Islamic equity screening relies on multiple binary rulebooks that often classify the same firm differently. This paper develops a Continuous Shariah Compliance Index (CSCI) on $[0,1]$ that embeds the published business-activity and financial-ratio thresholds of six leading standards in a single transparent measure. Using CRSP/Compustat U.S. equities from 1999-2024 with lagged accounting inputs and monthly portfolio formation, we document four results. First, existing binary standards map to distinct regions of a common compliance scale, so firms that receive the same pass/fail label can still differ materially in compliance strength. Second, CSCI-threshold portfolios provide a transparent way to vary compliance intensity while retaining economically meaningful diversification, although baseline risk-adjusted performance declines modestly as thresholds tighten. Third, the September 2023 DJIM/S&P methodology change admits firms with materially lower CSCI scores than firms that remained compliant under both the old and new rules. Fourth, in cross-sectional return tests, CSCI is not reliably associated with higher expected returns once standard characteristics are controlled for. The main contribution of CSCI is therefore measurement and portfolio design rather than the discovery of a new priced factor.

We study multipartite entanglement in typical pure states holographically dual to pure BTZ black holes, using multi-entropy and its ``genuine'' version. In the bulk, these quantities are computed by minimal geodesic networks (so-called Steiner trees). We find that at sufficiently high temperature, the genuine tripartite multi-entropy exhibits a volume-law scaling in sharp contrast to vacuum AdS$_3$, where the genuine contribution is universal and size-independent. Moreover, we find another phase: once one subsystem exceeds half of the total system, the leading genuine tripartite entanglement vanishes and reduces to that for global AdS${}_3$. This transition is indeed consistent with recent arguments for distillable EPR pairs in tripartite Haar-random states. Motivated by finite-cutoff holography, we further study the radial cutoff dependence of multi-entropy and show that genuine multi-entropy acquires nontrivial size dependence even for the tripartite case in AdS${}_3$. As a byproduct, we also observe an intriguing ``area-law'' contribution to multi-entropy that is relevant to vacuum AdS${}_3$.

We outline the design and characterization of a laser pulse shaper, which creates an ``ultraslow optical centrifuge'' - a linearly polarized field whose polarization vector rotates with arbitrarily low angular acceleration. By directly recording this rotation in time with nonlinear cross-correlation, we demonstrate the tunability of such centrifuge (both in terms of its initial and its final rotational frequencies) in the range of accelerations which are three orders of magnitude lower than those available with a conventional centrifuge design. We showcase the functionality of the ultraslow centrifuge by spinning CS$_2$ molecules in a molecular jet. Utilizing the extremely low angular acceleration to control molecular rotation inside viscous media is a promising application for this unique optical tool.

The present article has addressed the finite magnetic field extension of the previous work by Cho et al. (Phys. Rev. B 108, 235172, 2023) on microscopic calculation of shear viscosity for electron fluid in graphene system. Our calculation is based on the kinetic theory approach in the relaxation time approximation. In the absence of a magnetic field, transport is governed by a single shear viscosity coefficient, whereas the application of a finite magnetic field induces anisotropy, giving rise to five independent shear viscosity coefficients associated with distinct velocity gradient tensors. These coefficients can be physically categorized into perpendicular, parallel, and Hall components relative to the magnetic field direction. When the scattering time equals the cyclotron time, the perpendicular component is suppressed by 80% and the parallel component by 50% and the Hall effect can reach maximum. Corresponding magnetic field strength for electron fluid in graphene is around 0.01-0.1 Tesla, and the same for non-relativistic electron fluid and ultra-relativistic quark fluid are around 10 Tesla and 10^{14} Tesla respectively. They may be considered as the required magnetic field strength in three different fluid systems to observe noticeable magnetic field response in their shear viscosity coefficients.

The BRST Noether theorem, or ``Noether's 1.5 theorem'', asserts the triviality of the BRST Noether current. We provide two proofs of this theorem that are both valid without restriction on the structure of the gauge theory, extending thereby previous proofs holding in the case of gauge theories for which the solution of the master equation is linear in the antifields. We also relate explicitly the BRST Noether current to the BRST master current appearing in the master equation.

