We revisit the golden sieve, a self-referential deletion process on increasing sequences of positive integers introduced by the author in 2002. Applied to the natural numbers, the sieve produces the Wythoff pair as a Beatty partition. For arithmetic progressions $a\mathbb{N}+b$, we establish a connection with the $(j,x,y,z)$-hiccup sequences recently studied by Fokkink and Joshi and with Fraenkel's complementary partitions. We further introduce an extraction sieve that also produces hiccup sequences, and whose action on arithmetic progressions is governed by an explicit affine transformation of hiccup parameters.

The Newns-Anderson Hamiltonian is widely used to describe adsorption at gas-solid interfaces, yet its construction typically relies on simplifying assumptions such as constant coupling and the wideband limit approximation. Here, we present a first-principles approach to construct Newns-Anderson Hamiltonians by applying projection operator diabatisation to Hamiltonian matrices obtained from Kohn-Sham density functional theory calculations. We demonstrate this method for chemisorbed hydrogen on three fcc metal(111) surfaces: Al, Cu, and Pt. To validate the electronic coupling between adsorbed hydrogen and the metal surface, we compute the projected density of states, electronic tunnelling lifetimes, and vibrational lifetimes from the constructed Newns-Anderson Hamiltonians and find good agreement with reference calculations. Analysis of the chemisorption function reveals that the wideband limit approximation is valid for H/Al(111) but has limited applicability for H/Cu(111) and H/Pt(111).

We investigate the nature of quantum phase transitions in a (1+1)-dimensional field theory composed of $N$ copies of the Ising conformal field theory interacting via competing relevant perturbations. The field theory governs the competition between a mass term and an interaction involving the product of $N$ order-parameter fields, which is realized, e.g. in coupled Ising chains, two-leg spin ladders, and SO($N$)-symmetric spin chains. By combining a perturbative renormalization group analysis and large-scale matrix-product state simulations, we systematically determine the nature of the phase transition as a function of $N$. For $N=2$ and $N=3$, we confirm that the transition is continuous, belonging to the Ising and four-state Potts universality classes, respectively. In contrast, for $N \ge 4$, our results provide compelling evidence that the transition becomes first order. We further apply these findings to specific lattice models with SO($N$) symmetry, including spin-$1/2$ and spin-$1$ two-leg ladders, that realize a direct transition between an SO($N$) symmetry-protected topological phase and a trivial phase. Our results refine a recent conjecture regarding the criticality of transitions between SPT phases.

Task-based models of AI and labor hold organizational structure fixed. We introduce agent capital: AI that reduces coordination costs, expanding spans of control and enabling endogenous task creation. Five propositions characterize how coordination compression affects output, hierarchy, manager demand, wage dispersion, and the task frontier. The model generates a regime fork: the same technology produces broad-based gains or superstar concentration depending on who benefits from coordination compression. Simulations with heterogeneous workers confirm sharp regime divergence. Economy-wide inequality falls in all regimes through employment expansion, but the manager-worker wage gap widens universally. The distributional impact hinges on who controls organizational elasticity.

We investigate the electronic structure and Fermi surface of Co$_{1/3}$TaS$_2$ using angle-resolved photoemission spectroscopy (ARPES) combined with theoretical modeling beyond standard density functional theory (DFT+U). A shallow electron pocket, the so-called $β$ feature, is observed at the Fermi level near the corner of the superlattice Brillouin zone, representing the first experimental observation of this feature in an intercalated TaS$_2$ compound. Similar pockets have been reported in $X_{1/3}$NbS$_2$ ($X$ = Co, Cr, Ni), where their surface versus bulk origin remains actively debated. Because conventional DFT+U does not capture this feature, we employ cluster perturbation theory (CPT) to incorporate an explicit treatment of strong electron correlations ($U$) on the Co sites. CPT successfully reproduces the $β$ feature, demonstrating its origin from correlation-driven bulk states rather than surface effects. To further substantiate this conclusion, we studied a reduced Co-content sample, Co$_{0.22}$TaS$_2$, where the reduced charge transfer modifies the Co-derived states near the Fermi level. Its electronic structure remains largely similar to that of pristine 2H-TaS$_2$, showing only a minor overall energy shift and lacking the $β$ feature, consistent with disrupted long-range Co ordering and modified orbital character near the Fermi level. We demonstrate that the $β$ feature arises from strong local correlations on the Co sites and requires long-range crystallographic order among intercalated Co atoms to maintain coherence. These results highlight the importance of strong electronic correlations in magnetically intercalated transition-metal dichalcogenides and provide a microscopic understanding of features not captured by conventional DFT+U.

