Papers

A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of $k$-uniform intersecting families on $2k+1$ and $2k+2$ elements as $k\to\infty$. For $n=2k+2$, we prove that almost all intersecting families are trivial and that their number is \[ (2k+2+o(1))\,2^{\binom{2k+1}{k-1}}. \] Together with Yang's recent result for $n\ge 2k+3$, this settles a conjecture of Balogh, Garcia, Li, and Wagner. For $n=2k+1$, almost all intersecting families are nontrivial. We prove that, as conjectured by the same authors, a typical intersecting family is close to a full star: its members outside the star form components of size at most two in the graph joining sets that intersect in $k-1$ elements. We also obtain an asymptotic formula for the number of intersecting families in this case, with an explicit second-order term in the exponent. Our proof combines Sapozhenko's graph container method and stability in Kneser graphs to control families far from every star, and a polymer model and cluster expansion to enumerate families close to a fixed star.

This paper presents a decentralized power-optimal coordination framework for magnetically actuated spacecraft swarms. Swarms that form large space structures overcome the aperture limit set by the launch vehicle and hold their shape on solar-generated power alone. Magnetic actuation is propellant-free and generated by a magnetorquer, which is commonly used for attitude control. However, every spacecraft interacts with every other within range, and its effect depends on the actuation power and a carrier frequency. We therefore design a decentralized power-optimal framework to jointly derive the interaction graph, frequency grouping, and controller gains. Our decentralized controller preserves angular momentum, which is a nonholonomic constraint. Then, this framework for connected groups whose memberships overlap across carriers guarantees that the relative position errors, the absolute attitude errors, and the imbalance of the reaction-wheel momenta converge to the desired states under the decentralized power-optimal allocation. A closed-loop simulation of a thousand spacecraft with the complete alternating-current interaction confirms the framework. A fast approximate integration with a proven error bound extends the framework to a long-horizon orbital reconfiguration held with high precision.

On-policy self-distillation has recently emerged as an effective approach for improving language-model reasoning by supervising students with a frozen or EMA version of themselves that receives privileged information. Its application to multimodal large language models (MLLMs), however, remains largely unexplored. Recent approaches use privileged visual information, such as image crops corresponding to a question, to improve fine-grained perception, but their gains are confined to tasks that benefit from such visual zooming and require either human-annotated grounding data or external teacher models. We introduce a different form of on-policy self-distillation for MLLMs that provides the teacher with textual, spatially grounded guidance identifying the visual elements relevant to a query. We use procedurally generated scenes with automatically available object identities and spatial coordinates, enabling scalable and annotation-free post-training. The teacher uses this spatial guidance to locate and integrate evidence from multiple relevant image regions, while the student learns to reproduce the resulting behavior from the image and question alone. Our approach consistently improves performance on counting, document and chart understanding benchmarks across multiple models. Importantly, although post-training uses only synthetic scenes, the resulting improvements transfer to real-world perception benchmarks, yielding a 3.23-point gain in average performance across CVBench, V*, ZoomBench, BLINK, HR-Bench, and MME-RealWorld. These results show that spatially grounded privileged information can induce broader perceptual capabilities through on-policy self-distillation, enabling substantial synthetic-to-real transfer beyond the task and data distribution used for post-training. Project page: https://github.com/sirkosophia/Where-OPD

A comparative explainability framework is presented to audit DeBERTa-v3 under zero-shot classification of medical abstracts. The work addresses the disagreement problem in Explainable Artificial Intelligence, where different attribution methods produce divergent explanations for the same input and prediction. A natural language inference engine is implemented over the Medical Abstracts corpus with five enriched hypotheses per diagnostic category and a balanced sample of one thousand texts per class. Five explanation methods are compared: SHAP and LIME as model-agnostic approaches, occlusion and Input x Gradient as deep-learning-specific approaches, and Attention x Gradient as a transformer-specific approach. Explanations are standardized through top-token attribution, and pairwise agreement is quantified using the Jaccard index. High predictive accuracy is achieved across well-defined clinical domains, whereas performance degrades under high semantic ambiguity. Explanatory stability directly mirrors predictive certainty, exhibiting strong convergence in univalent categories and a marked drop under diagnostic uncertainty. Furthermore, qualitative error auditing uncovers three systemic failure mechanisms: lexical hypersensitivity, semantic overlap, and loss of attribution coherence. The results support the combined use of several explanation methods and quantitative agreement metrics when auditing transformer-based models in medical text classification, and suggest prioritizing specific clinical ontologies over broad diagnostic labels.

