta,c)$. We believe this is the first such general result on structured robust covariance estimation in the Frobenius norm.

Several models for chirality-induced spin selectivity (CISS) in charge transfer have been proposed, but rarely compared directly with experiments. This step is essential for their validation and future progress of the field. Here we investigate a broad class of radical pair states associated with different microscopic models by computing the related electron-paramagnetic resonance spectra. This allows us to evidence peculiar signatures of different radical pair states, to study their experimental accessibility and identify conditions to verify proposed microscopic models.

We study a broad class of anisotropic Dirichlet problems driven by the nonhomogeneous operator \[ \mathcal L_Au=-\sum_{i=1}^N\partial_i\!\left( a_i(|\partial_i u|^{p_i})|\partial_i u|^{p_i-2} \partial_i u\right), \] where the functions $a_i$ satisfy structural growth and monotonicity conditions of $(p_i,q_i)$-type. This framework contains, among other examples, the anisotropic $\vec p$-Laplacian, the orthotropic $p\&q$-Laplacian, and anisotropic operators combining several directional growth regimes. We first establish a comparison principle adapted to this lack of homogeneity. For power nonlinearities, including the singular range, we then obtain nonexistence, existence, singular uniqueness, comparison alternatives, and multiplicity results according to the position of the reaction exponent relative to the anisotropic growth and critical exponents. In particular, an intermediate power regime exhibits nonexistence for small positive parameters and two distinct nontrivial solutions for large parameters without additional restrictions on the smallest effective exponent. At partial resonance, $r=p_1<p_N$, we identify an exact positive parameter threshold: there are no nontrivial solutions at or below it, whereas each parameter above it admits infinitely many pairs of solutions converging strongly to zero. We also treat concave–convex reactions with subcritical or critical growth and prove the existence of nontrivial nonnegative weak solutions in the corresponding parameter ranges. The arguments combine a convexity method of Díaz–Saa type, anisotropic Sobolev embeddings, direct minimization, Ekeland's variational principle, mountain-pass techniques, and concentration–compactness.

In this paper, we characterize surjective nonlinear composition operators acting between the space of sequences of bounded variation and the space of bounded sequences. We show that their generators exhibit highly pathological behavior. To complement this, we construct explicit examples, confirming that such exotic objects do in fact exist.

Effective vacuum sectors in modified theories of gravity are often represented by anisotropic stress-energy tensors, even when the underlying source is geometric rather than material. This raises a broader question: can such effective stresses be characterized intrinsically through a covariant constitutive structure, rather than modeled phenomenologically as exotic fluids? We address this question in pseudo-complex general relativity (pcGR), where a constrained variational formulation identifies the effective vacuum source as geometric in origin but does not uniquely determine the corresponding geometric constraint sector. For the static pseudo-complex Schwarzschild solution, we show that the effective vacuum tensor separates naturally into an isotropic contribution and a purely transverse contribution associated with the intrinsic geometry of the two-dimensional subspace orthogonal to the timelike and radial directions. This decomposition leads to a constitutive formulation based on the transverse metric. Its infinitesimal deformations split uniquely into trace and traceless shear modes, with the static solution having only an isotropic transverse response. Within the class of local constitutive variables that depend algebraically on the transverse metric and whose first variation generates the trace projector with stateindependent normalization, the logarithmic transverse area strain is the minimal scalar constitutive coordinate.

This paper investigates the boundary stabilization of a linear hyperbolic system with a zero characteristic speed. We address a case that cannot be stabilized by proportional boundary feedback and develop a backstepping design for the critical setting where the zero-speed subsystem has no intrinsic dissipation. To this end, we construct a Volterra transformation leading to a nonstandard integro-differential target system. Its asymptotic stability in L2 is established by both the Lyapunov-LaSalle method and a spectral semigroup argument. The backstepping kernel equations are explicitly solved, yielding an explicit boundary feedback law that asymptotically stabilizes the original system. Numerical simulations are provided to illustrate the effectiveness of the proposed control design.

