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arXiv2026

When quantum thermal states look classical

At high temperature, quantum Gibbs states retain several classical features of the maximally mixed state: the absence of entanglement, the absence of magic, analyticity of the partition function, correlation decay, and algorithmic tractability. We prove new and sharp bounds showing that these features persist down to finite temperatures independent of system size, but fail at distinct inverse-temperature scales, forming a hierarchy of classical-to-quantum transitions. Our results hold for long-range Pauli interactions with bounded strength at every site. Despite such all-to-all interactions, we show that the death of entanglement occurs at constant temperature, resolving an open question of Rouze, Franca and Alhambra (STOC'25). We give a polynomial-time classical algorithm that prepares Gibbs states up to the death of entanglement transition. Notably, this is asymptotically colder than temperatures at which quantum Gibbs samplers are known to mix quickly, as well as the original separability temperature of Bakshi et al. (FOCS'24), which we improve to be tight up to constants. At asymptotically even colder temperatures, we show that the Gibbs state remains in the thermodynamic infinite-temperature phase. We give polynomial-time classical algorithms for estimating thermal expectations up to the phase transition (despite both entanglement and magic), and as a corollary resolve a correlation decay conjecture of Harrow, Mehraban and Soleimanifar (STOC'20).

quant-ph↗

arXiv2026

Optimal mean width and metric entropy estimates for convex bodies

We show that for any $n \geq 1$ and any convex body $K \subset \mathbf{R}^n$, there exists $T \in \mathrm{SL}(n)$ such that \[ \mathrm{diam}(TK) \lesssim \sqrt{n} \mathrm{vr}(K), \quad \mbox{and} \quad M^\ast\Big(T(K-x) \cap r\mathrm{vr}(K)\,B^n_2\Big)\lesssim \mathrm{vr}(K) \sqrt{\log(\mathrm{e} r^2)}, \] for every $x \in K$ and every $r \geq 1$. Above, $M^\ast(\cdot)$ denotes the spherical mean width and $\mathrm{vr}(\cdot)$ denotes the volume radius. As a consequence, we establish for any convex body $K \subset \mathbf{R}^n$ that \[ 1 \leq \inf_{T \in \mathrm{SL}(n)} \, \frac{M^\ast(TK)}{\mathrm{vr}(K)} \lesssim \sqrt{\log(\mathrm{e} n)}. \] The estimates above are sharp, up to universal constants, as they are attained for the crosspolytope and any regular $n$-simplex. Up to universal constants, our results show that all quermassintegrals, and the logarithm of the covering numbers for all scales, simultaneously for $K$ and the polar body $K^\circ$, are maximized by the simplex and crosspolytope. Our proof makes use of Eldan's stochastic localization. To establish the results, we work with an extension of Bobkov's maximal Gaussian measure position to possibly non-symmetric convex bodies. Our results imply that this position is an optimal "regular" Milman position, thereby improving a result of G. Pisier. We establish a non-symmetric analogue of the strong Gaussian (B)-theorem which may be of independent interest.

math.MG↗

arXiv2026

Cost-Effective Automated Judging of Natural-Language Mathematical Proofs

Grading natural-language mathematical proofs is a recurring cost in evaluating math-reasoning systems, and frontier LLM judges are expensive. We ask whether cheap open-weight models can serve as reliable judges given a candidate proof, a ground-truth proof, and a human-grading rubric. On a 200-instance validation sample of IMO-GradingBench, two of three cheap judges (GPT-OSS-120B, DeepSeek-V4-Flash) and their three-model consensus are statistically no worse than the frontier (Claude Opus 4.7, Gemini 3.1 Pro) on agreement with human pass/fail decisions, at 4-100$\times$ lower cost. On the full 1000-instance benchmark, the choice of consensus rule over the three judges is a precision/recall dial: unanimous (all-three-pass) rules reach the highest precision (0.855), majority vote the highest recall (0.912); across four replicate runs the unanimous rule is also the steadiest. No rule won outright; the dial replicated on a held-out 600-instance split and on the independent ProofBench. In this domain, cheap judges are competitive with the frontier at one to two orders of magnitude lower cost, and unanimity is the right setting when false positives are costly.