Motivated by applications in random geometry, we investigate the notion of subordination of snakes in the discrete setup. More precisely, given a random walk $W$ indexed by a tree $T$ and with steps in $\{...,-1,0,1\}$, we consider its subordinate tree obtained by contracting every edge of $T$ that does not lead to a new record of the walk $W$. When the underlying tree $T$ is a Bienaymé-Galton-Watson tree, we characterize the distribution of this subordinate tree. In particular, when $T$ has a critical offspring distribution in an $\alpha$-stable domain of attraction with $\alpha\in(1,2]$, and under a light tails assumption on the steps, we prove that the associated subordinate tree is itself a Bienaymé-Galton-Watson tree with an offspring distribution in an $\frac{\alpha+1}{2}$-stable domain of attraction. Along the way, we obtain the asymptotic tail of the maximal displacement of the critical branching random walk $W$ in this stable regime, under minimal assumptions. Finally, we use these results to prove scaling limit statements about the subordinate tree.

Existing genome-wide association studies (GWAS) of brain imaging provide predefined or deep-learning-derived imaging phenotypes, yet these phenotypes come from either volumetric scans or cortical surface meshes, so each captures only part of the heritable variation in brain anatomy. Here we introduce MEVA (Mesh-Enhanced Volumetric Autoencoder), a self-supervised framework that encodes voxel-level image intensity together with cortical mesh geometry, including curvature and cortical thickness at each surface vertex, into one shared set of imaging features. Combining the mesh and volumetric inputs in MEVA yields modest performance gains in age and sex prediction over models that use either input alone. When these features serve as phenotypes for GWAS in the UK Biobank, they reveal more genome-wide significant loci than features learned from volumes alone or from meshes alone. These results suggest that adding cortical surface geometry to volumetric self-supervised learning captures additional heritable variation and so increases the number of loci detected.

We introduce information-theoretically private quantum protocols for two-party Hamming distance when both parties must output the same estimate. Classically, for input length $n$, information-theoretic protocols require $\Omega(\sqrt{n})$ error under pure differential privacy and $\Omega(\sqrt{n}/\log n)$ error under strong approximate differential privacy, whereas computational security permits $O(1)$ error. In Klauck's honest, nonpreemptive, message-preserving model, we give an $O(n)$-communication quantum protocol with pure $\varepsilon$ quantum differential privacy (QDP) and expected error at most $\frac{2}{\sinh \varepsilon}+\gamma$, for every $\gamma>0$. For approximate $(\varepsilon, \delta)$ QDP, an exact hockey-stick divergence calculation yields strictly smaller error, while preserving the $O(1)$-versus-$\Omega(\sqrt{n}/\log n)$ separation for $\delta=o(1/n)$. Thus, quantum communication achieves $O(1)$ information-theoretic error, matching the accuracy available classically only under computational assumptions. The main construction uses a guarded coherent round trip and an equal-Gram rigidity principle that prevents an honest player from retaining input-dependent complementary information. We separate this model from weaker prescribed-channel privacy, which already admits an exact classical realization, and from fully retention-robust security, against which measurement-and-abort attacks remain possible. Therefore, we identify preservation of non-orthogonal quantum messages as a resource for privacy.

Cosine transforms see frequent use in image and video compression due to their ease of computation and high energy compaction. This has sparked interest within the optics community in computing cosine transforms optically for compression. When imaging in the Fourier plane to capture a cosine transform, one encounters a phase retrieval problem: optical fields have both magnitude and phase, but cameras only capture the magnitude of the field. Cosine transforms restrict the phase retrieval problem to a binary domain as opposed to the unit circle, but general phase retrieval solutions do not leverage this. In this work, we demonstrate that sign errors in phase retrieval for cosine transforms result in a significant increase in the magnitude of the Hessian of the transform at the error. Motivated by this, we develop an algorithm to solve this binary phase retrieval problem by minimizing the curvature of the reconstructed cosine transform. We demonstrate via computational experiments that given the magnitude of the cosine transform of an image, we can consistently reconstruct the original image within a multiscale structural similarity index of 0.95. We show that the solve time grows approximately linearly with the number of pixels solved.

