Merging galaxy clusters are a promising laboratory for measuring the self-interaction cross-section (SICS) of dark matter. However, previous studies have focused on galaxy-mass offsets, which numerical simulations have shown to be intrinsically small because galaxies remain tightly coupled to the dominant dark matter potential even with significant self-interaction. Their interpretation is further complicated by unknowns of the merger phase, geometry, and initial conditions. In this paper, we overcome these obstacles by introducing the shock-to-shock distance, traced by double radio relics, as a merger chronometer that time-stamps the post-pericenter dynamical phase. Because the propagation speed of merger shocks is nearly independent of the SICS, while the halo-to-halo distance is depressed by SIDM-induced drag, the ratio of the two distances translates directly into a constraint on sigma/m. Applying this method to a gold sample of eleven cluster mergers hosting symmetric double radio relics, we determine a 68% upper limit on the SICS of sigma/m < 0.22 cm^2/g. This is the first constraint from cluster collisions that fully marginalizes over mass uncertainty, viewing angle, collision speed, merger phase, impact parameter, and gas profile slope.

In the hierarchical paradigm of galaxy formation, central massive black holes (MBHs) are expected to coalesce after the merger of their host galaxies. One of the main goals of the Laser Interferometer Space Antenna (LISA) is to constrain the origin and growth of MBHs through their merger rates and mass distribution. Predicting MBH merger rates requires not only tracing their statistical population from large to small physical scales (kpc to sub-pc) but also modelling their formation, accretion, dynamics, mergers, and their galactic physical processes across cosmic time. This project is the result of a large collaborative effort undertaken by the LISA Astrophysics Working Group, bringing together its collective expertise on MBH formation, evolution, and modelling, to build a comprehensive understanding of MBH merger rates across cosmic time. The project compares various theoretical predictions of MBH merger rates, quantifies the spread, and evaluates the global astrophysical uncertainties of the LISA event rates. To build a unique and complete view, our work is based on about 20 semi-analytical models and cosmological simulations from the literature, all employing distinct approaches to modelling MBH and galaxy physics. To compute the merger rates, we also incorporate delays arising from the dynamical phase of MBH hardening to coalescence. We present the expected LISA merger rates given current galaxy formation models and discuss how the merger rate depends on model assumptions, such as the seeding model and the resolution of cosmological simulations.

Local kinetic constraints in quantum many-body systems can generate slow dynamics or complete many-body localisation. Here we focus on a modification of the quantum East model: Inspired by random matrix theory, we randomise the connectivity in Fock space (rendering it nonlocal in real space) while preserving its organisation into neighbouring magnetisation sectors. We find that there is still a transition between two distinct phases, one delocalised and the other localised. We conclude that, for East-type constrained models, the essential ingredient is the structure of the graph in Fock space rather than geometric locality of spin flips.

Levitation of microscopic objects in vacuum combines exceptional environmental isolation with precise control of their dynamics, pushing the limits of sensing and macroscopic quantum physics. In particular, magnetic levitation allows a large range of particle sizes, while avoiding detrimental effects from high-intensity optical trapping beams and electric field noise. However, existing diamagnetic and Meissner levitation approaches are typically constrained by low mechanical eigenfrequencies, limited integrability with other systems due to bulky coils or magnets, and, for Meissner levitation, the need for cryogenic operation. Here, we demonstrate a room-temperature on-chip magnetic levitation platform capable of stably levitating a nanogram (6.5 micrometer radius) ferromagnetic microsphere. The platform is scalable and tunable, and supports librational modes with eigenfrequencies exceeding 10 kHz. Further miniaturization and coupling to solid-state spin qubits could enable cooling to the quantum ground state. Beyond quantum experiments, this architecture enables integrated precision sensing and studies of isolated ferromagnet thermodynamics.

We propose a unifying framework for characterizing pure and mixed state phases of matter across equilibrium, non equilibrium, and metastable regimes. We introduce the concept of locally stable states, defined by the operational property that any local operation (including post selection) can be reversed by a local channel. We prove that local stability is equivalent to a state being short range correlated, defined by the decay of both correlations and conditional mutual information. We demonstrate that these properties are invariant under locally reversible channels, thus defining locally stable phases. Furthermore, we prove that local stability implies both the decay of a family of nonlinear correlators, including the fidelity correlator, and the decay of correlations in the canonical purification, thus bridging the gap between mixed and pure states. Along the way, we establish two results which may be of independent interest: we show that post-selection on locally stable (short range correlated) states can be implemented via local channels and that quantum Markov chains can be characterized by the local computability of nonlinear observables.