cs.CL↗

arXiv2026

Separability of Subsets in Infinite Groups

We study the separability of subsets of an infinite group. Given an infinite group $G$ and subsets $A,B\subset G$, we say that $A$ and $B$ are separated in $G$ if there exists an infinite symmetric subset $X\subset G$ with $e\in X$ and $XAX\cap B=\varnothing$. Question 17.102 of the Kourovka Notebook raises a natural cardinal question: if $A$ and $B$ are disjoint and $|A|,|B|<|G|$, must they be separated? We first give a negative answer, constructing two essentially different families of counterexamples, based respectively on rigid binary relations and on torsion-free groups whose square sets have cardinality smaller than that of the group. On this basis we introduce the $W$-witness set $W_G(A,B)=\{g\in G:\{g,g^{-1}\}A\{g,g^{-1}\}\cap B=\varnothing\}$, obtain a necessary and a sufficient condition for separability given by its size, and study the critical range. The central result of this paper is a necessary and sufficient characterization of separability: taking the inverse pairs $\{g,g^{-1}\}$ as vertices, we construct the conflict graph $Γ_{A,B}$, and $A$ and $B$ are separated if and only if the associated graph contains an infinite independent set; separability is thereby converted into an ordinary graph-theoretic problem. From this we further derive the case of finite subsets, the case of Abelian groups, and several cardinal criteria, and we obtain, via Ramsey's theorem, a structural dichotomy for the non-separable case. In addition, we study the range of cardinalities of counterexample groups and the possible sizes of $W$-witness sets, and prove that no uniform necessary and sufficient criterion can depend only on $|A|$, $|B|$ and structural invariants of the group, independently of the specific position of $A$ and $B$.

math.GR↗

arXiv2026

Search for a top-philic Z' boson decaying into a $\mathrm{t\bar{t}}$ pair in a final state with jets and an electron or muon in proton-proton collisions at $\sqrt{s}$ = 13.6 TeV

A search for a top-philic Z' boson in a final state with jets and an electron or muon is presented. The search is based on a sample of proton-proton collision data collected at $\sqrt{s}$ = 13 TeV by the CMS experiment at the CERN LHC during 2016$-$2018, corresponding to an integrated luminosity of 138 fb$^{-1}$. The top-philic Z' boson is produced in association with a top-antitop quark pair ($\mathrm{t\bar{t}}$) and decays into a $\mathrm{t\bar{t}}$ pair, as it couples exclusively to top quarks. The analysis aims to identify a heavy Z' boson that produces Lorentz-boosted top quarks, whose hadronic decay products are merged into large-radius jets. A machine-learning algorithm is employed to identify such jets. The distribution of the invariant mass of the two top quark candidates with the highest transverse momentum is used as the discriminant variable in a Z' boson mass range of 0.5$-$3 TeV, with intrinsic widths of 4, 10, 20, and 50% relative to its mass. The results obtained are found to be in agreement with the standard model background prediction. Upper limits at 95% confidence level are set on the production cross section of the Z' boson, for each of the decay widths as a function of its mass. These results represent the most stringent constraints to date on the existence of a top-philic Z' boson.

hep-ex↗

arXiv2026

An Identifiability Theory of Masked Prediction: Mode Blindness and Mask Schedules

Masked prediction learns to infer missing variables from visible context selected by a mask schedule. When does small excess risk guarantee recovery of the true joint distribution? We study this question on finite product spaces under masked-block log loss, with conditionals induced by a single joint distribution. To quantify recovery, we introduce an $\varepsilon$-identifiability modulus measuring the worst-case error among joint distributions with population excess risk at most $\varepsilon$. For data with separated modes pinned down by sufficiently large visible contexts, schedules that always retain such contexts can permit nonvanishing mode-weight errors while incurring exponentially small excess risk. An information decomposition explains why: the loss captures mode-weight mismatch only where the visible context leaves the mode uncertain. We prove two-sided bounds showing that, over a fixed range of mode weights, sensitivity to mode reweighting is governed by residual mode uncertainty averaged over the mask schedule. Assigning schedule mass to low-visibility masks that retain this uncertainty yields recovery bounds within the reweighting family. Beyond this family, positive full-mask probability characterizes uniform control of joint KL divergence by excess risk. Exact calculations and controlled gradient optimization validate these predictions.