Determining whether an algebraic number field admits a power integral basis is a classical problem, but it can be difficult for fields defined by polynomial compositions and dynamical iterates. In this paper, we study the monogeneity of compositions $f(h(x))$, where $f(x)$ and $h(x)$ are monic polynomials over an arbitrary Krull valuation ring and $h(x)$ is a trinomial. We derive explicit formulas for the discriminant of the composition and use them to characterize when $f(h(x))$ generates a monogenic field. In particular, we relate the monogeneity of the composition to that of $f(x)$ and to explicit square-free conditions on the critical values of $h(x)$. We further extend the results to polynomial iteration and obtain a criterion for the monogeneity of binomial iterates, yielding new infinite families of monogenic fields. Finally, we give quantitative results and illustrate our criteria with several examples.

Transparent and specular surfaces pose a serious challenge to LiDAR-based SLAM and navigation because laser returns may pass through glass, leaving collision boundaries absent from the map. Prior work attempts to reconstruct the missing surfaces, but inaccurate obstacle placement can create the opposite failure: contamination of traversable free space. Recognizing this dual requirement, we present GlassGuard, a navigation-oriented framework for reconstructing planar architectural glass from complementary visual and LiDAR evidence. We formulate success in terms of both glass coverage and free-space contamination and apply this principle throughout proposal verification and global map construction. A foundation vision model provides glass-instance masks, structural 3D cues generate metric plane hypotheses, and depth-free 2D projective geometry checks their orientations before they enter a consolidated global map. We evaluate GlassGuard in nine building-scale scenes spanning diverse glass structures, spatial scales, and lighting conditions, with more than one hour and 2.1 km of real-world robot traversal. GlassGuard achieves 85% of total glass coverage for its panoramic version. Under identical pinhole inputs, GlassGuard achieves 82% total coverage, compared with at most 61% for the evaluated baselines, while producing 5-17x fewer false voxels per frame. Qualitative examples with a navigation planner illustrate the reconstructed planes blocking paths through glass while leaving traversable routes open. The project page is available at https://glassguardproject.github.io/.

We construct, through Hodge-theoretical methods, what we call the $E_r$-Jacobian torus, the $E_r$-Albanese torus and the $E_r$-Albanese map of any compact complex manifold that is either {\it page-(r-1)-$\partial\bar\partial$} or {\it $E_r$-sGG}. These two classes of manifolds, the former of which is contained in the latter, have been introduced recently by both authors jointly with J. Stelzig, respectively by the first-named author. A Hodge theory is also developed for the latter class of manifolds. We then apply our results to give structure theorems and an identity of algebraic dimensions in terms of the $E_r$-Albanese map and torus. Other applications result in cohomological and metrical theorems for the $6$-dimensional sphere when it is equipped either with a hypothetical complex structure or with the complex structures very recently claimed to exist in the literature. For example, we show that in the latter case no strongly Gauduchon metric exists on $S^6$.

We study resonance expansions and localised long-time dynamics for three-dimensional semiclassical Dirac operators. For smooth Hermitian matrix-valued perturbations which are analytic outside a compact set, we first obtain exponential cut-off resolvent bounds away from the resonance set and polynomial bounds in thin resonance-free rectangles adjacent to the real axis. Combining these estimates with a local $\Oc(h^{-3})$ upper bound on the number of resonances, a pigeonhole selection of admissible contours, almost-analytic functional calculus, and a Cauchy–Green deformation, we derive resonance expansions for spectrally localised propagators. Positive-energy resonances contribute for large positive times, whereas negative-energy resonances contribute for large negative times. We then specialise to mass-type perturbations $\mbf{\beta}V(x)$ in a nontrapping energy window. Using the logarithmic resonance-free region established in our earlier work together with a quantitative FBI-transform/escape-function argument, we obtain a polynomial bound for the continued cut-off resolvent and deduce rapid local-energy decay. We subsequently refine this polynomial estimate by a finite-time propagation argument in the undistorted region. Introducing the maximal positive-sheet connection time $T_\chi$ associated with the spatial cutoff, we prove a geometric continued-resolvent estimate and, for every fixed $M$ below the resonance-free depth constant, derive the explicit local-energy decay threshold $T_N>T_\chi+(N+1)/M$. The same analysis gives a rigorous microlocal obstruction to a uniform $[h\log(1/h)]^{-1}$ bound for the full distorted resolvent and identifies a finite-flight-time amplification mechanism for the sandwiched continued resolvent at logarithmic depth.

