Papers

We formulate the low-energy theory of superconducting pairing in the geometric algebra $Cl_{3,0}$. Conventional BCS mean-field theory and its extensions ($s$-, $p$-, $d$-wave, etc.) have algebraic counterparts in an eight-dimensional real multivector space. The construction is a two-level projection: matrix blocks are represented as Clifford multivectors, and their grade-0 projected geometric products extract the scalar blocks from which Hamiltonians are built. The result is an algebraic reconstruction, physically equivalent to the standard complex-matrix formulation. In this reconstruction, the dichotomy of pairing channels acquires an algebraic root: it follows from the fermionic identity $B^\dagger(V)=-B^\dagger(V^{\mathsf T})$, and the Fierz rearrangement is realized by the map $\Phi:G\mapsto Ge_{31}$ (with $e_{31}^2=-1$). This yields a kernel/matrix dictionary: at the effective-kernel level grades $0\oplus3$ correspond to spin singlets and grades $1\oplus2$ to triplets; at the pairing-matrix level the singlet plane is $\mathrm{span}\{e_2,e_{31}\}$. The BCS limit has a double grade identity: kernel grade 0 (scalar glue) and vertex grade 2, a bivector ($i\sigma_2\leftrightarrow e_{31}$). We state three algebraic results: (i) a decomposition theorem for BdG quantum geometry with a null-texture criterion, applied to the chiral-state controversy in CsV$_3$Sb$_5$; (ii) a Fierz sign duality in the four-dimensional Minkowski algebra, with a chirality-reversal extension to two-node Weyl-semimetal models; (iii) an error bound and condition-number structure for tomographic reconstruction under a linear-response probe model. Illustrations include a glue-generation mechanism, a one-loop algebraic renormalization flow whose feeding constant coincides in magnitude with the $\mathfrak{su}(2)$ Casimir, closed-form $\mu=0$ inter-node spectra, and a UTe$_2$ two-phase analysis.

In this paper, we study the Fisher–KPP equation \[ \partial_t u = \alpha\Delta_{\TT_{q+1}}u+\beta u(1-u), \] on the homogeneous \((q+1)\)-regular tree \(\TT_{q+1}\), with \(q\geq2\), for nontrivial, compactly supported radial initial data. We first obtain an explicit representation of the fundamental solution of the corresponding linearized problem, together with sharp two-sided pointwise estimates. These estimates identify the propagation threshold \[ \beta>\alpha(\sqrt q-1)^2 \] and determine the associated critical speed \(c_\ast\) and decay rate \(\lambda_\ast\). In this regime, we prove that, for every fixed \(\theta\in(0,1)\), the outermost \(\theta\)-level set satisfies \[ \kappa_\theta(t) = c_\ast t-\frac{3}{2\lambda_\ast}\log t+O(1), \qquad t\to\infty. \] Therefore, while the branching geometry affects the threshold for propagation and the values of the critical parameters, the classical Bramson factor \(3/2\) persists. Our results therefore extend classical propagation and logarithmic-delay phenomena, previously known for one-dimensional continuous and discrete Fisher–KPP models, to homogeneous trees.

Modeling spatiotemporal coupling is a key challenge in building physical intelligence across scales, from microscopic to macroscopic. Existing models capture such structure broadly through physics-motivated dynamical formulations or learning-motivated architectures. The former provide stronger priors but may constrain flexibility, whereas the latter are more flexible but leave the spatiotemporal coupling largely implicit. We therefore seek an approach that combines flexible learning with an explicit geometric bias for jointly modeling time and space. To this end, we propose Minkowski Positional Encoding (MinkowskiPE), which uses joint temporal and spatial coordinates to parameterize Lorentz transformations applied to query and key features. With MinkowskiPE, the query-key attention score depends on position only through the relative spacetime displacement between the two tokens and is therefore invariant to global translation of the coordinates. This paradigm retains the standard dot-product attention interface and remains compatible with efficient attention implementations. We evaluate MinkowskiPE on microscopic molecular dynamics and macroscopic video prediction tasks, achieving the best results on all nine multi-trajectory molecular evaluations and reducing KTH video-prediction MSE by 9.9% relative to the best baseline while using roughly one-tenth as many parameters.