A two-dimensional crystal melts via the proliferation and unbinding of topological defects, yet quantitatively predicting the melting temperature $T_m$ in real systems is challenging. Here we resolve this discrepancy in quantum Hall electron bubble phases by combining Corbino-geometry transport experiment in an ultraclean GaAs/AlGaAs quantum well for Landau levels 2 to 5 with Hartree--Fock elasticity and the full Kosterlitz--Thouless--Halperin--Nelson--Young melting criterion including the finite-temperature renormalization-group calculation. The theoretically obtained $T_m$ quantitatively captures the measured solid-liquid phase transition boundaries across all probed ranges, validating the bubble-crystal interpretation and establishing defect--mediated melting as a predictive framework for strongly interacting electronic solids. This agreement further supports using bulk transport to probe the energetics of topological defects and screening in quantum Hall physics, and the approach is readily extendable to other electronic crystals, including the generalized Wigner crystal in moiré Chern bands.

We give an alternative proof of a result of Miyazaki and Yasuhara that there exists links that are not smoothly slice in $S^2 \times S^2$. We discuss potential applications to the detection of exotic $S^2 \times S^2$. This is a follow-up note to a similar paper for the $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$ case.

In this paper, we consider a once-reinforced random walk on the half-line, and give the limiting behaviors of all the moments of its range.

We establish two new inequalities, the weighted strong monogamy (WSM) and the maximum residual strong monogamy (MRSM), which sharpen the generalized Coffman-Kundu-Wootters inequity for multiqubit states. The WSM inequality distinguishes itself from the strong monogamy (SM) conjecture [Phys. Rev. Lett. 113, 110501 (2014)] by using coefficients rather than exponents to modulate the weight allocated to various m-partite contributions. In contrast, the MRSM inequality is formulated using only the maximum m-partite entanglement. We find that the residual entanglement of the MRSM inequality can effectively distinguish the separable states. We also compare the tightness of various SM inequalities and provide examples using a four-qubit mixed state and a five-qubit pure state to illustrate the MRSM inequality. These examples characterize the trade-off relations among entanglement components involving varying numbers of qubits. Our results provide a rigorous framework to characterize and quantify the monogamy of multipartite entanglement.

We construct the electron star solution in asymptotically global AdS spacetime, and investigate its stability properties, both locally under perturbations and globally with respect to the Reissner-Nordström black hole and thermal AdS metrics. We interpret the resulting phase diagram as that of a holographic metal confined to a finite volume. We identify a quantum critical point at finite chemical potential, around which the different phases are organized.

Multivariate shortfall risk measures provide a principled framework for quantifying systemic risk and determining capital allocations prior to aggregation in interconnected financial systems. Despite their well established theoretical properties, the numerical estimation of multivariate shortfall risk and the corresponding optimal allocations remains computationally challenging, as existing Monte Carlo based approaches can be numerically expensive due to slow convergence. In this work, we develop a new class of single and multilevel numerical algorithms for estimating multivariate shortfall risk and the associated optimal allocations, based on a combination of Fourier inversion techniques and randomized quasi Monte Carlo (RQMC) sampling. Rather than operating in physical space, our approach evaluates the relevant expectations appearing in the risk constraint and its optimization in the frequency domain, where the integrands exhibit enhanced smoothness properties that are well suited for RQMC integration. We establish a rigorous mathematical framework for the resulting Fourier RQMC estimators, including convergence analysis and computational complexity bounds. Beyond the single level method, we introduce a multilevel RQMC scheme that exploits the geometric convergence of the underlying deterministic optimization algorithm to reduce computational cost while preserving accuracy. Numerical experiments demonstrate that the proposed Fourier RQMC methods outperform sample average approximation and stochastic optimization benchmarks in terms of accuracy and computational cost across a range of models for the risk factors and loss structures. Consistent with the theoretical analysis, these results demonstrate improved asymptotic convergence and complexity rates relative to the benchmark methods, with additional savings achieved through the proposed multilevel RQMC construction.