Percolation theory from statistical physics has been applied to several aspects of tumor progression. Tumor growth on percolation clusters has been used to model spatial expansion, vascular percolation to describe nutrient supply and transport related percolation to investigate drug and gene delivery. At the molecular level, mutational percolation has been employed to account for the emergence of malignant phenotypes, while inverse percolation to represent treatment-induced structural disruption. We examined whether tumor containment can be interpreted as an anti percolation problem, in which spatial expansion depends on the formation of a connected malignant domain. We implemented a spatial simulation with biologically scaled parameters to represent tissue heterogeneity, local growth, cell movement and clearance. We measured both total malignant area and connectivity metrics, including the largest connected component and the probability of forming a spanning cluster. Our results indicate that tumor size and spatial connectivity are partially independent, with configurations of similar size showing different connectivity patterns. A transition from fragmented to connected structures emerged within a limited parameter range, consistent with a threshold like behavior. Incorporating spatial connectivity into quantitative analysis, our approach provides a complementary way to characterize tumor organization. Potential applications include integration of structural descriptors into computational models of tumor growth, design of experimental systems to probe spatial organization and interpretation of therapeutic approaches via connectivity-based metrics.

Baryons, as three-quark bound states, require a covariant treatment in the intermediate-energy regime where perturbative QCD is no longer applicable and where nonperturbative correlations dominate. This article reformulates the content of the CERN Baltic Conference 2025 presentation on baryonic bound states in the non-local Nambu--Jona-Lasinio (NJL) model. We review how the relativistic Faddeev approach reduces the three-body quark problem to an effective quark--diquark bound-state problem, describe the scalar and axial-vector diquark channels, and show how the resulting quark--diquark Bethe--Salpeter equation can be written as an eigenvalue problem for the baryon mass. The non-local NJL framework, motivated by QCD-based nonlocal interactions and Dyson--Schwinger considerations, provides a compact description in which baryon masses and form factors are extracted from the numerical solution of coupled integral equations.

Agentic security systems increasingly combine LLM planners with tools that can discover, validate, and report vulnerabilities. This creates an asymmetric control problem: the system should retain strong offensive capability inside an authorized engagement, while the same capabilities must be denied outside scope. Existing guardrails provide useful policy controls, but they do not make this boundary a first-class formal contract over observable effects. We introduce alignment contracts, a framework for specifying and enforcing behavioral constraints over observable effect traces. A contract defines scope, allowed and forbidden effects, resource budgets, and disclosure policies. We give the language finite-trace semantics, characterize satisfaction as a safety property with finite violation witnesses, develop refinement and one-way composition rules for modular contract engineering, and show that admissibility checking is decidable. We instantiate the framework for web-focused agentic security workflows and show how the same structure extends to other effect profiles. Under an explicit Effect Observability Assumption, where all $\SigmaEff$-effects are mediated, the soundness theorem quantifies over the agent model and gives guarantees for mediated $\SigmaEff$-effects, including enforcement soundness for monitor-realized traces. We also state an assumption-lifted adaptation result and formalize limits through undecidability transfer and observability-boundary theorems. A Lean 4 artifact checks the formal core theorems used by the paper.

Magog matrices, introduced by Holmlund and Striker in 2025, provide a matrix model for totally symmetric self-complementary plane partitions (TSSCPPs), as a natural analogue of alternating sign matrices (ASMs). In this paper, we develop several new combinatorial representations of magog matrices, mirroring classical representations of ASMs. Specifically, we define magog analogues of corner-sum matrices, height-function matrices, fully packed loop configurations, and vertex models, and establish explicit bijections among all of these objects. These constructions provide new structural insight into the combinatorics of TSSCPPs and illuminate parallels and differences between the ASM and TSSCPP frameworks.

A scheme is proposed for realizing simultaneous negative permittivity and negative permeability based on quantum coherence in a four-level dense atomic system here.Under some parametric conditions the system shows that simultaneous negative permittivity and negative permeability(i.e.Left handness) can be achieved in a wider frequency band because of quantum coherence.And the novelty properties of gain and dispersion near the resonance frequency may have some potential applications.