Deriving simple and accurate three-flavor neutrino oscillation probability formulas in matter is critical for understanding the phenomenology of upcoming experiments. In this article, we extend the Jacobi diagonalization method for deriving effective parameter mappings for neutrino oscillations in matter. The key insight behind the extension is to purposefully parameterize the mixing matrix, $U_{\text{PMNS}}$, using particular arrangements of Euler rotations. This novel extension allows for deriving simple and accurate effective mixing parameter mappings for exotic types of matter potentials. First, we revisit the Jacobi diagonalization method for neutrino oscillations with standard matter effects and use this to demonstrate our generalization. Subsequently, we apply the Jacobi diagonalization method to scalar non-standard interactions (SNSI) and derive effective parameter mappings for all three single diagonal coupling cases. The resulting formulas reveal that all diagonal SNSI couplings could give rise to a scalar MSW (SMSW) effect, that is an energy independent resonant enhancement analogous to the MSW effect. Finally, we study the accuracy of the derived SNSI formulas for various contemporary and upcoming experiments. For $\delta m_{ee}$, the accuracy is excellent for all considered experiments. For $\delta m_{\mu\mu}$ and $\delta m_{\tau \tau}$ the accuracy is worse since $A_{CC}$ must be neglected, and first order perturbative corrections are applied to amend this.

We give sufficient conditions ensuring that, at every fixed positive time, the scalar backward component of a Markovian forward-backward stochastic differential equation with multidimensional forward process admits a density with respect to Lebesgue measure. Existing density criteria for BSDEs often obtain Malliavin non-degeneracy through sign or monotonicity assumptions. We develop a different route for a scalar backward component with an arbitrary-dimensional forward state. Under regularity assumptions and a structural compatibility condition on the generator, the terminal condition is only required to be non-constant. The key idea is to deduce Malliavin non-degeneracy from deterministic rigidity of the critical set of the decoupling field. We combine Malliavin calculus with unique continuation and backward uniqueness for the associated semilinear parabolic equation. Unique continuation precludes the spatial gradient of the decoupling field from vanishing on a set of positive measure unless it vanishes identically on that time slice, while backward uniqueness propagates such vanishing to the terminal time. Along the way, we establish a unique continuation property from sets of positive measure and a backward uniqueness result on the whole space for the linear parabolic systems arising from differentiated semilinear equations.

Density waves in gravity-driven granular pipe flow reduce transport efficiency and produce large fluctuations in the local solid fraction. Helical wall textures suppress these instabilities but at the expense of reduced throughput. Using discrete element simulations (DEM), \RefA{we find that this homogenization is associated with} spiral particle motion induced by the helical wall texture. Quantifying this motion by the mean angular velocity reveals a strong correlation between spiral motion, flow homogenization, and transport performance. Motivated by this insight, we introduce a helical pipe whose centerline follows a helical path while maintaining a constant circular cross section. This design generates stronger spiral motion than conventional helical wall textures. For suitable design parameters, the helical pipe simultaneously enhances flow homogeneity and increases the mean flow velocity beyond that of a plain straight pipe. These results support spiral motion as a useful diagnostic and design variable for simultaneously improving flow homogeneity and transport efficiency within the investigated parameter range.