The material in planet-forming disks determines the composition of planets; hence, it is crucial to understand the physical and chemical processes that set the abundance and distribution of key volatiles. James Webb Space Telescope observations of disks around very low mass ($\sim0.1~\rm{M}_\odot$) stars (VLMSs) have revealed their hydrocarbon-rich inner regions (e.g., \ce{C2H2}), with column densities significantly higher than predicted. We employ chemical kinetics models using the physical structure of the inner disk around an M~dwarf star with an X-ray luminosity of $L_\mathrm{X}\sim10^{29}~\mathrm{erg~s^{-1}}$. We adopt initial abundances that mimic the effects of carbon enhancement and oxygen depletion (C/O from 0.44 to 87.47) and quantify how the abundances and distributions of key volatiles respond. The column density and number of molecules ($\mathcal{N}$) of hydrocarbons and oxygen-bearing species are highly sensitive to the C/O ratio, with the largest increases in hydrocarbons occurring when carbon increases by a factor of 2, and/or oxygen decreases by a factor of 10, relative to solar. In the IR-emitting region ($T_\mathrm{gas}>200~\mathrm{K}$), a range of C/O ratios can reproduce the observed $\mathcal{N}$ and ratios relative to \ce{CO2}. The disk-integrated molecular ratio with respect to \ce{CO2} is highly sensitive to the underlying C/O ratio. However, our results apply only to a source with a single X-ray luminosity value at the middle of that observed for VLMSs; hence, a degeneracy between the stellar $L_\mathrm{X}$ and the C/O ratio cannot be discarded. Nonetheless, our findings support that an enhanced C/O is required to drive the hydrocarbon-rich chemistry observed in the inner disks around VLMSs.

Safety cases, structured arguments that a system is acceptably safe, are becoming central to the governance of AI systems. Yet, traditional safety-case practices from aviation or nuclear engineering rely on well-specified system boundaries, stable architectures, and known failure modes. Modern AI systems, such as generative and agentic AI, are the opposite. Their capabilities emerge unpredictably from low-level training objectives, their behaviour varies with prompts, and their risk profiles shift through fine-tuning, scaffolding, or deployment context. This study examines how safety cases are currently constructed for AI systems and why classical approaches fail to capture these dynamics. This study introduces comprehensive taxonomies for AI-specific claim types (assertion-based, constraint-based, capability-based), argument types (demonstrative, comparative, causal/explanatory, risk-based, and normative), and evidence families (empirical, mechanistic, comparative, expert-driven, formal methods, operational/field data, and model-based). It then proposes a reusable safety-case template, each of which follows a predefined structure of claims, arguments, and evidence tailored for AI systems. Each template is illustrated by end-to-end patterns that address distinctive challenges, such as evaluation without ground truth, dynamic model updates, and threshold-based risk decisions. The result is a systematic, composable, and reusable approach to constructing and maintaining safety cases that are credible, auditable, and adaptive to the evolving behaviour of generative and frontier AI systems.

The interplay between band topology and light in condensed materials could unlock intriguing nonlinear optical phenomena, enabling modern photonic technologies such as quantum light sources and sub-wavelength topological lasers. Here, we unveil that a buckling-tuned topological transition in ferroelectric bismuth monolayer unleashes a giant second-harmonic generation. Using first-principles calculations, we surprisingly find that ferroelectric bismuth monolayer with a buckling parameter, $Δh$, has a large susceptibility $χ^{(2)}$ on the order of $10^{7}$ $\mathrm{pm}^2/\mathrm{V}$, exceeding monolayer MoS$_2$ by about two orders of magnitude. When $Δh$ is engineered to the critical window where Dirac electrons emerge, a low-frequency resonance appears, boosting $χ^{(2)}$ by an additional order of magnitude. We show that this enhancement is localized on the Dirac cones and dominated by intraband modification contributions. Based on an extended Dirac model, we establish that this enhancement physically originates from the ultralight effective masses $m^{*}$ of Dirac electrons through scaling with the Fermi velocity $v_F$ and band gap $E_g$. Our findings provide a general paradigm for achieving exceptional second-harmonic generation via engineering topological criticality, and could serve as an experimental signature of Dirac electrons in topological materials.