Reward models underpin the alignment of large language models, yet the dominant designs reduce each prompt–response pair to a point estimate or to a distribution from a fixed parametric family. This is at odds with human preference, which is inherently multimodal: the same response can be reasonably judged in many ways, and no single family covers all of them. To better fit this structure, we introduce DRM, a Diffusion Reward Model that recasts reward modeling as conditional density estimation over $p(\mathbf{r}\mid x,y)$. Conditioned on a frozen LLM encoder, a lightweight Diffusion Transformer denoises Gaussian noise into a reward vector, placing no parametric assumption on the output distribution and naturally representing its multimodal structure. A single architecture handles both multi-attribute regression and pairwise preference data, and at inference $N$ samples form an empirical reward distribution that can be aggregated into a scalar, a variance, or quantiles. Across five benchmarks, DRM matches or surpasses baselines under matched data and backbone, stays competitive with much larger discriminative, distributional, and generative RMs despite its modest training scale, and recovers multimodal reward structure where conventional heads collapse to a point. Uncertainty-aware rejection and lower-confidence-bound (LCB) aggregation further demonstrate that DRM can exploit distributional information beyond a scalar reward to improve reward-model decisions. Downstream RLHF experiments additionally show that using DRM as the training-time reward leads to improved policy performance, directly validating the practical benefit of diffusion-based reward modeling for RLHF training.

Understanding quantum phases of matter has long relied on physicists' intuition and mathematical tools such as symmetry and topology. Remarkably successful as these approaches have been, they provide no universal way to explore a Hamiltonian space whose organizing principle is not known in advance. In this work, we introduce a fully autonomous system combining differentiable programming and unsupervised learning for quantum phase discovery. The search evaluates ground-state data along an adaptive trajectory rather than on a predetermined parameter grid. We demonstrate the system with three different solvers and benchmark it against random sampling at equal ground-state-evaluation budgets. On a generalized cluster chain hosting up to $200$ distinct phases, the search finds up to $25$ more phases at the same budget, and matches random sampling given thirty times its budget. On a $50$-parameter Chern insulator, it reaches sectors not obtained by the simple harmonic constructions considered here, in a family whose inverse problem remains open, while recovering all sectors found by sampling. Our results establish autonomous, gradient-driven exploration of Hamiltonian space as a practical route to discovering quantum phases without phase labels or a prescribed target phase.

A sub-grid closure of the scalar-dissipation rate for the Large Eddy Simulation (LES) of turbulent premixed flames is assessed experimentally. Hotwire anemometry is performed to characterize the background turbulent flow, and the Rayleigh scattering measurements are conducted to acquire instantaneous temperature fields, which are used to obtain the scalar dissipation rate. Either methane or hydrogen-enriched (40% by volume) methane is used as the fuel. The Karlovitz number varies between 1.7 and 34.5. The LES filter size is changed from 0.1 to 3.0 times the thermal thickness of the corresponding laminar flames. Both Gaussian and top-hat filtering methods are used. Statistical analyses are performed to investigate the characteristics of the measured, resolved, and unresolved scalar dissipation rate as well as how these are influenced by the normalized filter size, filter type, and the Karlovitz number. It is shown that the algebraic model allows for acceptable prediction of the Favre-filtered, temporally and spatially averaged scalar dissipation rate across the examined test conditions. The experimental measurements are used to gain insight into a key parameter used in the algebraic model. While differences between the previously proposed and our measured value of the model parameter for small Karlovitz number flames are reported, good agreement between our measured values and previously suggested values is obtained for relatively larger Karlovitz number flames. This finding is shown to be consistent across the examined filter sizes and filtering methods, despite the differences in the examined flame configuration of the present experimental study and those of past numerical investigations. This motivates future investigations to assess use of the algebraic model for simulating turbulent premixed flames with an extended range of operation.