We present a framework for propagating systematic data mismatches into parameter biases and posterior deformations beyond the Gaussian Fisher approximation. Using the Derivative Approximation for Likelihoods (DALI), we derive analytic expressions for the maximum a posteriori (MAP) point and posterior mean, including a semi-analytic response expansion in systematic amplitude. We validate the framework against direct numerical calculations and Markov Chain Monte Carlo (MCMC) sampling in a controlled nonlinear two-parameter benchmark up to a covariance-weighted data-space mismatch of $d_{data}=5$. At $d_{data}=5$, Fisher predicts the wrong direction of the shift in one parameter and differs from the numerical result by a Fisher-metric distance of 23.1, whereas the analytic DALI MAP prediction remains within 0.091, with componentwise errors of 0.04% and 0.46%. The response expansion reproduces the posterior mean with errors of 0.02% and 0.18%, while the DALI posterior closely reproduces the displacement and deformation of the fully sampled nonlinear posterior. Mismatches of equal significance but different directions produce markedly different parameter shifts and posterior deformations, while multiple additive mismatches can produce nonadditive parameter responses through nonlinear inference. By reconstructing the posterior under systematic mismatch, the framework captures the nonlinear response and non-Gaussian geometry of parameter bias: how mismatches move and reshape the posterior, why their direction matters, and how their combined effects can be dissected to understand the interplay between multiple systematics.

Schur sampling is a fundamental primitive for extracting permutation-invariant information from many-body quantum systems and has broad applications in quantum information science. Circuit-based implementations typically rely on coherent representation-theoretic operations such as Clebsch-Gordan transforms, generalized phase estimation, or quantum Fourier transforms over the symmetric group. We show that unitary Schur sampling of arbitrary permutation-invariant mixed states on $n$ $d$-dimensional qudits can instead be implemented using only pairwise SWAP tests, namely, two-qudit projective measurements onto the symmetric and antisymmetric subspaces. Our protocol achieves error $\epsilon$ in diamond distance using $O(n\min\{n,d\}^{3}\log^2(n/\epsilon))$ random SWAP tests. The central idea is to repeatedly identify and extract the largest antisymmetric subsystem. This turns unitary Schur sampling into a search for antisymmetry and gives the algorithm a natural interpretation as a stochastic traversal of a Young diagram. The protocol preserves the $\mathrm{SU}(d)$-irrep state while preparing a canonical pure state in the multiplicity subsystem. As an application, we consider quantum purity amplification (QPA), in which multiple noisy copies of a pure quantum state are combined to produce a state of higher purity. We show that the optimal purified state can be obtained by retaining a single designated output qudit and discarding the rest.

The prime graph of a finite group $G$ is the graph $\Gamma(G)$ with vertex set the set of prime divisors $\pi(G)$ of $|G|$ and an edge between vertices $p, q\in\pi(G)$ if and only if there exists an element $g\in G$ with order $o(g) = pq$. Given a finite nonabelian simple group $T$, a group $G$ is $T$-solvable if there exists a composition series of $G$ such that every composition factor is either abelian or isomorphic to $T$. In this paper, we introduce the prime graph of an infinite group, the graph $\Gamma(G)$ with vertex set $\pi(G) = \{o(g):g\in G\text{ and }o(g)\text{ is prime}\}$ and an edge between $p,q\in\pi(G)$ if and only if there exists an element $g\in G$ with order $pq$, and generalize several results on the prime graphs of finite solvable and $T$-solvable groups to results on the prime graphs of members of certain classes of infinite solvable and $T$-solvable groups.

We prove near-optimal lower bounds for preprocessing attacks on quantum cryptography in the random oracle model. Specifically, we show that a $T$-query adversary with $S$ qubits of non-uniform advice can recover a random key $k$ from the $n$-qubit binary phase state $|\psi_k\rangle \propto \sum_{x} R(k,x) |x\rangle$ with probability at most $O(\frac{T^2 + \sqrt{ST}}{N})$ for $N=2^n$. In contrast, the best known bound for post-quantum one-way functions is $O(\frac{T^2 + ST}{N})$, with a trivial attack at $S = N$. This demonstrates a new advantage of quantum cryptography over classical cryptography: $n$ qubits of communication suffice for security against preprocessing attacks with space up to $N^2$ rather than $N$. Our methodology is simple: express the optimal preprocessing attack as the operator norm of a random matrix, and bound this value in expectation over the random oracle via the trace-moment method. These trace moments have a natural interpretation using compressed oracles [Zhandry, Crypto 2019], which we then analyze. This can be viewed as a simplification and generalization of the approach of Liu [Eurocrypt 2023] for proving the security of post-quantum cryptography against preprocessing attacks. We also prove the following results: (1) We tighten Liu's analysis of post-quantum PRGs in QROM, achieving a distinguishing advantage bound of $O(\frac{T^2}N + \sqrt{\frac{ST}N})$. (2) For unitary synthesis, we extend the one-query lower bound of Lombardi-Ma-Wright [STOC 2024] to hold against adversaries that can make one arbitrary function query along with polynomially many (adaptive) queries to the random oracle, either before or after the function query. This also interprets the original LMW24 result in terms of compressed oracles. (3) Finally, we prove a tight $O(\frac{\sqrt{S}}N)$ bound for the pseudorandomness of random binary phase states against space $S$ distinguishers.