We introduce a quantum algorithm to perform the Laplace transform on quantum computers. Already, the quantum Fourier transform (QFT) is the cornerstone of many quantum algorithms, but the Laplace transform or its discrete version has not seen any efficient implementation on quantum computers due to its dissipative nature and hence non-unitary dynamics. However, a recent work has shown an efficient implementation for certain cases on quantum computers using the Taylor series. Unlike previous work, our work provides a completely different algorithm for doing Laplace Transform using Quantum Eigenvalue Transformation and Lap-LCHS, very efficiently at points which form an arithmetic progression. Our algorithm can implement $N \times N$ discrete Laplace transform in gate complexity that grows as $O((log\,N)^3)$, ignoring the state preparation cost, where $N=2^n$ and $n$ is the number of qubits, which is a superpolynomial speedup in number of gates over the best classical counterpart that has complexity $O(N\cdot log\,N)$ for the same cases. Also, the circuit width grows as $O(log\,N)$. Quantum Laplace Transform (QLT) may enable new Quantum algorithms for cases like solving differential equations in the Laplace domain, developing an inverse Laplace transform algorithm on quantum computers, imaginary time evolution in the resolvent domain for calculating ground state energy, and spectral estimation of non-Hermitian matrices.

The population of Earth-impacting meteoroids and its size-dependent orbital elements are key to understanding the origin of meteorites and informing on planetary defence efforts. Outstanding questions include the role of collisions in depleting meteoroids on highly evolved orbits and the relative importance of delivery resonances. Those depend on size, with current dynamical models considering only asteroids larger than 10m in diameter. Based on 1,202 sporadic meteoroids observed by the Global Fireball Observatory, we created a debiased model of the near-Earth meteoroid population in the 10g - 150kg in size (approximately 1cm - 0.5m) as they dynamically evolved from the main asteroid belt onto Earth-crossing orbits. The observed impact population is best matched with a collisional half-life decreasing from 3Myr for meteoroids of 0.6kg (7cm) or higher, to 1Myr below this size, extending to the model lower bound of 10g. Placing our results in context with near-Earth object models for larger sizes, we find that the inner main belt continues to dominate feeding the small 1m to 10m diameter population primarily via the $ν_6$ secular resonance and the 3:1J mean motion resonance. We also evaluated the potential significance of physical processes other than collisions on Earth-impacting meteoroids, such as low-perihelion disruptions from thermal stresses.

The ultra-relativistic (Carrollian) regime of gravity has recently emerged as a fertile framework for exploring holography, non-Lorentzian symmetries, and geometric limit of General Relativity. In this letter, we establish the presence of a non-vanishing torsion within three-dimensional Carrollian gravity by constructing the Carrollian Mielke-Baekler (C-MB) gravity theory in its Chern-Simons formulation, obtained as the ultra-relativistic limit of the relativistic Mielke-Baekler model. The resulting C-MB theory features non-zero temporal torsion and curvature, together with spatial curvature, providing the most general three-dimensional Carrollian gravity model with these properties. Temporal torsion affects non-affinity of null generators and boundary dynamics. Several known ultra-relativistic gravity theories arise as particular limits of this framework, highlighting its unifying character.

Multipartite entanglement is a crucial resource for quantum technologies; however, its scalable generation in noisy quantum devices remains a significant challenge. Here, we propose a low-depth quantum neural network architecture with linear scaling, employing a novel approach to introducing activation functions for entanglement engineering. As a testbed to demonstrate the clear advantage unlocked by the introduction of nonlinear activations, we run a Monte Carlo sampling over $10^5$ circuit topologies for pure noiseless states. Subsequently, we focus on the noisy scenario; we employ the experimentally accessible Meyer-Wallach global entanglement as a scalable surrogate optimization cost and certify entanglement via bipartite negativity. For 10-qubit mixed states, the optimized circuits generate substantial entanglement across the bipartitions. Lastly, the presence of genuine multipartite entanglement is certified with semi-definite programming. These result establish an experimentally motivated and scalable framework for engineering multipartite entanglement on near-term quantum devices, highlighting the combined role of nonlinearity and circuit topology scaling up to 20 qubits readily.