Software Bills of Materials (SBOMs) are increasingly mandated by regulators, yet existing sharing mechanisms impose a binary choice between full disclosure and full opacity. This exposes software suppliers to attacks that can be deduced from the SBOM only, such as the presence of a vulnerable dependency. Conversely, software consumers can be fooled by software suppliers who modify or misrepresent published SBOMs. We present zkSBOM, a privacy-preserving SBOM sharing mechanism designed to address these threats. zkSBOM uses zero-knowledge sets to cryptographically commit to the components within an SBOM. Software consumers can query for known vulnerabilities and receive a cryptographic proof confirming whether the artifact described by the SBOM is affected, without revealing any additional SBOM content. We conduct a security analysis of zkSBOM by quantifying expected leakage from inclusion and exclusion proofs. We demonstrate real-world feasibility by applying it to realistic scenarios and evaluating its operation requirements. Our evaluation demonstrates that zkSBOM is a strong, secure, and privacy-preserving mechanism for SBOM sharing, protecting software suppliers and software consumers from one another.

We show that, under additivity, the maximal von Neumann algebra extension of $\mathcal{A}(O)$ inside $B(\mathcal{H})$ whose inner automorphisms are non-signalling with respect to all spacelike-separated regions is $\mathcal{A}(O')'$. Consequently, $\mathcal{A}(O)$ is maximal with respect to this property if and only if essential duality holds. The proof is purely algebraic. When essential duality fails, we construct a proper extension all of whose inner automorphisms, and more generally all normal completely positive maps admitting Kraus operators in the algebra, are non-signalling. Under essential duality, any proper extension necessarily admits a signalling operation. An entropic formulation using Araki relative entropy provides a quantitative diagnostic of signalling, though it is not used in the proof. Additional structural results include the wedge-intersection identity $\mathcal{A}(O')' = \bigcap_{W \supset O}\mathcal{A}(W)$ and equivalent characterisations of essential duality. These results identify essential duality as an operational maximality condition within the given representation.

We present EPITIME (EPidemic Integral models TIMe profile Explorer), a computational framework for the simulation of two classes of integral epidemic models: an age of infection model and an information dependent behavioural model. The framework combines structure preserving Non-Standard Finite Difference discretizations with modular implementations in MATLAB and Python, together with routines for parameter handling, input validation, performance assessment, and graphical interaction. The proposed methods preserve key qualitative properties of the continuous problems, including positivity, boundedness, invariant regions, and correct long term behaviour, independently of the time step. We outline the numerical schemes for both model classes and their main analytical properties, including first order convergence. We then describe the software architecture and illustrate its use through numerical experiments on asymptotic behaviour, inverse reconstruction of an infectivity kernel from COVID 19 incidence data, and behavioural dynamics under different memory kernels. Overall, EPITIME provides a reliable and accessible computational environment for the numerical study of renewal epidemic models.

Integrity of audit logs produced by Internet of Things (IoT) devices is a prerequisite for post-incident forensics, regulatory compliance, and operational accountability. While blockchain-backed logging infrastructures can satisfy this requirement, they introduce consensus overhead, network dependencies, and deployment complexity that are often prohibitive at the IoT edge. This paper presents a lightweight and evaluated integrity verification pipeline that combines Merkle-tree commitments with resource-aware adaptive chunking to provide tamper evidence without relying on distributed ledger technologies. The proposed pipeline operates in three stages: (i) resource-aware batch ingestion via adaptive chunk sizing, (ii) Merkle-tree construction with O(logn) inclusion proof generation, and (iii) deterministic single-entry verification against a trusted root anchor. We further report an implementation audit that identified and corrected two evaluation defects: a double-counting bug in tampering metrics and a redundant full-tree reconstruction during batch appends. Using the corrected implementation, five-run benchmarks on synthetic IoT log datasets demonstrate throughput exceeding 130,000 logs/s for 100,000 records. The system achieves per-entry verification latency of approximately 22 ms, proof generation latency of 22 ms, an average proof size of 1,006 bytes, and peak memory usage below 5 MB. Tampering detection achieves perfect precision, recall, and F1-score (1.0) across corruption ratios ranging from 1% to 50%.