Multimodal learning beyond two modalities commonly leverages a specific modality (e.g., text) to bind other modalities. However, how to establish a more balanced representation space that approximates shared semantics while respecting the holistic geometry of $n$-modal data remains challenging. In this work, we present BaryBind, which aims to transport the specific modality towards the Wasserstein barycenter (WB) optimized across all modalities and introduces a volumetric alignment objective to establish a unified semantic space around the WB embedding. Specifically, we project specific modalities to the WB, which minimizes the average Wasserstein distances to multimodal distributions and serves as the anchor for subsequent alignment. We then construct a barycenter simplex, whose volume is taken as a similarity metric for global alignment centered at the WB. Experiments show that BaryBind achieves competitive performance in text-video-audio retrieval, classification, videoQA, and cross-modal generation tasks, along with robustness under modality absence and scalability to more than three modalities. Code is released at https://github.com/xl-tang3/BaryBind.

This paper introduces oracle-parametrized automated market makers (OP-AMMs), i.e., automated market makers whose quoted price depends jointly on the pool reserves and an external oracle price. In doing so, we extend the information-agnostic AMM framework to settings, such as tokenized securities, for which price discovery occurs off-chain. Under a strict oracle-contraction condition, we show that the quoted price of any OP-AMM interpolates between the oracle price and an implicit autarkic price determined by the pool reserves. We then derive a general loss-versus-rebalancing (LVR) decomposition that separates the residual exposure to market lags from the losses induced by oracle errors. This analysis is further extended to stale, discrete-update oracles and to sandwich attacks around oracle updates. Using this framework, we find conditions under which OP-AMMs simultaneously increase local capital efficiency and reduce normalized LVR relative to information-agnostic AMMs. However, sufficiently noisy or stale oracles can reverse these gains. A counterfactual backtest using one-second SPY NBBO data is provided to demonstrate these trade-offs. In particular, we map the Pareto-efficient frontier of oracle-parametrized constant function market maker (OP-CFMM) designs across stylized oracle regimes.

We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, uniformly equi-integrable subset of $L^{q_c}$. The main difficulty is the possible concentration of the critical gradient energy. We overcome it through a two-scale blow-up argument: the first scale produces a small Hamiltonian coefficient, while energy normalization at the second scale yields strong endpoint compactness, allowing a parabolic Liouville theorem to rule out concentration. The resulting uniform little-Hölder estimate, combined with critical Gagliardo–Nirenberg interpolation, permits absorption of the nonlinear term in the parabolic Calderón–Zygmund estimate. As an application, we construct bounded strong solutions for $f\in L^{q_c}$ and continuous initial data $u_0$, and prove subsequential strong interior convergence of smooth approximations in $W^{2,1}_{q_c}$.

Electrolytes under an external magnetic field exhibit a Hall conductivity, much like metals and semiconductors. However, existing theories fail to quantitatively account for their measured Hall conductivities. To make progress toward such a theory, we develop a stochastic density functional framework for electrolytes in a magnetic field. This framework, which we dub odd SDFT, extends the celebrated stochastic density functional theory (SDFT) to incorporate transverse mobilities and chiral time-correlated noise. Using this framework, we calculate the Hall conductivity in the absence of hydrodynamic solvent effects through three approaches: (i) direct response, (ii) Green-Kubo relation, and (iii) a hybrid fluctuation-dissipation relation. We further show that odd SDFT applies more broadly to fluids with broken time-reversal and mirror symmetry, including odd-diffusive fluids and chiral active matter. Our results generalize the relaxation correction of Debye-Hückel-Onsager theory to magnetized electrolytes and provide a foundation for incorporating additional effects mediated by the solvent that could quantitatively reconcile theory and experiments.

Translating natural-language access-control requirements into policies requires careful reasoning about permissions, constraints, and exceptions, and even frontier LLMs often produce policies that violate the intended authorization semantics. We construct CedarInstruct, to our knowledge the first dataset that supports both training and semantic evaluation for formally verifiable Cedar policy synthesis. It contains 5,800 scenarios across 44 domains and 1,408 representing a single synthetic organization, each with a verified target policy and an executable verification plan. On this data we introduce RAISE, which trains policy synthesizers from formal verification in two stages, verified supervised fine-tuning (SFT) followed by a reinforcement learning (RL) stage that learns from verifier signal. We find that SFT succeeds largely by letting models express authorization logic they already have, since untrained models rarely write valid Cedar but often reason correctly when they do. After SFT, how the verifier's information is used matters more than how much of it is used. Of six RL instantiations that consume progressively richer verifier signal, only RAISE-OC improves meaningfully on SFT; it turns failed checks and symbolic counterexamples into guided exploration and learns from the result with off-context GRPO. With about 5.4K verified scenarios and LoRA fine-tuning, RAISE-OC trains Qwen3.5-9B to surpass zero-shot GPT-6 Astra and Claude Opus 5 by 13.33 and 16.26 percentage points in semantic success on held-out scenarios, and training transfers to the independently constructed CedarBench.

