Papers

Corporate and investment banking teams use presentations to support credit decisions and advise clients on financing and transactions. Producing these decks requires reconciling financial data, tracing sources and turning analysis into a recommendation. We retrospectively study the development of an agentic harness combining a 27B language model, financial calculations, narrative templates and validation checks. LLM judges guide engineering changes and assess the resulting decks, raising the question of whether higher scores reflect better documents or changes in grading. In shared-session text-only grading with template markers removed, five judges score the complete system 20.4 to 33.6 points out of 95 above the same model generating directly from a short prompt. Every judge scores the system higher on all seventeen development deliverables. Margins against direct Opus generation from a short prompt range from -4.7 to +0.8 points. Judges agree on broad progress across development rounds but agree less on final-deck rankings than on pooled scores. Repeated grading also shifts scores on unchanged decks, making small improvements difficult to distinguish from judge variability.

Coding agents solve repository-level tasks through sequences of actions, where a single erroneous action can misdirect subsequent decisions and increase recovery costs. Existing approaches use execution feedback for recovery or specialized checks to block errors, but deciding before execution whether intervention will benefit eventual task completion remains challenging. To address this challenge, we propose HiSentinel, a hindsight-distillation framework that trains lightweight 0.6B and 1.7B sentinels to select pre-execution interventions aimed at improving task completion rather than correcting every imperfect action. A privileged teacher uses recorded execution outcomes as evidence for intervention judgments, which are distilled into a causal student that receives only the pre-action context and proposed action. Beyond identifying whether and when to intervene, the sentinel must also provide actionable feedback that helps the coding agent recover or obtain necessary human input. To support these capabilities, we introduce SWE-Intervene, an action-level dataset constructed from software-engineering trajectories that annotates whether an action should be allowed, autonomously redirected, or paused for human assistance, together with corresponding intervention feedback. Across SWE-bench Verified Mini and Ask or Assume, HiSentinel consistently improves task completion across Sentinel scales and coding-agent families, with gains of up to 14% and 10%, respectively, while maintaining competitive token consumption. These results demonstrate that lightweight pre-execution intervention can effectively prevent error propagation and improve the reliability of autonomous coding agents.

In this work we propose a region of interest (ROI)-aware mode selection and task assignment (RAMSTA) framework for distributed integrated sensing and communication (ISAC) systems. In realistic ISAC use-cases, the sensing performance requirement can be location dependent. Vehicular targets at a traffic intersection, unmanned aerial vehicle approaching a restricted airspace or a sparsely populated areas may require different levels of detection or localization accuracy. This warrants the design of location sensitive resource allocation to account for the varying sensing performance requirements. Thus, we design a sigmoid-based ROI proximity parameter to tune the communication-sensing trade-off in a weighted sum rate and position posterior Cramér-Rao lower bound minimization problem. The resulting mixed-integer non-linear program is solved by applying a penalized convex-concave procedure. We show that the ROI sensitivity thresholds allows the adaptive variation of the communication-sensing trade-off with respect to the predicted location of the target and relative to a predefined ROI.

Single-step retrosynthesis models are commonly evaluated by their ability to recover recorded reactions. In practice, chemists may need to choose among several precursor sets for the same product, for example to preserve a particular motif. Recovering a recorded answer alone does not establish this ability to follow a preference. Satisfying such requests requires both coverage of relevant alternatives and control over which alternatives are favored. We introduce Route-Instruction Grounding and Steering (RIGS), a two-stage framework for instruction-conditioned retrosynthesis. Stage A trains a language projector, teaching it which alternatives an instruction favors or discourages. Stage B uses the projector learned in Stage A to steer a frozen generative model through lightweight residual adapters. We construct nested one-to-many training supports by pairing each product with increasing numbers of candidate precursor sets. Extensive experiments demonstrate that broader support helps the model generate a wider range of alternatives, and RIGS can learn to guide generation according to instructions. The relationship between coverage and control is consistent across model scales but non-monotone.

