arXiv:2605.16193v1 Announce Type: new Abstract: Large language models (LLMs) are increasingly used to simulate human opinions and survey responses, but their ability to reproduce population responses across cultures remains limited. Existing persona-based prompting methods typically rely on sociodemographic or personality traits, which are only indirect proxies for the values that shape human responses. We propose a value-based persona construction method that derives textual descriptors from survey responses capturing core cultural dimensions. By sampling value profiles from target populations and aggregating LLM responses across personas, we obtain population-level predictions grounded in observed value distributions. We further introduce a calibration procedure that improves response diversity while preserving estimated opinions. We show that our approach reduces prediction error across countries, with the largest improvements observed in underrepresented populations. This substantially narrows the performance gap between countries aligned with dominant LLM priors and those that are less represented in training data, while also yielding response distributions that closely match human diversity.
Science Journals
arXiv:2604.18145v2 Announce Type: replace Abstract: Automated medical report generation for 3D PET/CT imaging is fundamentally challenged by the high-dimensional nature of volumetric data and a critical scarcity of annotated datasets, particularly for low-resource languages. Current black-box methods map whole volumes to reports, ignoring the clinical workflow of analyzing localized Regions of Interest (RoIs) to derive diagnostic conclusions. In this paper, we bridge this gap by introducing VietPET-RoI, the first large-scale 3D PET/CT dataset with fine-grained RoI annotation for a low-resource language, comprising 600 PET/CT samples and 1,960 manually annotated RoIs, paired with corresponding clinical reports. Furthermore, to demonstrate the utility of this dataset, we propose HiRRA, a novel framework that mimics the professional radiologist diagnostic workflow by employing graph-based relational modules to capture dependencies between RoI attributes. This approach shifts from global pattern matching toward localized clinical findings. Additionally, we introduce new clinical evaluation metrics, namely RoI Coverage and RoI Quality Index, that measure both RoI localization accuracy and attribute description fidelity using LLM-based extraction. Extensive evaluation demonstrates that our framework achieves SOTA performance, surpassing existing models by 19.7% in BLEU and 4.7% in ROUGE-L, while achieving a remarkable 45.8% improvement in clinical metrics, indicating enhanced clinical reliability and reduced hallucination. Our code and dataset are available on GitHub.
arXiv:2507.05961v2 Announce Type: replace-cross Abstract: Low-temperature spin dynamics can become trapped in long-lived patterns shaped by the geometry of the interaction network. Here we introduce Chladni states: spin configurations obtained by binarizing the eigenmodes of the interaction Laplacian. These graph-spectral patterns organize the metastable configurations reached by Ising systems under non-ergodic relaxation. The resulting Topological Mode Decomposition provides a compact way to monitor and reconstruct frozen spin configurations in ferromagnets, frustrated antiferromagnets, and spin glasses.
arXiv:2512.19053v2 Announce Type: replace-cross Abstract: A detection method for high-frequency gravitational waves using two-dimensional ion crystals is investigated. Gravitational waves can resonantly excite the drumhead modes of the ion crystal, particularly the parity-odd modes. In the optical dipole force protocol, entanglement between the drumhead modes and the collective spins transfers the excitation of the drumhead modes to the rotation of the total spin. Furthermore, gravitational wave detection beyond the standard quantum limit becomes possible as a squeezed spin state is generated through this entanglement. The sensitivity gets better with a larger ions crystals as well as a larger number of the ions. Future realization of large ion crystals can significantly improve the sensitivity to gravitational waves in the 10 kHz to 10 MHz region.
arXiv:2605.09421v3 Announce Type: replace Abstract: Code authorship attribution (CAA) supports software forensics, plagiarism detection, and intellectual property protection. However, existing supervised CAA approaches suffer from scarce training data and closed-world assumptions: they require sufficient labeled code from fixed candidate-author sets, making training difficult in low-data cases and predictions unreliable for open-world test pairs with unseen samples, or heterogeneous code pairs. Large language models remove task-specific training, but direct prompting depends on costly expert-designed prompts, can hallucinate over complex heterogeneous code pairs, and rarely yields auditable evidence traces. We propose MACAA, a belief-revision-based multi-agent framework for training-free code authorship verification. MACAA comprises a Coordinator and four Expert Agents analyzing layout, lexical, syntactic, and programming-pattern evidence. The Coordinator gathers expert signals for expansion, discounts unreliable evidence through contraction, and resolves conflicts through revision to preserve belief consistency, replacing direct LLM judgment with auditable hypothesis refinement. MACAA achieves 89.15\% F1 on same-language benchmarks and 80.00\% on mixed cross-language pairs, outperforming the baselines overall in both same-language and cross-language evaluations.