Hydrogen-diesel dual-fuel (H2DF) combustion engines offer a promising pathway for decarbonising hard-to-electrify transport sectors, yet their highly nonlinear dynamics and coupled process variables demand constraint-aware control strategies. Model Predictive Control (MPC) meets these requirements but requires an online optimisation in every combustion cycle, which limits deployment on low-cost embedded hardware. This paper trains a feedforward deep neural network (DNN) by behaviour cloning (BC) to imitate an MPC expert, using 86,000 engine cycles of demonstration data collected at 1500 min-1 on a modified Cummins QSB 4.5-litre hydrogen dual-fuel engine. Two variants, one with process feedback and one without, are validated experimentally. Both track unseen fast-transient load steps (3-8 bar indicated mean effective pressure, IMEP) with normalised root mean square error (NRMSE) values of 7.80% and 9.03% against the expert's 8.01%, while keeping mean NOx and particulate matter emissions at or below those of the expert. Inference takes 2 ms or less on a Raspberry Pi 400 (ARM Cortex-A72 at 2.2 GHz), including 1 ms communication latency, compared to up to 7 ms for the MPC expert, a 3.5x speedup. Open-loop profiling on a low-cost ESP32 microcontroller at 180 MHz gives 4.3 ms per inference, 4x faster than required for the 18 ms cycle window. Beyond the training range the cloned policy saturates its controls but exceeds the pressure-rise-rate limit. To the authors' knowledge, this is the first experimental BC controller for cycle-to-cycle combustion control of an internal combustion engine (ICE), trained from demonstrations recorded on the engine itself.

Colossally abundant numbers have large sums of divisors relative to their size. They also admit a geometric description using supporting lines of a planar convex hull. Starting from this description, we change coordinates to obtain nested subsets of these numbers, which we call colossally abundant numbers of higher order. Their definition does not depend on the Riemann hypothesis. We give conditions under which Robin's inequality holds on any one of these subsets if and only if the Riemann hypothesis is true. Some of the resulting families have infinite sets at every level but an empty intersection. For two classes of coordinates, we determine how fast the least members grow. We also prove that every fixed positive integer divides all members of sufficiently high order. Under a concavity assumption, the least members form a divisibility chain; for power abscissas, successive quotients have unbounded numbers of prime factors, with an explicit limsup growth rate. In a second class of families, every global maximum of the normalized divisor sum is retained, and the least members tend to infinity if and only if the Riemann hypothesis is true. We give numerical examples and explain how the Ramanujan–Nicolas bounds produce terminating chains.

Charge regulation refers to the ability of ionizable biomolecules and their counterparts to adjust their ionization state in response to external perturbations, including changes in pH, ionic strength, or electrostatic interactions with neighboring charged species. Here, we use explicit-ion molecular dynamics / charge-regulation Monte Carlo simulations to compute the interaction force between two spherical, charge-regulated nanoparticles as a function of separation, and compare symmetric (chemically identical) and asymmetric (chemically complementary) surface configurations against the constant charge approximation. We find that charge regulation produces up to an order-of-magnitude stronger short-range attraction than constant charge for asymmetric nanoparticle pairs, particularly at low electrolyte concentration, and that charge regulation qualitatively alters the interaction between symmetric, chemically identical nanoparticles, driving a crossover from net repulsion to net weak-attraction as ionic strength decreases. This transition is not observed under the constant charge assumption which shows that charge regulation is not a minor correction but can change both the strength and the sign of nanoscale electrostatic interactions, with implications for the assembly and stability of charge-regulated colloidal systems

Fix a prime $\ell$. Let $K = \mathbb{F}_q(t)$ be a global function field such that $\gcd(q,6) = 1$ and $q \equiv 1 \pmod \ell$. Let $E$ be a non-isotrivial elliptic curve over $K$. Given a fixed monic polynomial $Q$ over $\mathbb{F}_q$, and assuming some mild conditions on $E$, we show that the rank of $E$ does not change with respect to a positive proportion of $\mathbb{Z}/\ell \mathbb{Z}$ extensions $K(\sqrt[\ell]{f})/K$, as $f$ varies over the set of monic polynomials over $\mathbb{F}_q$ such that $f \equiv A \pmod Q$ for any given polynomial $A$. We obtain this result by combining analytic and probabilistic techniques to study the distribution of certain prime Selmer groups of auxiliary abelian varieties constructed from these polynomials $f$.