In this paper, we explore the high-frequency properties of eigenfunctions of point perturbations of the Laplacian on a compact Riemannian manifold. These systems cannot be obtained as the quantization of a classical Hamiltonian, as the effect of the perturbation amounts to prescribing certain boundary conditions on a discrete set of points. We are interested in understanding to what extent the high-frequency behavior of eigenfunctions is governed by the global dynamics of the geodesic flow in the manifold (the classical flow corresponding to the unperturbed Laplacian). We prove that as soon as the Laplacian is perturbed by a finite set of point scatterers satisfying a \emph{non-focality} condition, namely, that the family of geodesics starting from this set and coming back to it has zero measure, semiclassical measures corresponding to high-frequency sequences of eigenfunctions are invariant under the geodesic flow. Invariance may fail when the non-focality condition does not hold, as is shown in a companion article [arXiv:2601.19701]. Our results are based on a quasimode construction that requires improved estimates on the spectral function of the Laplacian on the set of scatterers.

From any graph $G$ arises a flat space wavefunction, obtained by integrating a product of propagators associated to the vertices and edges of $G$. This function is a key ingredient in the computation of cosmological correlators, and several representations for it have been proposed. We formulate three such representations and prove their correctness. In particular, we show that the flat space wavefunction can be read off from the canonical form of the cosmological polytope, and we settle a conjecture of Fevola, Pimentel, Sattelberger, and Westerdijk regarding a partial fraction decomposition for the flat space wavefunction. The terms of the decomposition correspond to certain collections of connected subgraphs associated to $G$ and its spanning subgraphs, reflecting the fact that the flat space wavefunction contains information about how $G$ is connected.

Galaxies undergo perturbations, either gravitational or hydrodynamic in origin, which can generate extragalactic structures such as rings and tails, where in situ star formation may take place. We selected a sample consisting of JO201 and JW100, undergoing ram-pressure stripping, and NGC 5291 and NGC 7252, formed through gravitational interactions, to investigate how different perturbation mechanisms influence dust content and star formation in extragalactic features. In both cases, star formation can be observed outside the main disks of the galaxies. We present new results of dust attenuation for JO201 and JW100, while for NGC 5291 and NGC 7252 we use results from our previous study, based on high-resolution observations obtained with the Ultraviolet Imaging Telescope onboard AstroSat. Dust attenuation is determined from the ultraviolet continuum slope ($β$) calculated using the FUV-NUV colour, and the star formation rates of the star-forming knots are corrected accordingly. It is seen that dust attenuation and dust-corrected SFR densities of the knots in the ram-pressure stripped tails of JO201 and JW100 are comparable to those in the collisional ring of the NGC 5291 system and the tidal tails of the NGC 7252 system. We conclude that, though the formation scenarios of the tails of JO201 and JW100, the NGC 5291 ring, and the NGC 7252 tails are different, their dust content and star formation activity are notably similar.

We construct bisymmetric, strictly increasing binary operations on real intervals which are not continuous. This answers a natural question in the theory of bisymmetric and mean-type operations by showing that continuity may fail for non-reflexive operations of the form \[ F(x,y)=f^{-1}(αf(x)+βf(y)), \] where $α,β>0$ with $α+β\neq1$. Our construction is based on a Cantor-type perfect set whose elements are linearly independent over a countable subfield of $\R$, which allows the generating function $f$ to map an interval bijectively onto a nowhere dense fractal-type set. As a consequence we obtain a noncontinuous associative and strictly increasing operation on an interval. We also extend the construction to the multivariate case. In the opposite direction we prove that if a symmetric bisymmetric strictly increasing operation is reflexive at two points of an interval, then it is automatically continuous on the segment between them and coincides there with a quasi-arithmetic mean.

For quantum spin models defined on a two-dimensional lattice, we look for the best numbering of the lattice sites (a layout) that, at fixed bond dimension and other parameters of the density matrix renormalization group (DMRG) algorithm, gives the lowest value of the variational energy, maximum entropy and truncation error. We consider the conjecture that the optimal layout is a Hamiltonian path, and that it optimizes a simply computable geometric cost function. Finding the minimum of such a function, which is a variant of the minimum linear arrangement problem, provides the DMRG with an efficient layout of the lattice and improves both accuracy and convergence time. We present applications to the antiferromagnetic and spin glass spin-1/2 models on the square and triangular lattices.