cs.LG↗

arXiv2026

ST-LoRA: Single Trajectory LoRA Ensemble for Uncertainty Aware Agricultural Segmentation

Reliable decision support in digital agriculture requires not only accurate predictions but also well-calibrated uncertainty estimates, particularly for dense prediction tasks such as semantic segmentation. Ensembles provide strong uncertainty quantification but are computationally and memory demanding, while single-model approximations often sacrifice uncertainty quality. We propose ST-LoRA, a parameter-efficient ensemble that builds diverse members from a single training trajectory by combining Low-Rank Adaptation (LoRA) with snapshot ensembling. All members share a frozen pretrained backbone and differ only in lightweight low-rank adapters, which sharply reduces trainable parameters, checkpoint storage, and I/O overhead. We evaluate SegFormer, Mask2Former, and EoMT on GrowliFlower-L (cauliflower, open field) and BUP20 (sweet pepper, glasshouse), covering in-distribution performance, calibration under covariate shift, and near- and far-out-of-distribution (OoD) detection, with BUTom21 (tomato) as near-OoD data. Extensive ablations show that feed-forward layers, not attention projections, are the critical LoRA target for dense prediction, and that the scaling ratio $α/r$ governs an accuracy--calibration trade-off. Against full-rank snapshot ensembles, ST-LoRA is competitive in segmentation quality, with architecture-dependent training time and energy savings. Against MC Dropout, DDU, and six post-hoc calibrators, it achieves the strongest far-OoD image-level detection and near-OoD pixel-level localization with low cross-seed variance, although full-rank ensembles remain better calibrated. These results show that LoRA-based ensembling offers a compelling efficiency--performance trade-off for agricultural vision systems.

cs.CV↗

arXiv2026

ChaosProbe: A Neurochaotic Lens on Frozen Transformer Input-Embedding Spaces

Transformer models are most often understood through what they do: their benchmark performance, generation quality, or behavior on downstream tasks. Yet frozen transformer input-embedding spaces may also be examined through their responses to a controlled deterministic probe before contextual computation or task-specific adaptation. Guided by this response-based view, we introduce ChaosProbe, a deterministic neurochaos-inspired method for constructing response-based fingerprints of frozen transformer input-embedding spaces. For each prompt-level embedding matrix, ChaosProbe applies a chaotic trajectory-based transformation and summarizes its Firing Rate and Entropy channel responses with complementary representation-level measures, producing a fixed-length signature for each model. In a bounded proof-of-concept study of $80$ neutral prompts and four pretrained models---GPT-2, DistilGPT2, BERT-base-uncased, and RoBERTa-base---Pearson correlation, Spearman correlation, and cosine similarity each recover all four same-family nearest-neighbor assignments and both expected mutual family pairs. Euclidean distance recovers three of the four assignments and one of the two mutual family pairs. Paired bootstrap resampling supports the stability of the Pearson and Spearman pairings over the observed prompt set, and signature-validity checks show that constant or collapsed responses do not dominate the reported fingerprints. These results provide a cohort-dependent proof of concept that deterministic neurochaotic response signatures can expose broad structure among frozen transformer input-embedding spaces.

cs.LG↗

arXiv2026

Fourier-Latent Diffusion for Constrained Generation of Triply Periodic Minimal Surfaces