Since the advent of knowledge distillation, KL divergence has been the standard loss in distillation. Recently, on-policy distillation (OPD) has emerged as an efficient post-training paradigm for LLMs. As a distillation method, OPD naturally inherits KL divergence as its standard loss. However, in this work, we find that KL divergence may not be necessary for OPD. We show that simply preserving the update direction is sufficient for effective OPD. As long as the update direction is toward the teacher, OPD works. More precisely, it is not the direction of every token, but the direction of a small subset of tokens where the teacher and student disagree strongly. We first show that simply assigning a reward of (+1) to tokens where the teacher probability is higher than the student probability and (-1) where it is lower, which merely encourages updates toward the teacher, reproduces almost the same training mode as OPD with reverse KL. We further show that only the direction of a small subset of tokens with large teacher-student disagreement is critical, and training works as long as their update direction is toward the teacher, even if other tokens are pulled away from the teacher. And as an application of these findings, we introduce Consensus Multi-Teacher On-Policy Distillation (C-MOPD) to improve Multi-Teacher On-Policy Distillation (MOPD). Unlike MOPD, which routes each sample to a single teacher and may cause capability conflicts across domains, C-MOPD lets every sample be supervised by all teachers. Experiments show that C-MOPD consistently outperforms MOPD on both math and code benchmarks. Our code is available at https://github.com/LeapLabTHU/KL-Free-OPD.

Conventional reflectance photoplethysmography (PPG) typically reduces tissue optical responses to one or a few detector channels. Matrix Pinhole Image Sensing (MAPIS) instead simultaneously samples multiple aperture-dependent optical signals. We analyzed 44 recordings of 30 seconds each from a single MAPIS device to characterize the spatial relationships among DC intensity, pulsatile AC, perfusion index (PI), and temporal coherence across Red and infrared channels. AC increased sublinearly with DC at both wavelengths, resulting in higher PI toward lower-DC apertures, consistent with a first-order relative path-sensitivity interpretation. A second counterintuitive spatial trend was observed in temporal coherence: lower-DC and lower-AC apertures exhibited higher autocorrelation and greater cardiac-periodic spectral concentration. IR signals also showed consistently higher temporal coherence than corresponding Red signals. These observations demonstrate that multi-aperture PPG resolves reproducible spatial relationships among optical intensity, fractional pulsatile sensitivity, and waveform coherence that are largely averaged together in conventional single-channel PPG. The present study establishes within-device relationships and motivates controlled studies of the underlying photon-path mechanisms.

Let $M$ be a closed manifold with a bracket-generating distribution $H$. We study the Morse index of exponentially subelliptic harmonic maps. After establishing finiteness, we estimate the index through two test spaces. The extrinsic one averages the index form over projected ambient parallel frames, reducing the estimate to a trace identity involving an extrinsic scalar with a quartic exponential term. The intrinsic one uses parallel sections of the pullback bundle, on which the index form descends to a coupled quadratic form that is exact when a global parallel frame exists. The bounds apply to round spheres, products of spheres, and convex hypersurfaces. On a closed Heisenberg nilmanifold, the intrinsic reduction yields a spectral formula for the index.

We study the Siegel–Veech transform on the Poincaré section for the horocycle flow consisting of lattice surfaces with a visible horizontal holonomy vector of length at most one. We compute its first moment and derive formulas for higher and factorial moments with respect to the section's induced ergodic invariant probability measure, which is not supported on a periodic orbit. We emphasize these formulas in the case of the square torus of unit area and, as an application, use its first-moment formula to recover the Boca–Zaharescu pair-correlation density for Farey fractions.