Within the standard framework in which electroweak sphaleron processes relate lepton and baryon number, we derive an upper limit on the amplitude of a chiral gravitational wave background produced prior to the electroweak epoch. This bound is independent of the production time of chiral GWs for superhorizon modes, while it becomes sensitive to the production time for subhorizon modes. For sufficiently high reheating temperatures, the bound becomes significantly more stringent than the conventional big bang nucleosynthesis constraints at frequencies above the MHz scale, thereby providing a powerful and \emph{model-independent} probe of parity-violating physics in the early Universe.

We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the Kronecker double cover of $G$. As an application we calculate the homotopy type for paths, cycles and the categorical product of two complete graphs.

We investigate low-energy $Nϕ$ scattering driven by a pole-enhanced triangle-like diagram, in which the two-Kaon-exchange contribution is promoted by the near-threshold $Λ(1405)$ pole in the $N\bar K$ subsystem. Using an unphysical Kaon mass motivated by lattice simulations, we evaluate the $Nϕ$ scattering length and find that this mechanism generates an attractive interaction with a magnitude of $-1.1$ to $-0.5\, \rm{fm}$. Spin-dependent effects are not treated explicitly and are expected to provide subleading corrections in the near-threshold region. We further analyze the low-energy behavior of the triangle-like diagram amplitude and show that the scattering length depends on the parameter $δ$, defined as the mass difference between the $K\bar K$ threshold and the $ϕ$ meson, and the pole position of $Λ(1405)$, where the $Λ(1405)$ plays a crucial role to understand $Nϕ$ interaction. Furthermore, by employing physical hadron masses, our calculated scattering length is found to be consistent with current experimental data, providing a unified description across both unphysical and physical mass regimes. This type of interaction differs from that associated with van der Waals-type forces or the long-range tail of two-pion exchange, highlighting the role of three-body dynamics encoded in the pole-enhanced triangle-like diagram in shaping the near-threshold $Nϕ$ interaction.

We investigate the feasibility of extracting infinite volume scattering phase shift on quantum computers in a simple one-dimensional quantum mechanical model, using the formalism established in Ref.~\cite{Guo:2023ecc} that relates the integrated correlation functions (ICF) for a trapped system to the infinite volume scattering phase shifts through a weighted integral. The system is first discretized in a finite box with periodic boundary conditions, and the formalism in real time is verified by employing a contact interaction potential with exact solutions. Quantum circuits are then designed and constructed to implement the formalism on current quantum computing architectures. To overcome the fast oscillatory behavior of the integrated correlation functions in real-time simulation, different methods of post-data analysis are proposed and discussed. Test results on IBM hardware show that good agreement can be achieved with two qubits, but complete failure ensues with three qubits due to two-qubit gate operation errors and thermal relaxation errors.

The temperature-driven shape dynamics of isotopic chains of Mo and Ru elements and their impact on decay modes have been investigated in a statistical theoretical framework with macroscopic-microscopic apporach. These isotopes located at the key points in r-process path are known for the rapid structural changes, shape instabilities and shape coexistence that impact the nuclear processes, decay modes and lifetimes. At high temperatures that may exist in stars or in various nuclear reaction processes, these nuclei undergo a variety of shape and deformation changes due to thermal shell quenching effects influencing the decay energies (Q value), and eventually life-time have been studied in detail. Our findings provide insight into the observed shift in the deformation, shapes and coexisting states due to the diminishing nuclear shell effects in hot nuclei, revealing that the structural changes influence the decay processes and significantly in the astrophysically relevant Mo-Ru region especially around A = 100.

It is known that there exist theoretical limits on the mass of compact objects in general relativity. One is the Buchdahl limit for an object with an arbitrary equation-of-state that turns out to be the limit for an object with uniform density. Another one is the causal limit that is stronger than the Buchdahl limit and is related to the speed of sound inside an object. Similar theoretical limits on the mass of compact objects in deformed Hořava-Lifshitz (HL) gravity are found in this \paper. Interestingly, the both curves of the uniform density limit and the sound speed limit meet the horizon curve at the minimum of the horizon, where a black hole becomes extremal, i.e., $M=q$, considering the Kehagias-Sfetsos vacuum that is an asymptotic flat solution in the HL gravity.