We present a generative framework for the constrained design of $D_{2h}$-symmetric triply periodic minimal surfaces (TPMS) with low residual mean curvature. Existing TPMS design pipelines either explore a limited set of canonical analytical families or rely on computationally expensive numerical procedures, making it difficult to efficiently generate diverse candidates under geometric and/or mechanical requirements. Our approach learns a distribution of solver-generated TPMS in a compact Fourier space where periodicity and $D_{2h}$ symmetry are guaranteed by construction. This representation eliminates the need for a learned geometric decoder and enables the training of an effective diffusion model that generates low-mean-curvature candidates conditioned on sparse geometric constraints, selected homogenized elastic properties, or their combination. A subsequent coefficient-space refinement further reduces the residual mean curvature. Experiments show that our approach outperforms trigonometric, SDF-based, and standard Fourier-space baselines and enables controllable, high-quality generation under geometric and low-dimensional mechanical constraints. Overall, the proposed framework provides a compact and efficient design space for generating near-minimal periodic structures under user-specified requirements.

cs.GR↗

arXiv2026

Discrete Unique Continuation on Simplex

For integers $N\ge0$ and $n\ge2$, let \[ Δ_N^{(n)} =\left\{α\in\mathbb Z_{\ge 0}^n: α_1+\cdots+α_n=N\right\}. \] We study discrete unique continuation for functions on this lattice simplex. For an integer $R\ge1$, let $g:Δ_{nR}^{(n)}\to\mathbb R$ satisfy \[ \sum_{i=1}^n g(β+e_i)=0, \qquad β\inΔ_{nR-1}^{(n)}, \] where $e_i$ is the $i$th standard basis vector. We prove that nonvanishing at the center implies \[ |\operatorname{supp} g|\ge c_n R^{\lceil n/2\rceil}. \] Here $\operatorname{supp} g$ is the set of points where $g$ is nonzero, and $c_n>0$ depends only on $n$. The exponent $\lceil n/2\rceil$ is optimal. The proof represents the values of $g$ as polynomial coefficients and uses a Pascal uncertainty principle, which bounds from below the total number of nonzero coefficients of a one-variable polynomial and its unit translate in terms of their degree.

math-ph↗

arXiv2026

Compact Hyperbolic Coxeter Six-dimensional Polytopes With Ten Facets

We show that, up to isometry, there is exactly one compact hyperbolic Coxeter 6-polytope with 10 facets, the polytope $P_{6,10}$ attributed to Bugaenko. Together with results of Felikson-Tumarkin ($d \ge 7$) and of Burcroff and Ma-Zheng ($d = 4, 5$), this completes the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets. The proof is computer assisted. Affine Gale duality applied to the complete database of order types on 10 points yields 387 combinatorial types, of which Lannér's classification excludes 83. For the remaining 304, an exhaustive search over Coxeter labellings, with no a priori bound on the dihedral angles, leaves a single realizable Gram matrix. Every rejection is certified in exact arithmetic, and completeness of the search is certified independently by DRAT proofs checked by drat-trim. The same code reproduces the known censuses in dimensions 4 and 5. Code, data and certificates are publicly available.

math.CO↗

arXiv2026

Contrasting anisotropic electron-phonon-spin coupling in Fe$_{3}$GeTe$_{2}$ and Fe$_{5}$GeTe$_{2}$: A helicity-resolved Raman study

Two-dimensional van der Waals ferromagnets Fe$_3$GeTe$ _2$ (F3GT) and Fe$_5$GeTe$_2$ (F5GT) exhibit pronounced magneto-optical responses, which open promising platforms for investigating the interplay among lattice, electronic, and magnetic degrees of freedom. Here, we present a comparative study of optical resonance-induced anisotropic electron-phonon coupling and its association with magnetic ordering in these systems using wavelength- and temperature-dependent helicity-resolved Raman spectroscopy. By resolving the doubly degenerate E modes under left- and right-circularly polarized excitations, we demonstrate that the temperature evolution of the chiral mode splitting ($Δf$) does not track the magnetization behavior, indicating that the helicity-dependent Raman response arises not solely from time-reversal symmetry breaking due to magnetic order, but also from spin-orbit-coupled electronic interactions. Notably, in F3GT, the out-of-plane magnetization indirectly governs the in-plane anisotropic electron-phonon coupling under optical resonance, whereas F5GT exhibits static anisotropic interactions. The Fano asymmetry parameter $1/q$ reveals mode- and temperature-dependent coupling strengths between phonons and the electronic continuum, with pronounced angular anisotropy in F3GT but isotropic behavior in F5GT--- a consequence of its multiple Fe sites and enhanced interlayer hybridization in the latter. Our results demonstrate the role of crystal structure and magnetic anisotropy in shaping the anisotropically coupled electron-phonon-spin dynamics in these layered metallic ferromagnets, and highlight Fe$_n$GeTe$_2$ as a versatile platform for microscopic insight into chiral light-matter interactions in layered metallic ferromagnets.