As LLM-generated text becomes increasingly human-like, accurately localizing LLM-authored spans in human-LLM co-authored documents is important for attribution and accountability in cases involving copyright infringement, fraud, and other harmful uses of AI-generated content. Sentence-level detectors provide local authorship evidence, but content variation can cause score fluctuations even among sentences from the same source, creating spurious boundaries. Recovering a coherent document partition therefore remains challenging when both the number and locations of authorship transitions are unknown. We propose Local-Evidence-Aware Change-Point Detection (LA-CPD), a structured method that transforms noisy sentence-level score sequences into coherent authorship segments. Given scores from a frozen local detector, LA-CPD combines a length-weighted within-segment residual with a windowed two-mean contrast to capture segment consistency and sustained changes around candidate cut points. Dynamic programming optimizes cut locations for each candidate count, while an AIC-style criterion selects the final partition, yielding sentence labels, authorship boundaries, and maximal LLM-authored spans. On a held-out human-LLM co-authored test set, LA-CPD outperforms WCP+AIC, increasing sentence-level accuracy from 0.747 to 0.796 while improving boundary localization and LLM-span delineation.

A diffusion model can predict a follow-up medical scan from a baseline, but a clinician needs a per-voxel map of where that prediction can be trusted. Many such maps approximate the diagonal of the Tweedie posterior covariance, and are evaluated against another approximation of it, so whether the estimator or the target limits them is unclear. We compute the exact diagonal on six checkpoints across fourteen model-corpus conditions. Hutchinson at M=200 tracks it at rank agreement of at least 0.92 everywhere, yet in four of the fourteen the exact diagonal is anti-correlated with the denoising error, reaching -0.13, so a faithful estimator reproduces that reversal. All four are real-image conditions; on the models' own samples the reversal does not appear, so evaluating on generated samples flatters this family. What limits these maps is the target, not the estimator. We then introduce Tweedie Probe-Tangent (T-PT), a gradient-free residual probe that corrupts one model-supported prediction repeatedly and measures the voxel-wise variance of the denoiser's response. T-PT reads a different functional of the same Jacobian, and its exact second-order form ranks with the diagonal wherever the diagonal reverses; at thirty probes it returns a map too unstable to reproduce that ranking, while Hutchinson at M=5 already reproduces it, so T-PT there is not evidence against the reversal. We offer it as an instrument, not a better approximation. On brain MRI at full resolution, where every Jacobian-based estimator we test runs out of memory, T-PT leads a twenty-chain Monte-Carlo ensemble on five of eight endpoints inside tissue and trails it on none, at 16x fewer network evaluations; over the whole volume the ensemble leads, and fifty chains close the tissue gap. On lung CT the ensemble is ahead throughout. Both lose most of their discrimination where the change is, which remains open.

We construct a two-parameter family of complex structures on the standard six-sphere, starting from a rational elliptic surface with singular fibers $III^*,I_1,I_1,I_1$. The resulting torus fibration over $\mathbb{P}^1$ has one multiple fiber of multiplicity four, one hexagonal fiber, and two rank-one degenerations. These complex structures are not biholomorphic to the original Alpöge–Claude examples.

Structural observations in polymer chemistry are often rationalized through underlying mechanistic insights. The study of families of mechanisms - and the phenomenology and lawlike behavior that can accompany them - is less common. Here we construct a theory that links a mechanistic feature - the number of nearest neighbors $k$ that influence monomer incorporation - with the accessible copolymer sequence space. We find that the reaction mechanisms can be mapped 1-to-1 to a regular de Bruijn graph of degree $k$. We calculate the periodic copolymer sequences as function of neighbor index $k$, and number of monomers $s$. We show that the number of periodic sequences grows explosively with $s,k$, bounded from below by $s!^{s^{k-1}}/s^k$. We find that periodic k-polymerization is governed by several simple laws describing allowed sequences, number of specific reactions, repeating unit length and the allowed and disallowed sequence coexistence in block copolymers. We formulate Simple Assembly Rules (SAR) that capture commonly observed behavior and design of literature systems, and characterize the possible k-polymerizations they are allowed to access ($k\leq2$) as function of symmetries and illustrate our findings with reported examples from the coordination chemistry and supramolecular chemistry literature. We discuss chemical-structural contributions by which nontrivial ($k>1$) sequences are formed, highlighting the importance steric hindrance, long-range interactions and degrees of freedom.