arXiv:2604.14692v2 Announce Type: replace Abstract: Video understanding requires identifying and reasoning over semantically discriminative visual objects across frames, yet existing object-agnostic solutions struggle to effectively handle substantial object variations over time. To address this, we introduce Chain-of-Glimpse, a search-guided progressive object-grounded reasoning framework that explicitly anchors each reasoning step to specific visual evidence regions, enabling compositional and multi-step decision-making. Formally, Chain-of-Glimpse formulates video reasoning as a step-by-step process that incrementally builds spatially grounded traces around task-relevant visual objects, thereby mitigating over-reliance on saliency-driven cues. Specifically, Chain-of-Glimpse features a search-guided controller, optimized via reinforcement learning with a format reward that significantly incentivizes grounding capability, to iteratively ground visual evidence regions and form reliable reasoning trajectories, yielding accurate and interpretable multi-step decisions. Extensive evaluations on both in domain NExTQA and out-of-domain Video-Holmes, CG-Bench Reasoning, and VRBench benchmarks demonstrate consistent performance gains, robustness and generalization of Chain-of-Glimpse across diverse video reasoning tasks.
arXiv:2511.02342v3 Announce Type: replace Abstract: Aerial manipulation combines the maneuverability of multirotors with the dexterity of robotic arms to perform complex tasks in cluttered spaces. Yet planning safe, dynamically feasible trajectories remains difficult due to whole-body collision avoidance and the conservativeness of common geometric abstractions such as bounding boxes or ellipsoids. We present a whole-body motion planning and safety-critical control framework for aerial manipulators built on superquadrics (SQs). Using an SQ-plus-proxy representation, we model both the vehicle and obstacles with differentiable, geometry-accurate surfaces. Leveraging this representation, we introduce a maximum-clearance planner that fuses Voronoi diagrams with an equilibrium-manifold formulation to generate smooth, collision-aware trajectories. We further design a safety-critical controller that jointly enforces thrust limits and collision avoidance via high-order control barrier functions. In simulation, our approach outperforms sampling-based planners in cluttered environments, producing faster, safer, and smoother trajectories and exceeding ellipsoid-based baselines in geometric fidelity. Actual experiments on a physical aerial-manipulation platform confirm feasibility and robustness, demonstrating consistent performance across simulation and hardware settings. The video can be found at https://youtu.be/hQYKwrWf1Ak.
arXiv:2605.16062v1 Announce Type: cross Abstract: Correlative nanoscale surface characterization benefits from simultaneously measuring electronic and structural properties in the same environment, a capability that is essential for modern-day materials science and semiconductor failure analysis. In-situ AFM-SEM measurements facilitated by self-sensing cantilevers offer great potential here; however, they are limited due to their inherent capacitive crosstalk. Here, we demonstrate for the first time the in-situ implementation of single-pass heterodyne Kelvin probe force microscopy inside a scanning electron microscope, using piezo-resistive cantilevers. We overcome the capacitive crosstalk prevalent in piezo-resistive cantilevers by demodulating excitation and detection to simultaneously map surface topography and contact potential difference for correlation with compositional analysis. We systematically compare different operational modes of this heterodyne technique, elucidating their spatial resolution, signal sensitivity, and signal-to-noise ratio. The integrated approach yields exceptional signal quality and reveals how electron beam scan parameters can directly influence surface potential contrast. We demonstrate this correlative analysis workflow on two-dimensional heterostructures and semiconductor circuits. This work establishes a robust and versatile correlative imaging mode for in-situ Kelvin force and topography imaging inside a scanning electron microscope for next-generation semiconductor device analysis and materials science.