Can simple processes appear highly complex? Gowers (Comb. Prob. Comp. '96) conjectured that repeatedly composing local random reversible operations can yield global permutations that are indistinguishable from random. In this work, we study the unitary quantum analog of this question, in an attempt to make new progress on this longstanding conjecture. Our first result shows that statistical moment matching in the form of unitary designs does not generically lead to pseudorandomness—even for the simplest quantum processes: for every fixed $t$, we give an efficiently samplable family $\{\nu_n\}_n$ of distributions on one- and two-qubit gates such that, after $T=O_t(n^2\log^2 n)$ independent steps, the resulting $n$-qubit ensemble is an approximate unitary $t$-design with negligible error $\exp(-\Omega(\log^2 n))$, yet an efficient quantum algorithm distinguishes it from random using only $O_t(\log^2 n)$ queries. This refutes the unitary analog of the Hoory–Magen–Myers–Rackoff conjecture (ICALP '04) for permutations. Our second result is a stronger separation between unitary designs and pseudorandom unitaries at polynomially bounded moments; our counterexample, however, requires highly structured ensembles, in contrast with the simple local walks from before. This suggests caution when using unitary designs to model information scrambling in black-hole physics, as even maximally scrambled systems can exhibit structure which is accessible to efficient experiments. Motivated by these findings, we then propose new conjectures for how pseudorandomness can plausibly emerge within simple quantum processes, such as random quantum circuits.

A longstanding question in quantum communication complexity is whether some task can be accomplished with a small amount of communication in the presence of entanglement, yet require much more quantum communication in the absence of entanglement. Separations of this nature were previously known for relational problems and, in the simultaneous message passing model, for partial functions. But it has remained unresolved whether any such separation exists for a total Boolean function. We resolve this question with an exponential separation in the one-way setting: we exhibit a family of total Boolean functions $f_n\colon \{0,1\}^n \times \{0,1\}^n \to \{0,1\}$ that can be computed with $O(\log n)$ bits of one-way classical communication given prior entanglement, but that require $\Omega(n^{1/3})$ qubits of one-way quantum communication without entanglement. Our function is a special case of the subgroup membership problem, first studied in the communication setting by Aaronson, Le Gall, Russell, and Tani.

Mechanistic interpretability aims to recover the internal computations responsible for model behavior. Progress in automated circuit discovery is often framed as a search problem: better attribution or optimization should identify better mechanisms. This assumes that the evaluation objective can recognize a better circuit once it is found. Yet standard intervention-based evaluation replaces signals outside a circuit to isolate its effect, changing the computational environment in which the retained components operate. We call this change context distortion. This raises the question of whether faithfulness rewards recovery of model behavior or compatibility with the altered context. We show that intervention-based faithfulness can prefer an equally sized circuit that reproduces the model's behavior less well. We call this reversal misranking, an objective-level recovery gap. To isolate evaluation from discovery, we construct equally sized candidates through controlled edits to human-identified circuits in pretrained language models, including IOI and Greater-Than, and benchmark circuits with known mechanisms. We find misranking even without a discovery algorithm. Misranking also occurs in real automated circuit-discovery settings. Among outputs of EAP, EAP-IG, ACDC, and Edge-SP, KL faithfulness under ordinary resampling prefers the behaviorally worse candidate in 13.2%-24.4% of candidate pairs across these methods on four pretrained-language-model tasks. Finally, restoring selected intact-model signals reverses 96 of 100 selected persistent misrankings, demonstrating that context distortion contributes to these failures. In light of these findings, we call for rethinking both how circuit discovery is performed and how its results are evaluated.