A unified linear theory is developed for the unsteady loading of finite-chord aerofoils, rigid or porous, in compressible subsonic flow. The formulation combines Possio's integral operator with a convective permeable boundary condition, allowing chordwise-varying admittance while retaining the wake and unsteady Kutta condition. The loading exponents at aerofoil edges and admittance discontinuities keep their incompressible form in terms of the local permeability parameter, so weighted-Jacobi collocation carries over. The solution is verified against published incompressible results and an independent compressible formulation. Gust and heave responses and their indicial counterparts reveal that permeability controls the sensitivity of unsteady loading to compressibility. As permeability increases, the material impedance rather than the surrounding flow sets the pressure jump, but only gradually: at Mach number 0.7 the gust load on a weakly permeable surface changes by a third to a half, and closed-form steady and high-frequency limits for resistive surfaces show that it vanishes only when the permeability parameter greatly exceeds the Mach number. The porous-to-impermeable load ratio, by which a treatment is judged, cannot be obtained by combining incompressible porous and compressible rigid theories, which misjudge it by up to a factor of two. Incompressible theory overestimates this ratio for an acoustically compact chord and underestimates it for a non-compact one, by factors of 1.4 to 4.3 at Mach numbers 0.5 to 0.7 and reduced frequencies 5 to 20, overstating the load reduction. Since compactness depends on Mach number times reduced frequency, the error persists at low speed, exceeding 10 per cent at Mach number 0.05 for reduced frequencies above 10.

Let $X$ be a smooth projective surface over $\mathbb{C}$ with an effective action of a finite group $G$, and let $L$ be a $G$-linearized line bundle. We develop a residue-theoretic and birational approach to computing the equivariant Euler characteristic $\chi_G(X,L)$. Every such $L$ admits an equivariant divisor presentation. Divisor peeling applies the divisor exact sequences one layer at a time, treating whole $G$-orbits of components together. It reduces the problem to the structure sheaf and lower-dimensional equivariant contributions. On a resolution of $X/G$, the natural extensions of the isotypic direct-image sheaves preserve Euler characteristics, but their induced connections need not be logarithmic, as an explicit $D_4$ calculation shows. Blowing up fixed points before taking the quotient reduces the local calculation to diagonal actions and cyclic Hirzebruch-Jung singularities. We obtain explicit correction class functions for the ADE singularities and a uniform procedure for general diagonal actions, including quasi-reflections. In the diagonal case, the natural extensions are logarithmic, and their residues are computed from monomial valuations on the Hirzebruch-Jung resolution. Combining these residue calculations with the branch-curve terms and divisor peeling gives a procedure for computing the equivariant Euler characteristic of every $G$-linearized line bundle. We compare the local classes with the holomorphic Lefschetz formula and the Kleinian coefficients of Lim and Rota, and give examples on $\mathbb{P}^2$, including one with fixed curves.

Reinforcement learning with verifiable rewards (RLVR) enhances reasoning in large language models (LLMs) through outcome-level feedback, yet recent approaches to finer-grained credit assignment often require auxiliary models, additional sampling, or privileged information. Although policy entropy provides a readily available signal, prioritizing uncertain positions under both reinforcement and penalization concentrates penalties where failed responses still retain alternatives for recovery, which can suppress opportunities for exploration. To address this, we introduce Entropic Advantage Policy Optimization (EAPO), an entropy-guided credit assignment method that treats success and failure asymmetrically. Specifically, motivated by the observation that success under uncertainty is less repeatable while confident failures tend to recur, EAPO couples normalized policy entropy with the sign of the response advantage to reinforce surprising success and correct repeated failure. It assigns stronger reinforcement to high-entropy decisions in successful responses and stronger penalties to low-entropy decisions in failed responses, while attenuating penalties at uncertain positions to preserve opportunities for recovery. By redistributing the response advantage across tokens, EAPO derives token-level credit directly from existing rollout signals without additional supervision. We validate EAPO on a range of reasoning tasks across both base and reasoning backbones, demonstrating that it achieves the best overall performance. We further show that EAPO promotes more effective exploration, broadening problem coverage and generating more diverse candidate answers.