Confocal microscopy provides optical sectioning for three-dimensional imaging but conventionally relies on point-by-point scanning and physical pinholes, limiting imaging speed. Here, we demonstrate that confocal sectioning can instead arise directly from quantum measurement, using spatial correlations in entangled biphoton states. Spatially resolved coincidence measurements in a widefield imaging system suppress contributions from out-of-focus planes, producing optical sectioning in parallel across the whole field of view without physical pinholes or mechanical scanning. In addition, this quantum approach obtains an axial response that, as a result of position-momentum entanglement, is $\sqrt{2}$-times narrower than that of classical confocal imaging. Our work establishes biphoton correlations as a mechanism for improved confocal imaging and enables parallel, pinhole-free optical sectioning without mechanical scanning.

Omni-modal large language models (OmniLLMs) enable unified audio-video understanding, but their long multimodal token sequences make deployment computationally expensive. Token compression reduces this cost, yet aggressive compression often lowers accuracy. Existing works predominantly focus on designing better compression mechanisms; however, adapting the underlying language model to reason effectively over the remaining compressed context remains under-explored. To address this, we propose CAFD (Compressed-Context Adaptation via Full-Context Distillation), a ground-truth-free self-distillation framework that adapts OmniLLMs to fixed compression pipelines without requiring reference answers, rationales, or correctness rewards. CAFD leverages the full-token view of the same multimodal sample as a source of privileged information: a full-context self-teacher provides soft target supervision to a compressed-context student along the student's on-policy trajectory. Evaluated on Qwen2.5-Omni-7B across five audio-video benchmarks, five compression pipelines, and five deployment budgets, CAFD demonstrates consistent gains, improving 120 out of 125 conditions with an average accuracy boost of 1.44 points and recovering 26.9% of the accuracy gap on average. These results demonstrate that the proposed ground-truth-free adaptation offers an effective and practical route to improving the accuracy-efficiency trade-off in deployed OmniLLMs.

We enumerate all families of involutions avoiding a classical pattern of length four together with 4321. While the enumeration of permutations avoiding two classical patterns is by now well understood, the case of pattern-avoiding involutions presents additional structural constraints that require dedicated techniques. In particular our main enumerative tool is Biane's bijection between involutions and labelled Motzkin paths. The paper provides exact bivariate generating functions (taking into account length and number of descents), and structural descriptions for all except one family. In the remaining case we provide a functional equation for the generating function and express it as an explicit continued fraction. We make extensive use of experimental and software-supported methods.

We show that circle-valued spectral values in Tate Rabinowitz Floer homology give rise to quantized energy levels, which recover the quantum mechanical spectrum of a system of uncoupled harmonic oscillators.

Continuum robots are well suited for gentle manipulation because of their inherent compliance and ability to adapt to complex environments. However, their continuously deformable structure makes accurate configuration estimation challenging, particularly when external vision systems are unavailable or obstructed. In this work, we present an embedded pose sensing framework that combines inertial measurement units (IMUs) and active magnetic fields to estimate the robot configuration without relying on external cameras. The angular measurements from the IMU and magnetic-field references are fused to improve local orientation estimation and reduce accumulated orientation error during operation. This pose sensing scheme achieves an update rate of 16.7~Hz, allowing real-time feedback. The proposed system is experimentally validated through closed-loop control, where the estimated robot configuration is used to maintain the end-effector at a desired position while interacting with an object. These results demonstrate the potential of distributed magnetic–inertial sensing for real-time pose estimation and closed-loop control of continuum robots.

We study scaling law of variational quantum algorithms by large-scale numerical experiments on combinatorial optimizations. On experiments up to n = 5000 qubits with O( log(n)) circuit depth with a classically simulable ansatz, the converged energies consistently differ by no more than 0.5% from the optimal or reference energies reported in the benchmarks.