Satellite megaconstellations in low Earth orbit interfere with ground-based optical astronomy, but the scale of that interference has been hard to measure. Here we report an uncued photometric survey from a mid-northern latitude site in late 2024 which recorded some 25,000 tracks of 5,118 individual satellites to a limiting visual magnitude of +9.0. The observed population is systematically brighter than the altitude-adjusted limit of order seventh magnitude recommended by the International Astronomical Union Centre for the Protection of the Dark and Quiet Sky from Satellite Constellation Interference (IAU CPS). Starlink satellites exceeded that limit on roughly a quarter of all geometrically possible passes, rocket bodies on about a fifth, and the emerging Guowang/Hulianwang constellation on more than half. OneWeb was the most compliant major operator (exceeding the IAU CPS limit on only about one in twenty possible passes), which shows that effective brightness mitigation is achievable with current technology. Satellite visibility is concentrated into the two hours after sunset and before sunrise and into limited regions of the sky. From the measured plane-of-sky satellite flux we predict that of order one in three 30 s twilight exposures at the Vera C. Rubin Observatory will carry a satellite trail.

We study whether small LLM agents can operate effectively under explicit wall-clock time budgets by both respecting the allocated runtime and using available time productively. We evaluate Qwen3.6-27B on five competitions from MLE-Bench Lite and Qwen3-4B on Zork I (Jericho), two agentic benchmarks where additional computational time can meaningfully improve performance. In the simplest setting, where the budget is stated only in the prompt, agents fail to translate the stated budget into controlled use of time. These failures arise from gaps in time awareness, since the harness provides no timing feedback, but also because they cannot reliably anticipate the duration of actions, and do not have a learned mapping from available time to an appropriate strategy. We investigate two complementary classes of interventions: harness-based mechanisms that expose timing information and enforce deadlines, and reinforcement learning with budget-aware rewards. Injecting timing information through the harness substantially improves budget adherence for Qwen3.6-27B without measurable loss in performance, while enforcement hooks tighten adherence further. RL with GRPO achieves near-perfect budget adherence on Zork I and generalizes to held-out budgets not seen during training, but does not improve task performance over the untrained harness on MLE-Bench. Once agents are made to respect the budget, they still fail to use additional time to improve task performance. RL-trained policies learn when to stop but often fill extra time with repeated actions, and GRPO training on multiple budgets tends to collapse toward the strategy learned for the shortest budget. Our results reveal a gap between time adherence and productive time allocation, which remains a central challenge for budget-conditioned agents.

Conventional decision tree algorithms produce effective, transparent models that can be audited, communicated, and deployed independently of the training data, but require learning every new task from scratch. In contrast, tabular foundation models demonstrate that meta-learning from a synthetic prior distribution enables strong in-context prediction for previously unseen tasks, especially in small-sample regimes. However, this approach does not produce a standalone model that can be inspected in isolation. We introduce MotherTree, a tabular transformer that meta-learns decision tree induction: given a training set for a new task, it outputs a hard, axis-aligned decision tree, equivalent in form to classically trained trees, in a single forward pass. MotherTree is pre-trained on a synthetic prior using stochastic gradient descent without requiring reference trees for supervision. On established benchmarks with controlled sample size, the approach is competitive with size-matched trees from common algorithms: recursive partitioning, gradient-based tree learning, globally optimal trees, and distillation from tabular foundation models. Notably, MotherTree consistently improves over from-scratch gradient-based learning and acts as a strong initializer: task-specific tuning of the generated tree outperforms the corresponding from-scratch learner on all benchmarks and sample sizes. These results show that meta-learning can provide effective inductive biases for learning stand-alone, small decision tree classifiers.

Norin and Steiner recently proved the Linear Hadwiger Conjecture. That is, there is an absolute constant $C$ such that every graph $G$ satisfies $\chi(G)\leqslant C\,\text{had}(G)$, where $\chi(G)$ is the chromatic number and $\text{had}(G)$ is the Hadwiger number of $G$. They estimate that their proof gives a non-optimised constant $C$ of order $10^{100}$. This paper uses AI assistance to optimise their argument, and shows that every graph $G$ satisfies $\chi(G)\,\leqslant\,1{,}118{,}430\,\text{had}(G)$.