Modeling invasive neural spike data is fundamental to advancing high-performance brain-computer interfaces (BCIs). However, existing approaches face critical challenges, including limited-scale heterogeneous data, cross-domain distribution shift, and the intrinsic spatiotemporal complexity of invasive neural signals. In this work, we propose UniBCI, a unified pretrained model for invasive Brain-Computer Interfaces. The model integrates three key components: (1) a context-conditioned spatio-temporal tokenization (CST) scheme that embeds neural signals together with metadata into a shared representation space; (2) a hierarchical Interval-Area Attention (IAA) mechanism that captures patterns of spike dynamics in slots via linear attention and locality dependencies via sliding-window attention; and (3) a scalable self-supervised masked signals reconstruction objective for learning generalizable neural representations from large-scale unlabeled data. We construct a pretraining corpus spanning multiple species, subjects, brain regions, and behavioral experiment paradigms. These heterogeneous recordings are standardize via our proposed unified normalization and tokenization. Comprehensive experiments demonstrate that UniBCI achieves SOTA performance across diverse downstream tasks while improving generalization. Moreover, the model achieves a strong balance between accuracy and efficiency, with fewer trainable parameters and lower inference latency. These results suggest that UniBCI provides a practical step toward general-purpose neural foundation models, enabling robust, scalable, and transferable representation learning for invasive neural data. The code for this paper is available at: https://anonymous.4open.science/r/UniBCI-C805.

In his 1972 book Science At the Crossroads, Helbert Dingle attacked the consistency of special relativity through a fallacious argument championed by the crank community even to this day. Dingle's affair is a curious chapter in the history of physics and, more generally, science. We briefly review Dingle's case from a historical and didactic perspective.

Entropic cosmology with the Rényi entropy of the apparent horizon $S_R=(1/α)\ln(1+αS_{BH})$, where $S_{BH}$ is the Bekenstein--Hawking entropy, is studied. By virtue of the thermodynamics-gravity correspondence a model of dark energy is investigated. The generalised Friedmann equations for the Friedmann--Lemaître--Robertson--Walker spatially flat universe with the barotropic matter fluid are obtained. We compute the dark energy density $ρ_D$, pressure $p_D$ and the deceleration parameter $q$ of the universe. At some model parameters the normalized density parameter of the matter $Ω_{m0}\approx 0.315$ and the deceleration parameter $q_0\approx -0.535$ for the current epoch, which are in the agreement with the Planck data, are found. Making use of the thermodynamics-gravity correspondence, we describe the late time of the universe acceleration. The entropic cosmology considered is equivalent to cosmology based on the teleparallel gravity with the definite function $F(T)$. The Hubble parameters are in approximate agreement (within $5$ percents) with the observational Hubble data for redshifts $0.07\leq z \leq 1.75$ at the entropy parameter $α\approx 0.305~GH_0^2$.

Sparsity in the delay-Doppler (DD) domain enables efficient channel estimation, but the realization-wise sparsity level is rarely known in advance, and it fluctuates. What if we could estimate the channel without ever knowing how many delays or Dopplers are active? This paper answers that question. We propose a sparsity-agnostic structured estimator that requires no prior knowledge of delay or Doppler sparsity budgets. The key idea is to exploit the Cartesian-product structure of DD support (active delays share a common Doppler set) and to select the support dimensions directly from the data via the Bayesian information criterion. We instantiate the framework on an affine frequency division multiplexing system, where the observation model naturally admits an on-grid DD representation. Numerical results demonstrate that it recovers the exact support with high probability and achieves near-oracle channel reconstruction accuracy, consistently outperforming fixed-budget baselines and sparse Bayesian learning. The approach is waveform-agnostic and offers a practical, adaptive solution for DD-domain channel estimation under unknown and time-varying sparsity.

Given that the restricted equivalence functions (REFs) can serve to measure the similarity of two fuzzy sets, this motivates the integration of REFs with similarity-based approximate reasoning systems to enhance inference capabilities. Therefore, this work primarily constructs hierarchical similarity-based approximate reasoning (SBAR) using REFs. Specifically, we first characterize REFs with a given aggregation function, then discuss the approximation equality of SBAR method proposed by Raha et al. with REFs. Finally, we suggest two REF-based hierarchical Raha's SBAR methods which efficiently restrain the explosion of fuzzy rules.