arXiv:2407.08094v3 Announce Type: replace-cross Abstract: We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. Subsequently, such differences are integrated, weighted by their associated uncertainties, using a maximum-likelihood formulation. This procedure can be seen as an extension to a multidimensional setting of the thermodynamic integration, a technique developed in statistical physics. The method leverages the manifold hypothesis, estimating quantities within the intrinsic data manifold without defining an explicit coordinate map. It does not rely on any binning or space partitioning, but rather on the construction of a neighbourhood graph based on an adaptive bandwidth selection procedure. BMTI mitigates the limitations commonly associated with traditional nonparametric density estimators, effectively reconstructing smooth profiles even in high-dimensional embedding spaces. The method is tested on a variety of complex synthetic high-dimensional datasets, where it is shown to outperform traditional estimators, and is benchmarked on realistic datasets from the chemical physics literature.
arXiv:2603.20030v2 Announce Type: replace Abstract: A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of an orientable surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].
arXiv:2505.03307v3 Announce Type: replace-cross Abstract: Simulating Clifford and near-Clifford circuits using the extended stabilizer formalism has become increasingly popular, particularly in quantum error correction. Compared to the state-vector approach, the extended stabilizer formalism can solve the same problems with fewer computational resources, as it operates on stabilizers rather than full state vectors. Most existing studies on near-Clifford circuits focus on balancing the trade-off between the number of ancilla qubits and simulation accuracy, often overlooking performance considerations. Furthermore, in the presence of high-rank stabilizers, performance is limited by the sequential property of the stabilizer formalism. In this work, we introduce a parallelized version of the extended stabilizer formalism, enabling efficient execution on multi-core devices such as GPU. Experimental results demonstrate that, in certain scenarios, our Python-based implementation outperforms state-of-the-art simulators such as Qiskit and Pennylane.
arXiv:2512.09502v2 Announce Type: replace Abstract: Diverse scientific and engineering research areas deal with discrete, time-stamped changes in large systems of interacting delay differential equations. Simulating such complex systems at scale on high-performance computing clusters demands efficient management of communication and memory. Inspired by the human cerebral cortex -- a sparsely connected network of $\mathcal{O}(10^{10})$ neurons, each forming $\mathcal{O}(10^{3})$--$\mathcal{O}(10^{4})$ synapses and communicating via short electrical pulses called spikes -- we study the simulation of large-scale spiking neural networks for computational neuroscience research. This work presents a novel network construction method for multi-GPU clusters and upcoming exascale supercomputers using the Message Passing Interface (MPI), where each process builds its local connectivity and prepares the data structures for efficient spike exchange across the cluster during state propagation. We demonstrate scaling performance of two cortical models using point-to-point and collective communication, respectively.
arXiv:2605.15403v1 Announce Type: new Abstract: Mixture-of-Experts (MoE) models rely on balanced expert utilization to fully realize their scalability. However, existing load-balancing methods are largely heuristic and operate on noisy mini-batch assignment statistics, introducing bias relative to population-level objectives. We propose $\phi$-balancing, a principled framework that directly targets population-level expert balance by minimizing a strictly convex, symmetric, and differentiable potential of the expected routing distribution. Using convex duality, we derive an equivalent min-max formulation and obtain a simple online algorithm via mirror descent, yielding an efficient EMA-based routing adjustment with negligible overhead. Across large-scale pretraining and downstream fine-tuning, $\phi$-balancing consistently outperforms prior Switch-style and loss-free baselines, demonstrating more stable and effective expert utilization.
arXiv:2512.04457v2 Announce Type: replace Abstract: Removing specific data influence from large language models (LLMs) remains challenging, as retraining is costly and existing approximate unlearning methods are often unstable. The challenge is exacerbated when the forget set is small or imbalanced. We introduce RapidUn, an influence-driven and parameter-efficient unlearning framework. It first estimates per-sample influence through a fast estimation module, then maps these scores into adaptive update weights that guide selective parameter updates -- forgetting harmful behavior while retaining general knowledge. On Mistral-7B and Llama-3-8B across Dolly-15k and Alpaca-57k, RapidUn achieves up to 100 times higher efficiency than full retraining and consistently outperforms Fisher, GA, and LoReUn on both in-distribution and out-of-distribution forgetting. These results establish influence-guided parameter reweighting as a scalable and interpretable paradigm for LLM unlearning.