In 2018, the concept of a fort in graph theory was introduced as a non-empty subset of vertices satisfying the condition that no vertex outside the set has exactly one neighbor in the set. Since then, forts have played a significant role in characterizing zero forcing sets, modeling the zero forcing number as an integer program, and generating lower bounds for the zero forcing number of Cartesian products. Recent research has focused on the number of minimal forts, defined as those for which no proper subset is a fort. Notably, it has been established that the number of minimal forts in any graph is strictly less than Sperner's bound, a famous bound due to Emanuel Sperner (1928) on the size of a collection of subsets where no subset contains another. Moreover, lower bounds on the number of minimal forts for several families of graphs were established, and it was shown that certain families have an exponential number of minimal forts. In this article, we provide a combinatorial-cut characterization of the minimal forts in trees. Using this characterization, we derive an upper bound on the cardinality of minimal forts and a lower bound on the number of minimal forts in trees. We also characterize the trees that attain this lower bound through a four-part equivalence theorem that provides a connection to other graph parameters, such as star centers, the fort number, and the zero forcing number.

We consolidate recent theoretical advances in Detector Error Model (DEM) estimation and formalize several algorithms to learn DEM parameters and structure from syndromes without using a decoder, demonstrating recovery of known DEMs from simulated syndromes with precision limited only by finite-sample effects. We then apply these algorithms to estimate DEMs from Google's 72- and 105-qubit chips. Using a likelihood function that is tractable for small DEMs, we show that DEMs estimated directly from syndromes agree more closely with unseen syndromes than DEMs trained to optimize logical performance, whereas the latter outperform the former as priors for decoders in logical memory experiments. We used a time-series of estimated DEMs to track both global error and specific local errors over the course of a QEC experiment, suggesting applications in online characterization. We employ a sequence of DEM estimation techniques to discover and quantify long-range detector correlations spanning the width of the 105-qubit chip, for which DEM analysis suggests correlated measurement errors rather than high-weight Pauli errors as the most likely explanation. Finally, we present two artifacts in repetition code syndromes that are \emph{not} well-modeled by a DEM: correlated flipping of pairs of adjacent detectors in many consecutive rounds of QEC, and signatures consistent with radiation events occurring more frequently than previously reported.

We consider the ground states of the nonlinear Schr{ö}dinger equation, which stand for radially symmetric and exponentially decaying solutions on the full space. We investigate their behaviors at both endpoint powers of the nonlinearity, up to some rescaling to infer non-trivial limits. One case corresponds to the limit towards a Gaussian function called Gausson, which is the ground state of the stationary logarithmic Schr{ö}dinger equation. The other case, for dimension at least three, corresponds to the limit towards the Aubin-Talenti algebraic soliton. We prove strong convergence with explicit bounds for both cases, and provide detailed asymptotics. These theoretical results are illustrated with numerical approximations.

We investigate matrices satisfying the Kreiss condition $$\|(zI-T)^{-1}\|\le\cfrac{K}{|z|-1}, \hspace{0.7 cm} |z|>1, $$ with $K$ lying arbitrarily close to $1.$ We provide lower bounds for the power growth of such matrices, which complement and refine related estimates due to Nikolski and Spijker-Tracogna-Welfert. We also study operators that satisfy a variant of the above Kreiss condition where $K$ is replaced by $1+ε(|z|)$, where the positive continuous function $ε(|z|)$ tends to $0$ as $|z|\to 1^+.$ We show that, if the spectrum of $T$ touches the unit circle only at a single point and the resolvent of $T$ satisfies a growth restriction along the unit circle, it is possible to choose $ε$ so that this Kreiss-type condition guarantees similarity to a contraction. At the core of our proof lies a positivity argument involving the double-layer potential operator. Counterexamples related to less restrictive choices of $ε$ are also provided.

We improve the theoretical predictions for rare $Z$-boson decays, $Z\to V\ell^+\ell^-$ ($\ell=e$ or $μ$), where $V$ denotes a heavy vector quarkonium including $J/Ψ$, $Ψ(2S)$, and $Υ(nS)$ with $n=1,2,3$. These processes are thought to be dominated by the electromagnetic fragmentation transition, i.e., $Z\to γ^*\ell^+\ell^-$ followed by $γ^*\rightarrow V$. The present study includes all of the relevant tree-level Feynman diagrams, which contribute to these decays in the standard model. Our analysis shows that, for the charmonium final states, the fragmentation transition almost saturates the whole contribution and the other diagrams can be neglected; while for the bottomonium final states, the inclusion of other diagrams can increase their branching fractions by $4\%\sim 9\%$. Further investigation of the differential distributions, especially the angular distributions, indicates that forward-backward asymmetries for final leptons in these processes would be zero in the standard model. Therefore, in future experimental facilities with large number of $Z$-boson events accumulated, studies of these rare $Z$ decays may help both to test the standard model and to probe its interesting extensions.