In the paper [E. Jiménez-Fernández, J. Rodr\'ıguez-López, E. A. Sánchez-Pérez, Fuzzy Sets and Systems 406 (2021),66-81], a McShane-Whitney extension theorem is presented for real-valued fuzzy Lipschitz maps between fuzzy metric spaces. Specifically, the codomain space is considered as a so-called Euclidean fuzzy metric space $(\mathbb{R},M_{ϕ,g},\ast).$ However, while the function $ϕ$ is only required to be increasing, some results of the paper implicitly assume that $ϕ$ is invertible, even though this is not explicitly stated. We propose here an alternative possibility that only requires $ϕ$ to be also left-continuous.

We show that every family of quasi-arithmetic means generated by (a subset of) $\mathcal{C}^1$ functions with nonvanishing derivative which is bounded (from below or from above) by a quasi-arithmetic mean, possesses the best (lower or upper) bound which is a quasi-arithmetic mean generated by a function belonging to the same family.

The transitivity of fuzzy relations plays an important role in fuzzy set theory, artificial intelligence, clustering and decision-making. However, it is often difficult for fuzzy relations to satisfy the transitivity property in many practical applications. This has motivated researchers to investigate the degree to which a fuzzy relation is transitive. Therefore, this work first investigates two different measures of T-transitivity for fuzzy relations using some well-known fuzzy implications. And then, the relationship between two different degrees of transitivity is investigated. Further, the concept of an ε-T-transitive fuzzy relation is introduced, and the aggregation functions that preserve the ε-T-transitivity of fuzzy relations are characterized. Finally, the ε-T-transitive fuzzy relation is utilized to make inferences and cluster objects. Compared to finding the T-transitive closure, it is reasonable to cluster objects using the ε-T-transitive fuzzy relation under the permissible error.

Topological indices are numerical invariants derived from molecular graphs and play an important role in characterizing chemical compounds and predicting their properties. Among the earliest descriptors are the classical Zagreb indices introduced by Gutman and Trinajstić in 1972. A more recent development is the hyper-Zagreb index ($HM$), defined as $HM(G)=\sum_{v_i v_j\in E(G)}(d_i+d_j)^2$, where $d_i$ denotes the degree of vertex $v_i$. In 2023, Hayat et al. posed an open problem concerning bounds on the $HM$ index under fixed vertex-connectivity or edge-connectivity, along with the characterization of the corresponding extremal graphs. In this work, the problem is resolved by determining the extremal graphs that maximize $HM$ index under these constraints. The investigation is further extended to several additional extremal problems, including graphs with a given number of leaves, chromatic number, and independence number. The associated extremal graphs are identified in each case. In addition, the chemical relevance of $HM$ is examined through QSPR studies. Finally, the conclusion is presented.

Point clouds can be regarded as discrete samples of smooth manifolds and are typically analyzed via the eigenfunctions of the Laplace-Beltrami operator. This paper extends manifold spectral analysis to the fractional domain, enabling continuous interpolation between the spatial and spectral domains for point cloud data. First, a point cloud manifold fractional harmonic transform (PMFHT) is proposed, with its fundamental properties rigorously derived, along with the associated convolution, correlation, and sampling theorems. These theoretical results establish a solid foundation for stable fractional-order spectral representation on manifolds. Second, within the PMFHT framework, two representative algorithms are developed. On the one hand, by integrating multi-order PMFHT with chaotic phase modulation, a point cloud encryption scheme is constructed, characterized by a large key space and high sensitivity to key perturbations. On the other hand, an optimal filter is designed in the fractional manifold spectral domain, leading to a maritime target detection method specifically tailored for point cloud data, which effectively suppresses sea clutter while preserving weak target energy under low signal-to-clutter ratio conditions. Finally, experiments on measured data validate the effectiveness of the proposed algorithms.

We introduce the inverse monoid of inner partial automorphisms of a semigroup -- a tool that associates to every semigroup an inverse semigroup. When the semigroup is a group, this inverse semigroup is isomorphic to the group of inner automorphisms with a zero adjoined. We then describe this structure for completely simple semigroups, the full transformation monoid, and the endomorphism monoid of a finite $G$-set when $G$ is a finite abelian group. The paper ends with some open problems.

Revisiting a $50$-year-old estimate of Choi, Erdős and Szemerédi, we show that if $A \subseteq \{1, 2, \ldots, 2n\}$ satisfies $|A| \ge n + 1.2 \cdot 10^8$, then there exist five distinct integers whose pairwise sums are all contained in $A$. In order to guarantee pairwise sums of three or four integers instead, we show that one can replace the constant $1.2 \cdot 10^8$ by $1$ or $3$ respectively, which are both optimal.