Verified solutions are not equally useful for preparing reasoning models for reinforcement learning (RL). We present a comprehensive study of route diversity, the variation in the sequences of reasoning steps in supervised fine-tuning (SFT) data, and propose a lightweight, rule-based fingerprint to select for it. From one pool at one budget, with matched training recipes and checkpoints, selecting diverse rather than similar routes improves post-RL problem coverage across puzzles and mathematics, including on problems harder than those seen in either training stage. In synthetic experiments, route-diverse SFT improves OLMo3-7B's pass@8 by 16.9 points on environments held out from SFT. In a single-model condition, where one model writes every candidate, diverse selection gains up to 6.2 points of mean pass@8 across 10 mathematics benchmarks. Pre-RL diagnostics suggest why: diverse SFT can produce both successful and failed attempts on more prompts despite slightly lower mean accuracy, giving group-relative RL more prompts with a learning signal. On 3 open-source corpora, our CPU-only selector, without model calls, outperforms more expensive alternatives in every comparison of mean post-RL performance. These results identify reasoning-route diversity as a practical criterion for selecting SFT data that better prepares models for RL.

Vertex Integrity is a classical graph parameter measuring the vulnerability of a network to vertex failures. We settle two fundamental questions concerning its exact exponential complexity. First, we prove that, unless the Exponential Time Hypothesis fails, Vertex Integrity has no $2^{o(n)}n^{O(1)}$-time algorithm. Secondly, we give a deterministic exact algorithm running in $O(1.9602^n)$ time and space, improving the straightforward $O(2^n)$ bound.

Modern multi-hop LLM agents are equipped with built-in mechanisms to detect errors in intermediate reasoning steps. Such errors trigger corrective actions from these agents, which mostly follow the paradigm of retrying the steps or the reasoning trajectories. Not only are these retries expensive, we present in this paper that they are also potentially unnecessary. To this end, we introduce Evidence-Inference Reconstruction (EIR), which uses structured state to guide one retrieval trajectory, accumulating source evidence in the process. We show that as long as the relevant evidence has been collected, EIR is capable of generating the correct answer in a single final model call even if erroneous evidence has been mixed in due to incorrect intermediate reasoning steps. In one evaluation, using Haiku 4.5 and GPT-4.1 Mini, we evaluate EIR on matched 1,000-question subsets of HotpotQA, 2WikiMultiHopQA, and MuSiQue, showing that EIR improves Answer F1, the overlap between the model's and the correct answer, over the baseline by 8.3–32.8 points, Agentic SSR by 10.6–29.1 points, and Reflexion by 1.1–15.9 points. Additionally, we show that EIR averages 4.85 total model calls per question, compared with 35.29 for Agentic SSR and 12.41 for Reflexion. Together, these results corroborate EIR's central premise: separating evidence retrieval from the final answer model call can improve answer accuracy while utilizing substantially less computation.

Phase diagrams and energy surfaces require low-energy states across a family of Hamiltonians, but solving each parameter point independently is prohibitively expensive. An important question is not how a single labeled eigenstate behaves under parameter variation, but how much information about the low-energy sector is contained in a small set of reference wavefunctions. We show that the physically relevant object is the isolated low-energy subspace itself: it remains well defined through degeneracies and level crossings, and it can be learned efficiently from nearby sampled states. Working in this subspace, we prove that a sampling pattern cancelling the first $q$ nonconstant response orders yields an energy-density error bounded by $d^{2(q+1)}$ for every retained level, where $d$ is the parameter-space sampling distance. For a gapped local ground state on $n$ sites, locality sharpens this to $d^2(nd^2)^q$, identifying $nd^2$ as the natural scaling variable governing the effectiveness of higher-order information from sampled wavefunctions. In a local analytic regime this gives a sampling cost $K=\mathcal{O}([\log(1/\varepsilon)]^D)$ at fixed parameter dimension $D$. Numerical results on Heisenberg and transverse-field Ising chains confirm the distance scaling law and the $nd^2$ collapse. Beyond characterizing learnability, the scaling behavior further provides a finite-size probe of phase transitions near criticality.