Electromagnetic duality in the toric code has order two at the level of anyon types and acts as a Hadamard-type logical transformation on the encoded quantum information. Here, we ask whether it can likewise be realized microscopically as an order-two Clifford operation. For the $\mathbb{Z}_2$ toric code, we prove that any locality-preserving Clifford realization of electric-magnetic exchange cannot have order two. Our proof is fully general and requires neither translation symmetry with restricted families of system sizes nor a fixed pairing between vertex and plaquette stabilizers. Geometrically, the proof locally emulates the unavoidable crossing between electric and magnetic strings, similar to a cross-cap in a non-orientable manifold. Extending the problem to the $\mathbb{Z}_N$ toric code, we find that an order-two Clifford realization exists for odd $N$, whereas it is impossible for even $N$. Our results show that the group law of an emergent electromagnetic duality need not lift faithfully to its microscopic Clifford realization.

We present a global determination of the six long-distance matrix elements (LDMEs) within non-relativistic QCD (NRQCD) appearing in the production of prompt $J/\psi$ up to $\mathcal{O}(v^2)$ using high-$P_T$ LHC data for direct and feed-down contributions from $\chi_c$ and $\psi(2S)$. Using a leading-power fragmentation framework at NLO+NLL, we fit $\psi(2S)/J/\psi$ yield ratios from ATLAS and CMS and $\psi(2S)$ CMS polarization data at $\sqrt{s} = 13$ TeV, injecting one decay width per charmonium. The fit LDMEs provide excellent predictions of the CMS and ATLAS cross-section data up to $P_T=360$ GeV for the $J/\psi$, as well as the $J/\psi$ polarization data from CMS. The obtained excellent agreement with production and decay data demonstrates that $\mathcal{O}(v^2)$ NRQCD is sufficient to describe $J/\psi$, $\psi(2S)$, and $\chi_c$ hadroproduction at large $P_T$.

We prove randomized matrix-vector query lower bounds for two normalized matrix-game geometries: a Euclidean unit ball against a probability simplex, with row norms at most one, and two probability simplices, with entries of absolute value at most one. Each query returns $(Ax,A^\top y)$ for arbitrary real vectors. The algorithm must return a feasible pair with full saddle-point gap at most $\varepsilon$, with probability at least $2/3$ on every admissible matrix. For sufficiently small $\varepsilon$, the worst-case query complexities are $\Omega(\varepsilon^{-2/3}/(\log^2(1/\varepsilon)\log\log(1/\varepsilon)))$ for ball-simplex games and $\Omega(\varepsilon^{-2/3}/(\log^{7/3}(1/\varepsilon)\log\log(1/\varepsilon)))$ for simplex-simplex games. The hard instances have dimensions of order $\varepsilon^{-2/3}$ and $\varepsilon^{-2/3}/\log^{1/3}(1/\varepsilon)$, respectively, and the bounds extend to larger dimensions. These lower bounds match the deterministic upper bounds of Karmarkar, O'Carroll, and Sidford up to logarithmic factors. The proof extracts a fresh Gaussian core after adaptive two-sided queries and uses uncertainty in its smallest singular value to establish linear-system solve hardness. Two reductions transfer this hardness to matrix games by converting a small full gap into a small residual, with an additional logarithmic normalization loss only for simplex-simplex games.

The quantum state preparation problem is to, given a description of a quantum state, efficiently generate a quantum circuit computing the state. We show that for quantum states described by weighted d-DNNF (deterministic, decomposable pseudo-Boolean circuits) a quantum circuit computing the state can be obtained in linear time up to complex arithmetic.

Symmetric private information retrieval (SPIR) lets an inquirer obtain one specific bit only, without leaking any information about its index to the database owner. With a single server, this is impossible for parties with unrestricted quantum power. We show that it becomes possible, with information-theoretic security, when the server and the user hold only noisy intermediate-scale quantum (NISQ) devices without long-term quantum memory and measure each qubit on arrival, while the eavesdropper remains all-powerful; this is a stricter form of the memory restriction of the bounded- and noisy-quantum-storage models. The server sends BB84 states one way; the user discriminates them unambiguously and so knows a random subset of the bits. After a joint random permutation into blocks, the user announces the conclusive locations of one privately chosen block, and the server hashes and takes a random parity of these locations in every block; the permutation makes it exponentially unlikely that one set of locations fits two blocks. We prove secrecy against the eavesdropper, database privacy through an explicit size rule for the announced set, and user privacy against a server who prepares the prescribed BB84 states and announces truthfully, exactly for single photons and up to a small multi-photon term over a tolerated noise channel. The only addition to BB84 equipment is a passive linear-optical measurement with one ancillary mode. For a $10^{4}$-bit database and database privacy $10^{-6}$, blocks of about a thousand qubits suffice over a tolerated noise channel for a user who applies the prescribed measurement; certifying an arbitrary memoryless user and committing to the permutation need larger auxiliary transmissions, which we quantify with finite keys. Database privacy extends to decoy-state weak coherent pulses, and simulations show that the bound is nearly tight.