cond-mat.str-el↗

arXiv2026

A Hesselink-type formula for the nilpotent cone of Lie algebra representations

A well-known result of Hesselink gives a formula for the $q$-character of the nilpotent cone of a semisimple Lie algebra in terms of a $q$-analog of the Kostant partition function. For a reductive Lie algebra $\mathfrak{g}$, we define a class of \emph{Hesselink-type representations}, for which we prove an analog of Hesselink's formula under certain additional freeness assumptions. Moreover, we prove that the formula still holds for certain examples of Hesselink-type representations without the freeness assumptions. Using these methods, we obtain $q$-character formulas for the nilpotent cone of a representation of a cyclic quiver with equal dimensions, a representation of a cyclic quiver with two vertices, and a representation of a product of copies of $\mathfrak{sl}_2$ we call an \emph{extended quiver representation of trivial type}. In the process, we give a general framework for proving formulas of this type for representations that are not necessarily Hesselink-type. We also provide some counterexamples to natural questions regarding Hesselink-type representations.

math.RT↗

arXiv2026

Gravitational Enstrophy: Local Geometric Origin and Energy-Cascade Constraints

Two-dimensional incompressible fluids conserve energy and enstrophy in the inviscid limit. Fjørtoft's argument uses this pair to constrain nonlinear energy transfer. We examine the corresponding question in General Relativity using the magnetic Weyl functional $\mathcal Z=\int B_{ab}B^{ab}\sqrtγ\,d^3x$. Its transverse radiativ e part carries an additional frequency-squared weight relative to gravitational-wave energy: $δ\mathcal Z_k\simeqω_k^2W_k$ in the radiation zone, with $W=8πG E_{\rm GW}$. We derive its balance law from the Bianchi identities, obtaining the algebraic cancellations, bulk sources and boundary fluxes. Conservation of positive e nergy and curvature norm through nonlinear transfer gives a Fjørtoft constraint when their weights are ordered. Resonant weak-wave interactions generally change the c urvature norm; $2\leftrightarrow2$ dynamics instead conserves energy and wave action. We examine the competition between transfer and linear losses in Kerr and AdS backg rounds, where long-lived or confined modes provide time for nonlinear evolution. In the hydrodynamic regime of AdS$_4$, a fluid-weighted magnetic Weyl functional inherit s fluid enstrophy conservation in the inviscid limit. We compute its proportionality coefficient to boundary enstrophy in closed form at leading gradient order, giving a bulk diagnostic for holographic turbulence.

gr-qc↗

arXiv2026

Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes

The set of $n\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\exp\{-\tfrac14 n^2\log n\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular. Its maximum absolute correlation is of order $\sqrt{\log n/n}$, its Frobenius distance is asymptotic to $\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. Distinct off-diagonal entries are exactly pairwise independent under every $\operatorname{LKJ}(η)$ law, which yields a Chen-Stein proof of the extreme-correlation point-process limit for the whole LKJ family and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. Finally, for a bounded, centered i.i.d. off-diagonal perturbation of any correlation matrix, the nearest-correlation projection removes a fraction of the squared perturbation that tends to one and recovers the original matrix in average squared error per entry.