Materials with a kagome sublattice have been heavily studied recently for their exotic electronic band structure, structural frustration, high-temperature charge order transitions, and unconventional electron-phonon coupling. In LaRu3Si2, it was proposed that electronic flat bands conspire with the characteristic phonon spectrum of the kagome lattice to drive enhanced superconductivity at Tc = 7 K. Here, we report bulk superconductivity in the structural analogue YRu3B2, which hosts a structurally pristine kagome lattice. We observe a superconducting transition at Tc = 0.7 K through magnetization, resistivity, and heat-capacity measurements in this novel kagome metal.

We investigate the fundamental cosmological implications of holographic dark energy using the Granda-Oliveros (GO) infrared cutoff, spatial curvature, and generalized entropies. We demonstrate that the GO cutoff establishes a geometric origin for phantom acceleration, inevitably leading to a big rip singularity without requiring exotic matter. Incorporating spatial curvature reveals that topology acts as a quantitative catalyst; positive curvature accelerates the singularity in closed universes, but cannot alter its fundamental behavior. Furthermore, we show that Kaniadakis generalized entropy modifications are structurally insufficient to prevent this finite-time divergence. To successfully soften the big rip and yield an asymptotic little rip, it is necessary (as first alternative) to integrate irreversible thermodynamical mechanisms, such as non-equilibrium particle creation. These macroscopic processes are sufficient to neutralize the geometric divergence of the GO cutoff, as we discuss in the work.

In this note we will illustrate a method for computing the $π_0$ of the effective log motive of a smooth and proper variety over a perfect field $k$ and show that it is $\mathbf{A}^1$-invariant. We will apply this to compute the first homotopy groups of $\mathbf{P}^1$ to show that the stripping functor from log motivic sheaves to (usual) Nisnevich sheaves with transfers is fully faithful.

Ambient-pressure superconductivity in nickelates has been capped at an onset transition temperature ($T_{c}^{onset}$) of ~50 K, a value that remains lower than the cuprate (~133 K) and iron-based (~55 K) counterparts, despite the promise shown under high pressure. Here, we report ambient-pressure superconductivity onset at ~63 K in epitaxial (La,Pr)3Ni2O7 thin films grown under compressive strain on SrLaAlO4 substrates. This $T_{c}$ leap is enabled by pushing our gigantic-oxidative atomic-layer-by-layer epitaxy (GAE) method into an extreme non-equilibrium growth regime. It simultaneously enhances kinetics via higher temperatures and achieves full oxygenation in situ without post-annealing. Synchrotron X-ray diffraction and scanning transmission electron microscopy confirm that this approach yields films of large-scale crystalline purity, overcoming the inherent metastability of the strained superconducting phase. Transport measurements reveal a zero-resistance temperature ($T_{c}^{zero}$) reaching ~37 K, while mutual inductance measurements demonstrate a robust diamagnetic transition starting at ~23 K. These films exhibit a systematic evolution in their normal-state resistivity-temperature curve: the power-law exponent $α$ evolves from Fermi-liquid-like ($α$ ~2) at lower $T_{c}^{onset}$ to strange-metal-like ($α$ ~1) in higher $T_{c}^{onset}$ samples, directly linking the enhanced superconductivity to non-Fermi liquid behavior. Mapping the vortex melting phase diagram by the mutual inductance technique further reveals 2D melting limit suppressed to near zero, which demonstrates significantly stronger interlayer coupling than that of cuprates. These results identify the nickelates as an ambient-pressure strange-metal high-temperature superconductors with strong interlayer coupling.

Computing haplotypes from sequencing data, i.e. haplotype assembly, is an important component of molecular and population genetics problems, including interpreting the effects of genetic variation on complex traits and reconstructing genealogical relationships. Assembling the haplotypes of polyploid genomes remains a significant challenge due to the exponential search space of haplotype phasings and read assignment ambiguity; the latter challenge is particularly difficult for haplotype assemblers since the information contained within the observed sequence reads is often insufficient for unambiguous haplotype assignment in polyploid genomes. We present pHapCompass, probabilistic haplotype assembly algorithms for diploid and polyploid genomes that explicitly model and propagate read assignment ambiguity to compute a distribution over polyploid haplotype phasings. We develop graph theoretic algorithms to enable statistical inference and uncertainty quantification despite an exponential space of possible phasings. Since prior work evaluates polyploid haplotype assembly on synthetic genomes that do not reflect the realistic genomic complexity of polyploidy organisms, we develop a computational workflow for simulating genomes and DNA-seq for auto- and allopolyploids. Additionally, we generalize the vector error rate and minimum error correction evaluation criteria for partially phased haplotypes. Benchmarking of pHapCompass and several existing polyploid haplotype assemblers shows that pHapCompass yields competitive performance across varying genomic complexities and polyploid structures while retaining an accurate quantification of phase uncertainty. The source code for pHapCompass, simulation scripts, and datasets are freely available at https://github.com/bayesomicslab/pHapCompass.