arXiv:2602.23410v3 Announce Type: replace Abstract: Brain foundation models have achieved remarkable advances across a wide range of neuroscience tasks. However, most existing models are limited to a single functional modality, restricting their ability to exploit complementary spatiotemporal dynamics and the collective data scale across different neuroimaging techniques. This limitation largely arises from severe semantic heterogeneity and resolution discrepancies among modalities. To address these challenges, we propose Brain-OF, an omnifunctional brain foundation model jointly pretrained on fMRI, EEG and MEG, capable of handling both unimodal and multimodal inputs within a unified framework. To reconcile heterogeneous spatiotemporal resolutions, we introduce the Any-Resolution Neural Signal Sampler, which projects diverse brain signals into a shared semantic space. To further manage semantic shifts, the Brain-OF backbone integrates DINT attention with a Sparse Mixture of Experts, where shared experts capture modality-invariant representations and routed experts specialize in modality-specific semantics. Furthermore, to explicitly internalize the characteristics of neural activity through self-supervised learning, we propose Masked Temporal-Frequency Modeling, a dual-domain pretraining objective that jointly reconstructs brain signals in both the time and frequency domains. Brain-OF is pretrained on a large-scale corpus comprising around 40 datasets and demonstrates superior performance across diverse downstream tasks, highlighting the benefits of joint multimodal integration and dual-domain pretraining.
arXiv:2602.06239v3 Announce Type: replace Abstract: We introduce PEPO (Pessimistic Ensemble based Preference Optimization), a single-step Direct Preference Optimization (DPO)-like algorithm to mitigate the well-known over-optimization issue in preference learning without requiring the knowledge of the data-generating distribution or learning an explicit reward model. PEPO achieves pessimism via an ensemble of preference-optimized policies trained on disjoint data subsets and then aggregates them through a worst case construction that favors the agreement across models. In the tabular setting, PEPO achieves sample complexity guarantees depending only on a single-policy concentrability coefficient, thus avoiding the all-policy concentrability which affects the guarantees of algorithms prone to over-optimization, such as DPO. The theoretical findings are corroborated by a convincing practical performance, while retaining the simplicity and the practicality of DPO-style training.
arXiv:2601.03707v2 Announce Type: replace Abstract: Existing UAV vision-and-language navigation (VLN) benchmarks rarely provide realistic aerial scenes, natural process-level instructions, and sufficient scale simultaneously, making it difficult to systematically train and evaluate UAV VLN agents under realistic settings. To address this, we propose \textbf{AirNav}, a large-scale benchmark built on real urban aerial data, comprising 137K navigation samples with natural and diverse instructions generated via a human--LLM collaborative pipeline with 10 user personas. We conduct a systematic evaluation of representative approaches on AirNav, ranging from traditional models to multimodal large language models (MLLMs), under unified metrics with open-source implementations. We further propose \textbf{AirVLN-R1}, trained via supervised fine-tuning (SFT) and reinforcement fine-tuning (RFT), achieving state-of-the-art performance with a 51.82\% success rate on the test-unseen split. Real-world experiments on a physical UAV platform provide preliminary evidence of sim-to-real transferability, and our dataset and code are publicly available.
arXiv:2512.21651v3 Announce Type: replace Abstract: Large Language Models (LLMs) deliver strong performance across a wide range of NLP tasks, but their massive sizes hinder deployment on resource-constrained devices. To reduce their computational and memory burden, various compression techniques have been proposed, including quantization, pruning, and knowledge distillation. Among these, post-training quantization (PTQ) is widely adopted for its efficiency, as it requires no retraining and only a small dataset for calibration, enabling low-cost deployment. Recent advances for post-training quantization have demonstrated that even near 4-bit methods can maintain most of the original model performance. However, 1-bit quantization remains particularly challenging. A common strategy in 1-bit quantization is to determine binary weights by matching full-precision parameters, following a weight-driven criterion. However, this objective is not directly aligned with the quantized model's objective, which is to preserve the model's output behavior under the impact of quantization. A natural alternative is to adopt output-driven criteria that minimize discrepancies in model outputs using calibration data. Surprisingly, naive output-driven approaches often perform even worse in the 1-bit regime. In this paper, we show that this failure arises from two fundamental issues: error accumulation across layers and, more critically, \emph{anisotropic distortion} of the representation space. Based on these insights, we propose a novel PTQ method for 1-bit LLMs that explicitly addresses these issues while maintaining computational efficiency. Extensive experiments demonstrate that our approach consistently outperforms existing 1-bit PTQ methods.