In this study, we present a quantum-statistical analysis of H$_2$ and LiH diatomic molecules within the Frost--Musulin potential framework. By combining the analytical bound-state approach to the radial Schrödinger problem with the near-equilibrium Pekeris representation, we obtain a validated rotation-vibration spectrum that reproduces a physically consistent ordering of energy levels. These bound states are subsequently combined with standard translational and rotational ideal gas contributions to construct the total partition function and the corresponding thermodynamic observables of the ground state. The resulting formulation captures the Gibbs free energy deviation function for both molecules with high quantitative accuracy and provides chemically plausible trends for heat capacity and enthalpy increase over a wide temperature range. At the same time, residual errors become increasingly pronounced in derivative-sensitive quantities, particularly at high temperatures; this indicates that the dominant limitations now stem not from the local bound-state spectrum itself, but from the neglect of inelastic rotational, continuity contributions and dynamics close to dissociation. Consequently, the present results define the potential model as a compact and analytically tractable representation of the bound region, recovering a significant portion of the observed thermochemistry whilst also delineating the regime where more comprehensive molecular statistical mechanics is required.

To enable fault tolerance on millions of qubits in real time, scalable decoding is necessary, which motivates this paper. Existing decoding algorithms (decoders), such as clustering, matching, belief propagation (BP), and neural networks, suffer from one or more of inaccuracy, costliness, and incompatibility, upon a broad set of quantum error correction codes, such as surface code, toric code, and bivariate bicycle code. Therefore, there exists a gap between existing decoders and an ideal decoder that is accurate, fast, general, and scalable simultaneously. This paper contributes in three aspects, including decoder, decoder architecture, and decoding simulator. First, we propose Lottery BP, a decoder that introduces randomness during decoding. Lottery BP improves the decoding accuracy over BP by 2~8 orders of magnitude for topological codes. To efficiently decode multi-round measurement errors, we propose syndrome vote as a pre-processing step before Lottery BP, which compresses multiple rounds of syndromes into one. Syndrome voting increases the latency margin of decoding and mitigates the backlog problem. Second, we design a PolyQec architecture that implements Lottery BP as a local decoder and ordered statistics decoding (OSD) as a global decoder, and it is configurable for surface/toric code and X/Z check. Since Lottery BP boosts the local decoding accuracy, PolyQec invokes the costly global OSD decoder less frequently over BP+OSD to enhance the scalability, e.g., 3~5 orders of magnitude less for topological codes. Third, to evaluate decoders fairly, we develop a PyTorch-based decoding simulator, Syndrilla, that modularizes the simulation pipeline and allows to extend new decoders flexibly. We formulate multiple metrics to quantify the performance of decoders and integrate them in Syndrilla. Running on GPUs, Syndrilla is 1~2 orders of magnitude faster than CPUs.

Recent explorations of topological aspects in black hole thermodynamics have achieved unprecedented progress. By utilizing topological numbers, different black hole systems can be categorized into distinct universality classes. This universal classification is particularly evident in thermodynamic limits, offering valuable insights for developing a comprehensive quantum gravity framework. This review highlights the latest advancements in this field. Specifically, we outline fundamental topological frameworks underlying black hole solutions, critical points, Davies points, and the Hawking-Page phase transition. For each scenario, we calculate the associated topological numbers and analyze their physical significance. Furthermore, we explore the practical implications arising from this research.

Privacy-preserving utility mining (PPUM) aims to hide sensitive high-utility patterns while preserving the utility of the sanitized database. In practice, however, many datasets are associated with taxonomic information, which makes the identification and processing of generalized items more challenging. To address this, we investigate the cross-level privacy-preserving utility mining (CLPPUM) problem and propose a method for protecting generalized items. Based on different victim item selection strategies, we develop three CLPPUM algorithms: minimum RGISU first (Min-RF), maximum RGISU first (Max-RF), and best NSC first (Best-NSCF). Furthermore, to enable efficient victim item identification, a novel dictionary structure named GI-dic is designed to accelerate the computation of required utility metrics. Experimental results on multiple datasets demonstrate that the proposed algorithms successfully hide all sensitive cross-level high-utility itemsets without introducing artificial itemsets. The results also show that our method performs well on sparse datasets, and both Min-RF and Best-NSCF consistently outperform Max-RF. Overall, Min-RF achieves the best performance, particularly when the minimum utility threshold is low and the dataset is dense.