Data selection is critical for training large language models on massive and heterogeneous corpora. Meta-learning for Training-data Selection offers a principled alternative to heuristic scoring by learning data weights from a target validation objective, but existing methods face a trade-off between fine-grained valuation and transferability to unseen data. A natural solution is to replace per-sample weights with a selection network. However, we find that directly incorporating such a network into existing MTS objectives leads to unstable optimization and poor generalization, caused by weight suppression and persistent reliance on easy-to-learn features. To address these issues, we propose Transferable Example Scoring and Selection (TESS), a scalable data-selection framework built on a Pointwise Value Matching objective (PVM). Experiments on LLM safety and targeted instruction tuning demonstrate strong transfer across datasets, from subsets to full corpora, and from smaller to larger models.

Despite progress in vision-language models, 3D spatial reasoning from 2D images remains challenging. Text-based methods describe intermediate geometry with discrete tokens, limiting fidelity for continuous spatial relations. Continuous latents offer richer representations, but a single latent type does not explicitly separate the cues needed across spatial tasks. Decomposed spatial latents address this by representing position, direction, and global geometry separately under geometric supervision. Yet the geometry representation can still collapse toward one dominant direction, and unrestricted attention can leave the latents underused during answer learning. We introduce GeoLatent, combining Common–Residual Geometry Alignment (CR-GEO) with routed optimization to structure the geometry states while promoting latent-mediated answer learning. CR-GEO separates shared from residual teacher geometry; routed optimization jointly trains geometry and language, temporarily directs visual answer learning through the latents, and restores full attention with geometry supervision. In controlled comparisons, CR-GEO raises geometry effective rank from 1.00 to 3.87, while blocking latent readout at the bottleneck lowers direction accuracy from 89.1% to 25.8% on 128 fixed questions. After recovery, the differentiated geometry representation and latent-mediated visual route remain available alongside direct image access. GeoLatent achieves 73.0% on SPAR-Bench and 72.1% on SPBench, outperforming previously reported methods on both.

A stationary scalar correlation energy admits an exact one-mode terminal representation even when the microscopic correlated return is strongly multimode. We test predictive realizations built from specified stationary information carriers in two complementary two-electron systems with the same Coulomb repulsion but different confinement: He-like ions and harmonium (Hooke's atom). For the Coulomb sequence, canonical RHF supplies the symmetry-faithful information carrier. The repeated same-mode return gives a He correlation error of $+0.1164$ m$E_{\mathrm h}$ and no more than $0.0157$ m$E_{\mathrm h}$ for Li$^+$–Ar$^{16+}$; H$^-$ is nevertheless bound and $93.3\%$ of its conventional correlation energy is recovered even though the RHF carrier is qualitatively strained. The dimensionless confinement factor $R_{CC}=D_{CC}/Z$ tends to $1.092394$. For harmonic confinement, an independently constructed response-compressed one-mode return gives $R_H=D_H/\sqrt{\omega}$; for $\omega\ge0.1$ it agrees with reference-inferred values within $0.21\%$ and tends to the literature high-density limit $1.309283$. Cioslowski and Pernal locate the qualitative crossover to the radially localized strongly correlated harmonium regime at $\omega_{\rm crit}\simeq0.04012$, below the quantitative domain used here. The H$^-$ endpoint and $\omega_{\rm crit}$ are therefore complementary carrier-validity boundaries: they limit the predictive realization, not exact terminal one-mode representability. A broad helium Bethe–Goldstone spectrum confirms that this terminal statement is not microscopic one-pole dynamics.