arXiv:2511.18719v4 Announce Type: replace Abstract: Reinforcement learning (RL) has become a powerful tool for post-training visual generative models, with Group Relative Policy Optimization (GRPO) increasingly used to align generators with human preferences. However, existing GRPO pipelines rely on a single scalar reward per sample, treating each image or video as a holistic entity and ignoring the rich spatial and temporal structure of visual content. This coarse supervision hinders the correction of localized artifacts and the modeling of fine-grained perceptual cues. We introduce Visual Preference Policy Optimization (ViPO), a GRPO variant that lifts scalar feedback into structured, pixel-level advantages. ViPO employs a Perceptual Structuring Module that uses pretrained vision backbones to construct spatially and temporally aware advantage maps, redistributing optimization pressure toward perceptually important regions while preserving the stability of standard GRPO. Across both image and video benchmarks, ViPO consistently outperforms vanilla GRPO, improving in-domain alignment with human-preference rewards and enhancing generalization on out-of-domain evaluations. The method is architecture-agnostic, lightweight, and fully compatible with existing GRPO training pipelines, providing a more expressive and informative learning signal for visual generation.
arXiv:2308.06822v3 Announce Type: replace Abstract: Federated Learning (FL) is a distributed learning paradigm that enables multiple clients to collaborate on building a machine learning model without sharing their private data. Although FL is considered privacy-preserved by design, recent data reconstruction attacks demonstrate that an attacker can recover clients' training data based on the parameters shared in FL. However, most existing methods fail to attack the most widely used horizontal Federated Averaging (FedAvg) scenario, where clients share model parameters after multiple local training steps. To tackle this issue, we propose an interpolation-based approximation method, which makes attacking FedAvg scenarios feasible by generating the intermediate model updates of the clients' local training processes. Then, we design a layer-wise weighted loss function to improve the data quality of reconstruction. We assign different weights to model updates in different layers concerning the neural network structure, with the weights tuned by Bayesian optimization. Finally, experimental results validate the superiority of our proposed approximate and weighted attack (AWA) method over the other state-of-the-art methods, as demonstrated by the substantial improvement in different evaluation metrics for image data reconstructions.
arXiv:2602.08843v1 Announce Type: cross Abstract: Many fundamental problems in computational geometry admit no algorithm running in $o(n \log n)$ time for $n$ planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees, onion layer decompositions, Euclidean minimum spanning trees, KD-trees, Voronoi diagrams, and decremental closest-pair. A classical result shows that, given $n$ points sorted along a single direction, the convex hull can be constructed in linear time. Subsequent works established that for all of the other above problems, this information does not suffice. In 1989, Aggarwal, Guibas, Saxe, and Shor asked: Under which conditions can a Voronoi diagram be computed in $o(n \log n)$ time? Since then, the question of whether sorting along TWO directions enables a $o(n \log n)$-time algorithm for such problems has remained open and has been repeatedly mentioned in the literature. In this paper, we introduce the Presort Hierarchy: A problem is 1-Presortable if, given a sorting along one axis, it permits a (possibly randomised) $o(n \log n)$-time algorithm. It is 2-Presortable if sortings along both axes suffice. It is Presort-Hard otherwise. Our main result is that quadtrees, and by extension Delaunay triangulations, Voronoi diagrams, and Euclidean minimum spanning trees, are 2-Presortable: we present an algorithm with expected running time $O(n \sqrt{\log n})$. This addresses the longstanding open problem posed by Aggarwal, Guibas, Saxe, and Shor (albeit randomised). We complement this result by showing that some of the other above geometric problems are also 2-Presortable or Presort-Hard.