The input-to-state stability of a class of infinite-dimensional second-order port-Hamiltonian systems on a one-dimensional spatial domain is analyzed under nonlinear dynamic boundary feedback and boundary disturbances. Using energy methods in the port-Hamiltonian framework, we obtain sufficient conditions that guarantee uniform input-to-state stability of the closed-loop system.

Progress in information processing relies on spintronics, where magnetic states serve as efficient carriers for data storage and transfer. In this work, we theoretically study magnon propagation in a bilayer composed of a ferro-magnet and an antiferromagnet. For this purpose, we probe the spin Seebeck effect by introducing a spatially varying temperature profile. This generates a local magnon excitation and a continuous magnon flux from hot to cold regions which we quantify through the resulting non-equilibrium magnon accumulation. Based on the chirality of these modes, we identify specific constraints for magnon modes traveling either from the ferromagnet into the antiferromagnet or vice versa. A key finding is the observation of a thermally triggered spin current in the antiferromagnet, a phenomenon typically absent in bulk antiferromagnets that obey time-reversal symmetry. These results provide important insights into the design of heterostructures for magnonic chirality-selective spin transport.

We introduce a simple reaction-network model of a quorum-sensing population that couples the cell density $x$ to the signal density $w$. In the four-channel cell-signal network studied here, scaling signal production and removal by the same factor $r$ leaves all deterministic equilibria, and their stability types, unchanged. Nevertheless, we show that $r$ shifts the quasipotential barrier $\Delta V(r)$ for rare transitions towards extinction, and thus, under metastable exit assumptions, the mean time to reach a fixed neighbourhood of the extinction state on the exponential scale $e^{N\Delta V(r)}$. We compute the barrier by minimization of the path action with the signal retained as a fluctuating coordinate, and we check it against mean extinction times obtained without the action solver, i.e., from the backward equation of the jump process on a truncated lattice and from exact stochastic simulation. Over a range of $r$, the minimum-action barriers satisfy $\Delta V(r)=\Delta V_\infty+O(1/r)$, where $\Delta V_\infty$ is obtained by eliminating the signal first, and at $r=1$ the barrier is 55% larger than $\Delta V_\infty$. A formal first-order expansion attributes this increase to the lag of the signal, of which the fluctuations of the signal remove a quarter. The saddle barrier $\Delta V(r)$ is also the least action needed to enter the basin of extinction, whereas the density threshold can be crossed more cheaply, by paths which keep the signal high and are usually followed by recovery. The mean extinction times of the backward equation, which include failed attempts, have a prefactor which tends to a constant, and they grow at a rate within $10^{-5}$ of $\Delta V(r)$ at every tested rate.

We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits $L^\infty$-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and positive boundary mean curvature, such examples exist with umbilic boundary in every dimension $N\ge22$ and with nonumbilic boundary in every $N\ge15$. For positive scalar curvature and minimal boundary, the corresponding ranges are $N\ge21$ and $N\ge15$. For every $N\ge9$, examples with either umbilic or nonumbilic boundary also exist when the scalar curvature is $N(N-1)$ and the prescribed boundary mean curvature is below a negative threshold depending on $N$. The construction uses polynomial metric perturbations and corrected bubbles on the half-space, followed by conformal compactification to the ball. We evaluate or estimate the correction solving the linearized Neumann or Robin boundary problem and its contribution to the quadratic term of the reduced energy. We prove that this quadratic term has negative, nondegenerate local extrema with respect to tangential translations and scale in every stated dimension.

We compute analytically greybody factors for d-dimensional large spherically symmetric black holes with Gauss-Bonnet corrections in the high frequency limit. Our calculations include both the eikonal limit, where the real part of the frequency of the scattered wave is much larger than the imaginary part, and the highly damped case,where the imaginary part of the frequency is much larger than the real part, addressing the emission of gravitons and test scalar fields, and yielding full transmission and reflection scattering coefficients. We consider tensorial, vectorial and scalar gravitational perturbations, obtaining the same result for the three cases in the highly damped limit, but not in the eikonal limit.