In this work, we analyze the fundamental question of what is the ensemble ground state of a general, finite, many-electron system at zero temperature, with a given, possibly fractional, electron number $N_{tot}$ and a given $z$-projection of the spin, $M_{tot}$, distinguishing between low- and high-spin cases. For the low-spin case, the general form of the ensemble ground state has been rigorously derived in J. Phys. Chem. Lett. 15, 2337 (2024), finding the presence of an ambiguity in the ground state. Here we further discuss this ambiguity, and show that it can be removed via maximization of the entropy. For the high-spin case, we find that the form of the ensemble ground state strongly depends on the system in question. Furthermore, we prove three general properties which characterize the ensemble, and narrow the list of pure states it may consist of. We relate the frontier Kohn-Sham orbital energies to total energy differences, providing a generalization of the ionization potential theorem to systems with arbitrary fractional $M_{tot}$. Furthermore, we derive expressions for new derivative discontinuities, which appear as jumps in the KS potentials when crossing a boundary in the $N_{\uparrow}$-$N_{\downarrow}$ plane. Our analytical results are supported by an extensive numerical analysis of the Atomic Spectra Database of the National Institute of Standards. The new exact conditions for many-electron systems derived in this work are instrumental for development of advanced approximations in density functional theory and other many-electron methods.

The classical quasi-shuffle algebra for multiple zeta values have a well-known Hopf algebra structure. Recently, the shuffle algebra for multiple zeta values are also equipped with a Hopf algebra structure. This paper shows that these two Hopf algebras are isomorphic utilizing quasi-symmetric functions. This Hopf algebra isomorphism is compared with with the well-known isomorphism between the shuffle Hopf algebra and quasi-shuffle Hopf algebra of Hoffman, Newman and Radford.

We investigate the observational tests of generalized mass-to-horizon entropic cosmology by incorporating large-scale structure growth data in addition to purely geometric probes. The theoretical framework is constructed from a generalized mass-to-horizon scaling relation, $M \propto L^n$, which implies a corresponding generalized entropic functional $S_n \propto L^{n+1}$. Within this setting, cosmic acceleration arises as an emergent phenomenon driven by an entropic force acting on the cosmological horizon. While earlier studies demonstrated that these entropic cosmologies can reproduce the background expansion history of the standard $Λ$CDM model, here we present a comprehensive observational analysis that jointly employs Pantheon+ Type Ia supernova data with SH0ES calibration, DESI DR2 baryon acoustic oscillation measurements, cosmic microwave background (CMB) distance priors, and a suite of cosmological structure growth observations. A Bayesian model comparison indicates that the entropic models are statistically preferred over the conventional $Λ$CDM scenario, thereby providing strong support for an entropic origin of the observed late-time cosmic acceleration in place of a fundamental cosmological constant.

We present a detailed evaluation of constrained minisuperspace path integrals in Jackiw-Teitelboim (JT) gravity and in biaxial Bianchi IX quantum cosmology, employing the Gelfand-Yaglom theorem to compute the relevant functional determinants. In both settings, integrating out the dilaton or a minisuperspace variable produces a functional delta that enforces the classical constraint equation, thereby localizing the remaining path integral onto classical configurations. The associated Jacobian, equivalently, the functional determinant of the operator obtained by linearizing the constraint about the classical solution, fixes the semiclassical prefactor and the correct measure. We evaluate this determinant exactly via the Gelfand-Yaglom method and obtain the fully normalized fixed-lapse propagators. We further extend the JT analysis to a quadratic dilaton potential $U(ϕ)=m^{2}ϕ^{2}$ and comment on the corresponding saddle-point structure of the lapse integral. Finally, we apply the same approach to Bianchi IX quantum cosmology and derive the fixed-lapse propagator, including its prefactor. Our results provide a systematic and broadly applicable prescription for treating constraint structures in gravitational path integrals using functional determinant techniques, with potential applications to a wider class of minisuperspace quantum cosmology and quantum gravity.

Understanding how much each variable contributes to an outcome is a central question across disciplines. A causal view of explainability is favorable for its ability in uncovering underlying mechanisms and generalizing to new contexts. Based on a family of causal explainability quantities, we develop methods for their estimation and inference. In particular, we construct a one-step correction estimator using semi-parametric efficiency theory, which explicitly leverages the independence structure of variables to reduce the asymptotic variance. For a null hypothesis on the boundary, i.e., zero explainability, we show its equivalence to Fisher's sharp null, which motivates a randomization-based inference procedure. Finally, we illustrate the empirical efficacy of our approach through simulations as well as an immigration experiment dataset, where we investigate how features and their interactions shape public opinion toward admitting immigrants.