Machine-learning interatomic potentials have become indispensable tools in atomistic simulations. The most popular models rely heavily on physical priors, such as locality, smoothness and symmetry, that are assumed to improve the accuracy and transferability of the trained models. However, recent evidence suggests that, in the data-rich regime, unconstrained models that learn symmetry from the data can achieve very competitive accuracy and computational efficiency, without sacrificing stability and generalization power. Many equivariant symmetric architectures historically rely on the assumption that the interatomic potential can be approximated well by a convergentcluster expansion, i.e., a decomposition in terms associated to atomic pairs, triplets, quadruplets, and higher-order tuples. We describe an architecture that incorporates locality, smoothness, and symmetry priors, but that – thanks to a computationally efficient tensor product formulation – does not assume low-order truncation of the body-ordered series.

Conformal prediction provides prediction sets with finite-sample guarantees, but the label verification required for calibration can be expensive. We develop a partial verification method that returns exactly the same prediction sets as complete verification. We characterize calibration certificates, the verified information sufficient to determine the conformal threshold, and design a procedure that coordinates verification across calibration examples. For finite thresholds at high coverage, its verification cost is less than twice the minimum certificate cost when candidates are checked in order. Across retrieval, mathematical solutions, and configuration evaluation, it reduces verification cost by 15-82% compared with verifying calibration examples one at a time, while producing identical prediction sets.

We prove an exact identity, as convergent logarithmic germs at large Euclidean momentum, between the equal-mass two-dimensional $(n-1)$-loop sunset Feynman integral, with $n\geq4$, and the genus-zero one-point descendant invariants of the local Calabi–Yau $n$-fold $X_n=\operatorname{Tot}(K_{F_n})$, where $F_n\subset(\mathbb P^1)^{n}$ is a smooth $(1,\ldots,1)$ hypersurface. Writing $s$ for the momentum variable and $y=-1/s$, the identity reads \[ I^{(n-1)}_{\circleddash}(s)=(-1)^{n-1}yA\,\Bigl[nR^{n-1}-\theta_Q{\mathcal D}_0^\Delta\,R^{n-2} +\sum_{j=0}^{n-3}c_jR^j\Bigr], \] where $R=\log y+O(y)$ is the local mirror coordinate, $A=\theta_yR$, $Q=e^R$, ${\mathcal D}_0^\Delta(Q)$ is the primary one-point potential, and each $c_j$ has the form \[ c_j=c_j^{(0)}+\sum_{a=0}^{n-2}\bigl(c_{j,a}\,{\mathcal D}_a^\Delta(Q) +c'_{j,a}\,\theta_Q{\mathcal D}_a^\Delta(Q)\bigr) \] with the coefficients in $\mathbb Q[\zeta(3),\zeta(5),\ldots]$ the odd zeta values coming from the Gamma class. The descendant series ${\mathcal D}_a^\Delta(Q)$ are given explicitly by the ambient hypergeometric series and the inverse mirror map. We also prove global univalence of the principal mirror map, with explicit upper and lower bounds on the $Q$-Taylor radii, extract the full primary evaluation divisor and its integral cover transform, and reduce the ambient endpoint constants to a finite universal Gamma-class part and $\lfloor(n-1)/2\rfloor$ real parameters.

Corrective feedback is among the best-evidenced drivers of second-language acquisition, yet corrections delivered during lessons rarely accumulate into an actionable view of grammar mastery. Prompted frontier models can provide such a view from learner–tutor lesson transcripts, but they are costly at scale. We close this gap by fine-tuning Qwen3.5 small language models (SLMs) on filtered and rebalanced teacher-generated supervision, then deploying an efficient 0.8B model in an end-to-end grammar mastery tracker for all English learners on our platform. Internalizing the annotation contract into adapter weights enables pairing the 0.8B model with a compact matched prompt rather than verbose instructions. On two human-curated benchmarks, both the deployed 0.8B model and a 4B reference comparator outperform prompted GPT-5.4 and GPT-5.6 Sol in precision and recall under nested matching criteria of increasing strictness: concept, evidence span, and correctness. The deployed 0.8B SLM reduces serving cost by approximately 16$\times$. A feature-level online experiment shows significant gains in learner engagement ($+15.8\%$) and key business metrics, including scheduled hours ($+2.1\%$) and GMV from new lessons ($+13.2\%$).