Every non-linear plane Cremona map can be decomposed into quadratic maps, and the minimum number of quadratic maps required is called its quadratic length. It is known that plane Cremona maps of degree 3 have quadratic length either 2 or 3. In this paper, we study the quadratic length of plane Cremona maps $\varphi$ of degree 4. Recall that either $\varphi$ is de Jonquières, i.e. it has a base point of multiplicity 3, or it is not de Jonquières. In the latter case, $\varphi$ has quadratic length 2, whereas in the former case we prove that $\varphi$ has quadratic length 3, 4, or 5, and we classify those maps that reach the upper bound.

Let $S$ be a family of sequences decreasing to zero. A set $E$ in a metric space is $S$-dominated if, for every $s\in S$, there exists a countable cover $\{E_n\}$ of $E$ such that $diam E_n<s_n$ for every $n$. We continue the study of families of dominated sets initiated in the previous paper. We look, e.g., into Cartesian products of dominated sets with strong measure and interaction of microscopic and porous sets. Based upon a tight relationship of Hausdorff measures and dominated sets, cardinal invariants of the ideals of dominated sets are determined and a related problem of Kwela is resolved.

Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity $\varepsilon$ from a number of channel uses proportional to $1/\varepsilon$. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.

The resolved Kennicutt-Schmidt (KS) relation is a power-law relation between the surface densities of the star formation rate ($\Sigma_\mathrm{SFR}$) and molecular gas ($\Sigma_\mathrm{mol}$) in star-forming galaxies on kiloparsec scales. Despite its apparent simplicity, it exhibits substantial scatter, suggesting the influence of additional physical parameters beyond gas surface density. We investigate whether molecular gas velocity dispersion ($\Delta v$) acts as a hidden parameter in the resolved KS relation in nearby galaxies. We use spatially resolved measurements of $\Sigma_\mathrm{SFR}$, $\Sigma_\mathrm{mol}$, and $\Delta v$ from the ALMaQUEST, EDGE-CALIFA, and PHANGS-ALMA surveys and from observations of the single galaxy M51. We identify two groups of galaxies based on the direction of the $\Delta v$ gradient with respect to the best-fit KS line: one in which the $\Delta v$ gradient is completely aligned with the KS line (group A) and thus does not contribute to the scatter, and one in which it has a component perpendicular to the line (group B) and contributes to the scatter. For each group, we fit the KS relation and examine the dependence of the residuals ($Res$) of the fit on $\Delta v$. In group B, we find a weak but statistically significant correlation between $Res$ and $\Delta v$, which is consistent across all surveys. A three-dimensional fit including $\Delta v$ yields $\Sigma_\mathrm{SFR} \propto \Sigma_\mathrm{mol}^{1.15} \Delta v^{-0.59}$. We conclude that there is a set of star-forming galaxies for which the velocity dispersion acts as a secondary parameter in the KS relation, with higher $\Delta v$ corresponding to lower $\Sigma_\mathrm{SFR}$ at fixed $\Sigma_\mathrm{mol}$. We find no obvious correlation between membership in this set and the other properties examined.

We investigate direct photon and dilepton production in a concurrent minijet-hydrodynamics framework that consistently incorporates event-by-event initial-state fluctuations and dynamical minijet-medium interactions. Produced in initial hard scatterings, minijets thermalize more slowly than the bulk medium and persist into the hydrodynamic stage, where they deposit energy and momentum into the evolving plasma, generating wakes that modify its spacetime evolution. Minijets modify the hadronic multiplicity and momentum anisotropy in a manner that can largely be absorbed by adjusting the shear viscosity and the normalization of the system's initial energy-momentum tensor. In contrast, electromagnetic probes are emitted throughout the evolution and remain directly sensitive to the underlying spacetime history. We perform a comprehensive calculation of direct photon and dilepton production with the minijets and bulk medium evolved simultaneously, using PYTHIA to produce minijets, and describing the underlying plasma with IP-Glasma initial conditions, hydrodynamic evolution with MUSIC and hadronic transport with UrQMD. We find that the minijet-induced modification of the hydrodynamic evolution produces measurable effects in both photon and dilepton observables, establishing electromagnetic radiation as a sensitive probe of the interplay between hard partons and the quark-gluon plasma.