math.PR↗

arXiv2026

Search for resonant production of lepton-enriched semivisible jets in proton-proton collisions at $\sqrt{s}$ = 13 TeV

This search targets the resonant production of lepton-enriched semivisible jets (SVJs) from a strongly coupled dark sector, using 138 fb$^{-1}$ of proton-proton collision data collected with the CMS detector at the CERN LHC at $\sqrt{s}$ = 13 TeV. Two scenarios are investigated: jets enriched in all lepton flavors (SVJ$\ell$ signature) and jets predominantly enriched in tau leptons (SVJ$τ$ signature). The analysis focuses on final states in which the missing transverse momentum is aligned with jets containing nonisolated leptons. A dual machine-learning strategy is employed, using a graph neural network for jet identification and a fully connected neural network that combines jet- and event-level information to enhance signal sensitivity and background estimation. The signal models assume a heavy Z' mediator with a benchmark coupling of 0.25 to standard model quarks, together with prompt decays of unstable dark hadrons. In the SVJ$\ell$ scenario, mediator masses up to 4.7 TeV are excluded at 95% confidence level, while masses between 1.8 and 3.5 TeV are excluded in the SVJ$τ$ scenario. These results provide the first experimental constraints on lepton-enriched semivisible jets.

hep-ex↗

arXiv2026

RA-CAD: Learning Post-Execution Critique for State-Aware Text-to-CAD Generation

Text-to-CAD generation translates natural-language design intent into editable and executable parametric computer-aided design (CAD) codes, reducing the expertise and effort required for manual modeling. Existing methods incorporate fixed, externally supplied, prompt-induced, or separately optimized critique mechanisms to optimize the generation process, but they do not necessarily optimize how feedback is interpreted and translated into effective corrective actions throughout the generation process. To bridge this feedback-utilization gap, we present RA-CAD (ReAct Agent for CAD), a state-aware agent that interacts with the CAD environment through a Generate--Execute--Critique--Rewrite loop. At each iteration, RA-CAD executes the current code and observes its outcome. Conditioned on the design instruction, current code, and execution feedback, the agent then generates an explicit post-execution critique as an intermediate policy action. This critique either validates the current result for termination or provides revision-oriented guidance that conditions the next rewrite. CAD Code Bootstrapping (CCB) first establishes fundamental parametric CAD coding capabilities through supervised fine-tuning. Feedback-Driven Agent Optimization (FAO) subsequently applies trajectory-level Group Relative Policy Optimization to both policy-generated code and critique sequences, assigning terminal F1 and Chamfer Distance rewards to the complete interaction trajectory. This formulation makes critique an outcome-aligned, learnable policy decision rather than an unoptimized auxiliary output. Experiments on CADFusion and Text2CAD show that RA-CAD achieves state-of-the-art execution validity and geometric quality compared with existing methods and strong proprietary language models, demonstrating the effectiveness of the proposed state-aware text-to-CAD agent.

cs.AI↗

arXiv2026

Online learning of quantum states under structure

Quantum state tomography is fundamental to quantum information processing but becomes infeasible at scale due to the exponential growth of the state space. Shadow tomography alleviates this challenge by focusing on predicting measurement outcomes rather than reconstructing the full state. Its online variant models adaptive and potentially adversarial measurement scenarios, where a learner sequentially predicts outcomes while competing with the best fixed quantum state in hindsight. We show that exploiting additional structure in the measurements leads to significantly stronger regret guarantees. In particular, under the assumption that the adversarial measurements have bounded Frobenius norm, we analyze online mirror descent and derive optimal regret bounds that depend on intrinsic structural properties, such as rank or sparsity in the standard basis, rather than the dimension of the measurement operators. As a complementary result, we also show that, even in the setting where adversarial measurements are known to be sparse in the Pauli basis commonly used in variational quantum eigensolvers and near-term quantum error mitigation, the underlying regret bound for learning quantum states is the same as that obtained in the generic setting, where the adversarial measurements are not known to possess any particular structure.

quant-ph↗