We denote functions mapping n to the Fourier coefficient of q^n in the expansion of a cusp form as Ramanujan functions. We empirically study the eigenvalues of determinants that represent values of these Ramanujan functions. In some cases, considered as point sets in the complex plane, they appear to oscillate as n increases. We look for regularities in this phenomenon and discuss the possibility of exploiting it to attack Lehmer's question about the existence of zeros of Ramanujan's tau function.

Type I X-ray bursts are thermonuclear flashes on the surface of accreting neutron stars, involving hundreds of nuclei and thousands of reactions with larger uncertainties in reaction rate. To investigate the impact of nuclear reaction rate uncertainties on type I X-ray burst nucleosynthesis, comprehensive Monte Carlo simulations were performed with temperature-independent and -dependent variations in reaction rates using the REACLIB and STARLIB libraries, respectively. A total of 1,711 $(p, γ)$, $(p, α)$, $(α, p)$, and $(α, γ)$ reaction rates are varied simultaneously along with their inverse reactions via detailed balance. For the first time, it has been found that Monte Carlo sampling with larger perturbations to these reaction rates may lead to multi-peak abundance distributions for certain isotopes, such as $^{64}$Zn and $^{55}$Co. These multi-peak structures arise not only from coupled reactions but also from single reactions in some cases. Our studies also confirm previously identified key reactions and provide more robust lists that deserve priority consideration in future studies.

An empirical baryon density profile can be included in the Schrödinger-Poisson (SP) equations to influence the fuzzy dark matter (FDM) soliton formation. However, to probe the effects of baryon on the other dynamical evolutions of the FDM soliton, its equation of motion (EoM) inside the corresponding FDM soliton is needed. In this paper, given an empirical baryon density profile, we first provide the cylindrical symmetric FDM soliton solution about the FDM density and the total potential of FDM and baryon. Then, instead of EoM, we build an analytical emulator (AE) for the baryon density distribution from the obtained FDM density and total potential by machine learning. Finally, we check that this AE works as well as an empirical baryon density profile for the FDM soliton formation, with the fractional errors $\lesssim0.04$. It should also work as a kind of baryon EoM for some other simple FDM soliton evolutions.

We prove best approximation error estimates for discontinuous piecewise polynomial approximation in fractional Sobolev spaces on non-Lipschitz meshes of non-Lipschitz domains. In particular, the boundary of the domain, and the boundaries of the mesh elements, can be fractal.

We aim to analyze the consistency of the deformation of the Heisenberg algebra in the setting of constrained Hamiltonian systems, providing a procedure to induce the deformation on the Poisson algebra after symplectic reduction. We investigate this in the context of the classical interpretation of Generalized Uncertainty Principle theories, treating two cases separately. For the first case, we consider a group action on the phase space together with a set of first-class constraints that can be interpreted as a momentum map. We furnish an explicit example in the case of rotational invariant deformed algebras. In the second case, we consider a single constraint provided by the Hamiltonian, which is a common instance in General Relativity, with straightforward application in cosmology.

Massive bosonic fields can trigger superradiant instabilities in rotating astrophysical black holes leading to gaps in their mass-spin distribution. For spin-2 fields, the instability timescale is orders of magnitude shorter than for any other superradiant mode, thereby yielding much stronger constraints. We consider a tower of ultra-light spin-2 fields arising from a warped compactification of a single extra dimension and translate superradiant constraints on their masses into constraints on the warping. As a concrete scenario we consider the 2-brane Randall-Sundrum model and find constraints on the size of the extra dimension and the curvature of $AdS_5$. We discuss the implications of these bounds for strongly warped throats and D-brane uplifts commonly used in attempts to realise metastable de Sitter vacua in string theory.