Hour-scale audio-visual question answering is constrained by a context dilemma: dense whole-recording encoding rapidly exhausts context limits, whereas uniform temporal compression severely dilutes fine-grained acoustic and visual evidence. We introduce LEAP, a framework where the model retrieves its own evidence without placing the whole recording in one context. LEAP divides a recording into fixed-duration blocks, applying a lightweight localization pass to each block to score short candidate windows. The highest-ranked windows are pooled and re-encoded in a single bounded answer pass. Consequently, the answer input and peak context remain independent of the recording duration. By decoupling evidence localization from reasoning, our framework can localize candidate temporal windows over pre-computed transcripts without decoding media frames, while preserving fine-grained visual and non-speech evidence by routing the final answering pass over raw audio-visual streams. LEAP trains both stages: a localization LoRA improves the selected windows, and an answer LoRA improves the answers read from the same windows. The block grid natively supports causal queries, enabling LEAP to support streaming inference without streaming-specific training. Across several AVQA benchmarks, LEAP improves over the Qwen3-Omni-30B-A3B baseline by 4.5-16.8%, and transfers to a second omni-modal backbone, MiniCPM-o 4.5, surpassing its published results by 3.1-13.0%.

This letter develops a data-based method for designing state feedback for a discrete-time linear system under bounded disturbances. For each admissible feedback gain and scalar design parameter, a Lyapunov equation determines an invariant ellipsoid: a region that the state cannot leave under the permitted disturbances. We optimise the gain and parameter to minimise a trace-based measure of the resulting output enclosure. At each parameter value, value iteration gives a lower cost bound, while separate controller evaluation gives an achievable upper bound. These bounds also let us assess whole parameter intervals without evaluating every point. We prove that the search stops after finitely many evaluations with a stabilising state-feedback gain whose objective is within any prescribed absolute tolerance of the infimum over the chosen ellipsoid family. Neither an initially stabilising gain nor attainment of the infimum is assumed. The guarantee assumes exact arithmetic and a sufficiently informative batch of exact measurements, including disturbances during data collection; the resulting feedback uses only the state. A position–velocity example illustrates the bounds, controller checks, and computational cost.

We study equilibrium configurations and linear perturbations of a Lorentz-violating Kalb–Ramond field with a quartic symmetry-breaking potential and a non-minimal Riemann coupling on a fixed Schwarzschild background. The electric equilibrium is determined algebraically by a characteristic function that can develop a finite-radius double root. Approaching this configuration, the electric monopole response scales as $|\widehat e(\omega,r_c)|\propto(\gamma_c-\gamma)^{-1/2}$, while the propagating monopole remains regular. For higher multipoles we obtain the exact axial zero-mode tower $\xi_{\ell n}=-(\ell+n+1)(\ell+n+2)/3$. At $\ell=1$ a complementary static mode gives a second low-frequency branch. Increasing the positive Riemann coupling causes two purely imaginary growing modes to collide and form an oscillatory unstable pair, with square-root real-part splitting. Independent numerical checks confirm that this behavior persists under changes of the equilibrium parameters. The corrected bare sign removes the monopole ghost, but a constrained high-momentum negative-energy direction remains on the asymptotically electric branch. We therefore distinguish the existence of classical growing modes from the stronger requirement of a positive-energy theory.