arXiv:2605.16181v1 Announce Type: new Abstract: Training data attribution (TDA) for music generation must answer two questions that copyright analysis requires, namely which training songs influence a generated output and along which musical aspects the influence operates. Existing methods reduce influence to a single scalar, without revealing which musical aspects are dominant in that influence. We propose ARIA, a framework that decomposes attribution along musical aspects (five for symbolic music, three for audio) and pairs the decomposition with reliability diagnostics computed from the segment-level score matrix. It measures within-group similarity among the top-K attributed tracks against random reference groups drawn from the training pool, and diagnoses the score matrix through its singular value decomposition and column statistics. On a symbolic-music model where attribution ground truth is available through counterfactual retraining, the reliability diagnostics rank four attribution methods identically to that ground truth. On an audio music generation model, ARIA reveals attribution behaviors that vary substantially across TDA methods, flags score matrices whose retrieved tracks are nearly identical across queries rather than reflecting per-query attribution, and characterizes embedding-similarity retrieval baselines by the musical aspect each encoder surfaces. Together, ARIA produces per-aspect attribution evidence aligned with the musical aspects considered under the idea-expression distinction in copyright analysis.
arXiv:2512.19179v3 Announce Type: replace Abstract: Efficiently harnessing GPU compute is critical to improving user experience and reducing operational costs in large language model (LLM) services. However, current inference engine schedulers overlook the attention backend's sensitivity to request-length heterogeneity within a batch. As state-of-the-art models now support context windows exceeding 128K tokens, this once-tolerable inefficiency has escalated into a primary system bottleneck, causing severe performance degradation through GPU underutilization and increased latency. We present CascadeInfer, a runtime system that dynamically reschedules requests across multiple instances serving the same LLM to mitigate per-instance length heterogeneity. CascadeInfer partitions these instances into length-specialized groups, each handling requests within a designated length range, naturally forming a pipeline as requests flow through them. CascadeInfer devises a dynamic programming algorithm to efficiently find the stage partition with the best QoE, employs runtime range refinement together with decentralized load (re)balance both across and within groups, achieving a balanced and efficient multi-instance service. Our evaluation shows that, under the same configuration, CascadeInfer reduces end-to-end latency by up to 67% and tail latency by up to 69%, while improving overall system throughput by up to 2.89 times compared to the state-of-the-art multi-instance scheduling systems.
arXiv:2212.13347v3 Announce Type: replace Abstract: We report a diffuse Maxwellian illumination scheme for wide-field retinal laser Doppler holography. Inserting an engineered diffuser in the illumination arm transforms a spatially concentrated near-infrared laser focus into an angularly diversified illumination pattern, thereby reducing local irradiance near the anterior segment while preserving coherent interferometric detection. This configuration allows the eyepiece to be positioned closer to the cornea, increasing the digitally reconstructed retinal field of view without producing a localized corneal hot spot. We compare three illumination geometries: focused non-diffuse illumination, diffuse illumination at the same cornea--eyepiece distance, and diffuse Maxwellian illumination. Diffuse Maxwellian illumination expands the retinal field of view while preserving Doppler contrast in broad and high-frequency fluctuation bands. Light-hazard assessment is limited to the current ophthalmic standards ISO 15004-2:2024 and ANSI Z80.36-2021. Based on measured beam profiles, the recommended operating power at 852 nm is set by the most restrictive relevant exposure condition among the assessed anterior-segment, iris, and retinal limits. These results support diffuse illumination as a practical route toward safer, non-mydriatic, wide-field Doppler holography of the human retina.
arXiv:2605.15372v1 Announce Type: cross Abstract: We derive an explicit formula for the intrinsic MacWilliams transform for permutation-invariant qudit codes. Such codes naturally live in symmetric power representations, where the relevant error sectors are determined by the irreducible decomposition of the conjugation action on the associated operator space. Using the multiplicity-free structure of this decomposition and the corresponding intertwiner algebra, we identify the intrinsic MacWilliams matrix with a finite Racah transform. The entries are given by a terminating hypergeometric series, and the rows of the matrix are Racah orthogonal polynomials with parameters determined explicitly by the block length and local dimension. Computing the spectrum of the degree-one twirl reveals that this spectrum lies on an affine quadratic lattice. Then we derive a tridiagonal multiplication rule from the representation theory of the adjoint sector. As consequences, we obtain closed-form orthogonality, detailed-balance, and involutivity identities for the transform. The resulting formula supplies an explicit MacWilliams matrix for computing linear programming bounds on permutation-invariant qudit codes.