In this paper, we study the multigraded Betti numbers of Veronese embeddings of projective spaces. Due to Hochster's formula, we interpret these multigraded Betti numbers in terms of the homology of certain simplicial complexes. By analyzing these simplicial complexes and applying Forman's discrete Morse theory, we derive vanishing and non-vanishing results for these multigraded Betti numbers.

We present an analysis of the covariance spectrum of the black hole X-ray binary MAXI J1820+070 during its hard state. For the first time, we extend coherence and covariance studies into the hard X-ray band up to 150 keV. We detect a clear drop in coherence above 30 keV on both short- and long-timescales relative to the 2-10 keV reference band. To investigate the origin of the coherent variability, we simultaneously fit the short- and long-timescale covariances and the time-averaged spectra with a Comptonization model. Surprisingly, the electron temperature associated with long-timescale variability is significantly higher than that on short timescales. Moreover, the temperature on long timescales remains relatively constant throughout the hard state, whereas the short-timescale temperature evolves with X-ray luminosity. We attribute the drop in coherence to multiple sources of seed photons, i.e., the blackbody and synchrotron photons. The independence between these two photon fields leads to the drop in coherence. To explain the lower electron temperature on short timescales, we propose a two-Comptonization framework in which short-timescale variability arises from a vertically extended central region, while long-timescale variability originates at larger radii. The elevated geometry of the inner region leads to illumination primarily by cooler outer-disk photons, yielding a lower electron temperature. In this case, the evolution of the height of the elevated region could explain the evolution of the electron temperature associated with the coherent variability throughout the hard state.

We explicitly compute the spectral metric, torsion and Einstein tensors for a nontrivial spectral triple on a noncommutative torus, with the Dirac operator related to the fully equivariant Dirac by a partial conformal rescaling (as introduced in [1]). The results show that the spectral triple has vanishing torsion and the Einstein tensor also identically vanishes.

A hybrid approach combining the Tabular Prior-data Fitted Network (TabPFN) with the Coulomb and Proximity Potential Model (CPPM) is developed to investigate $α$-particle preformation factors $P_α$ and their impact on $α$-decay half-lives. The TabPFN model, trained on 498 nuclei, accurately learns the relationship between nuclear structure properties and $P_α$, achieving a root mean square deviation of $σ_{\mathrm{rms}} = 0.211$. The predicted factors reveal clear odd-even staggering and shell closure effects, and exhibit linear correlations with both $Q_α^{-1/2}$ and the fragmentation potential $V_{\mathrm{frag}}$. When incorporated into CPPM calculations, the machine-learning-based $P_α$ values significantly improve half-life predictions. Similar improvements are also obtained when deformation effects are included in the potential barrier description. The capability of the model is further demonstrated through predictions for superheavy nuclei ($Z = 117$--120), suggesting $N = 184$ as a potential neutron magic number.

Nanostructured materials, critical for thermal management in semiconductor devices, exhibit a strong size dependence in thermal transport. Studying thermal resistance variation across grain boundaries is critical for designing effective thermal interface materials. Frequency-domain Thermoreflectance (FDTR)-based techniques can provide thermal resistance mapping at the micrometer (μm) scale. Scanning Thermal Microscopy (SThM) enables quantification of local thermal transport with significantly higher spatial resolution (<100 nm). However, challenges in quantifying the raw signal to thermal conductivity and surface sensitivity limit its widespread adoption for understanding nanoscale heat transport and defect-mediated thermal properties in nanostructured films. Here, we introduce a circuit-based probe thermal resistance (R_p) calibration technique independent of parasitic heat pathways, enabling accurate determination of probe heat dissipation and tip temperature rise, thereby allowing extraction of local thermal resistance. SThM achieved sub-100 nm spatial resolution in mapping thermal resistance across a 15 nm-thick Al film deposited via e-beam evaporation on SiO_2 substrate. The thermal resistance maps are converted to thermal conductivity using robust analytical and finite element models that account for tip-sample geometry, lateral heat spreading, and buried interface effects. Gaussian distribution fitting of pixel-level thermal resistance values yields k_Al = 45.1(+4.7/-3.6) W/(m.K) for the ultra-thin Al film (13-15 nm), representing a 5.3-fold reduction from bulk aluminum (237 W/(m.K)). These results agree with published experimental data and theoretical frameworks explaining thickness-dependent heat transport in ultra-thin metallic films.