Discovering properties of many-body quantum systems is challenging because of the large number of parameters to determine. Here, we show that correlations among quantum observables help reduce the complexity of quantum learning. A polynomial number of selected measurements can bound the values of exponentially many dependent yet unmeasured quantities, enabling efficient estimation of general functions of quantum states. We leverage this result to design correlation-informed learning algorithms that estimate key quantum resources, such as entanglement and magic, in systems of 100 qubits. They achieve a lower relative error than shadow tomography and neural-state methods by using information about a polynomial number of measured Pauli strings. These protocols are readily testable with today's quantum computers, as they do not require premeasurement entangling gates. Quantum constraints themselves are therefore a resource for exploring large quantum systems.

Checkpoint selection in domain generalization often relies on source-validation accuracy, yet the selected checkpoint need not provide reliable probabilities on unseen target domains. Source-target distribution shifts can alter accuracy rankings, while accuracy alone does not measure predictive probability quality. We identify an empirical selection opportunity within fixed training trajectories: reselecting among checkpoints with near-optimal source accuracy can improve mean target probability quality with small observed changes in mean target accuracy. We study accuracy-constrained reliability selection (AC), which retains checkpoints within a tolerance of the best source-validation accuracy and ranks them by source reliability. Our reference rule aggregates within-set normalized negative log-likelihood (NLL) and class-wise calibration error (CwECE) using $D_\infty$. AC uses no target data and requires neither additional training nor weight averaging. We evaluate five domain generalization training algorithms on three benchmarks, using PACS to develop the objectives and a 0.5-percentage-point tolerance. In exploratory aggregation comparisons on 360 OfficeHome and TerraIncognita runs, the reference rule reduces mean target soft-bin squared-gap ECE and CwECE by 0.240% and 0.182%, respectively, and NLL by 0.030 relative to Source-Acc. Mean target accuracy changes by +0.213 percentage points. These results identify opportunities for reliability-aware reselection, while the additional benefit of joint over single-objective ranking remains unresolved.

When designing machine learning models, desirable properties are often in tension: improving one behavior can impair another, so task progress can depend on alleviating the conflict. LLM-based AutoResearch systems, which iteratively edit model code and retain edits based on scalar task-performance feedback, have largely ignored this trade-off. We find that scalar feedback supports broad exploration early in search, but it does not reveal how edits affect competing behaviors. In matched-budget experiments, introducing competing-behavior feedback as task gains diminish increases the share of proposals that improve both behaviors and sustains progress beyond scalar-only plateaus. Obtaining this feedback for a given model requires identifying its competing behaviors and designing probes to measure them. To make competing-behavior feedback actionable, we introduce ConflictGuide. Its reusable ConflictGuide-Skill combines a literature-grounded taxonomy with model-specific evidence to identify competing behaviors and specify probes for a code agent to implement as metrics. Evolution proceeds in two stages: Stage I explores with task feedback; Stage II uses probe feedback to steer proposals toward conflict alleviation and retains marginal-gain edits only when probes indicate sufficient alleviation. Across five diverse model families, ConflictGuide reduces task and conflict-related errors by up to 28% and 14%, respectively, relative to scalar-only AutoResearch, with gains extending to other code agents.

Veterans face multiple hardships after they leave the military. Is food insecurity one of these hardships? This paper examines the long-term effects of military service on food insecurity using new data from the PSID. Military veteran-headed households experienced lower food insecurity than non-veteran-headed households, and participated less in federal nutrition assistance programs. We use the Vietnam-era lottery draft to identify causal effects, and find no evidence that military service adversely affects food insecurity. Our results suggest policies that assist currently enlisted service members, and veterans unlikely to head a traditional household, may most effectively tackle food insecurity among military veterans.

Grain boundaries fundamentally dictate the macroscopic properties of polycrystalline materials by breaking long-range symmetry. In ferroelectrics, these structural discontinuities are conventionally considered as detrimental features that induce depolarization fields and accumulate defects, thereby suppressing or pinning local polarization. Here, we demonstrate that the commensurability of grain boundaries inherently governs the polar-orthorhombic phase stability in polycrystalline Hf0.5Zr0.5O2 (HZO) thin films. Using depth-resolved multislice electron ptychography, we map the three-dimensional (3D) phase distribution across diverse boundaries with atomic resolution, revealing that highly commensurate grain boundaries effectively suppress the nonpolar-tetragonal phase by compensating for lattice mismatch while mitigating geometric frustration. Monte Carlo simulations uncover the atomistic and energetic origins of polar-phase stabilization, showing that commensurate grain boundaries favor the polar-orthorhombic phase more strongly than other grain boundaries. Guided by this principle, we epitaxially engineer HZO thin films to promote the preferential formation of commensurate grain boundaries, resulting in an approximately 60% enhancement in the remanent polarization of ferroelectric devices. These findings provide direct 3D experimental and theoretical evidence that establishes grain boundary commensurability as a critical degree of freedom for designing functional interfaces, guiding the construction of novel ceramics and polycrystalline ferroelectrics.

We construct modular-invariant torus partition functions with a unique vacuum, discrete energy levels and positive integer degeneracies by recursively repairing the Maloney–Witten–Keller modular completion of the Virasoro vacuum. Exact modular repairs and finite moment matching discretize successive spectral bands while preserving earlier levels and controlling convergence. For $c_L=c_R=c$ and $a=(c-1)/12$, the construction realizes primary dimension gaps $\Delta_1=(1+\kappa)a$ for sufficiently small fixed $\kappa>0$, and $\Delta_1=a+\delta$ for any fixed $\delta\ge0$, at every sufficiently large real $c$. Every nonvacuum primary satisfies $h,\bar h\ge(c-1)/24$. In the fixed-$\delta$ family, spectra can be chosen whose densities of states, smoothed with a fixed nonnegative normalized smooth kernel of compact support, match the correspondingly smoothed prediction of a single perturbative BTZ saddle through every fixed finite order in $1/c$. The count includes all spins and Virasoro descendants. Its logarithm reproduces the Bekenstein–Hawking entropy and its corrections at every fixed positive $E/c$, where $E=\Delta-c/12$. This includes $0<E<c/12$, where thermal AdS dominates the canonical ensemble.

Long-context failures of RoPE-based language models can arise from RoPE's intrinsic tradeoff between maintaining stable token preferences and distinguishing nearby positions. Determining which weakness to address, and how, requires a more precise characterization of RoPE's behavior in trained models across context lengths. We address a key limitation of prior theory by allowing unequal query-key scales across RoPE frequencies, which aligns well with practical empirical observations. Our theory makes both vulnerabilities measurable for individual heads and inputs, and quantifies how high-frequency components support positional sensitivity while potentially disrupting semantic stability. We also derive a theoretical context-length bound beyond which, under specified conditions, a fixed attention-score comparison cannot jointly avoid semantic reversal and positional insensitivity. Guided by our fresh theoretical insights, we introduce RoPE Profiler, a lightweight, plug-and-play diagnostic toolkit that augments existing evaluations with zero additional forward passes by reusing cached query and key activations. Reusing activations collected during evaluation, the toolkit incurs little overhead. It supplements standard benchmark scores with two diagnostic scores that reveal semantic and positional weaknesses and help users prioritize which aspect to address. Crucially, our evaluations across 49 long-context task settings reveal a distinct pattern where reasoning tasks predominantly suffer from semantic reversal, whereas retrieval tasks are primarily vulnerable to positional insensitivity. Guided by our theory and diagnostic profiles, targeted high-frequency rescaling achieves immediate gains without additional training, improving task accuracy by up to 20 percentage points on Qwen3-8B and 25 percentage points on Llama-3.1-8B-Instruct.