For high-throughput applications such as ultra-high-definition video streaming and immersive extended-reality, perceptual quality rather than bit-level accuracy defines the primary performance criterion and provides a more informative and spectrally efficient objective than strict bitwise reconstruction. This is particularly relevant in millimeter-wave (mmWave) and sub-Terahertz (sub-THz) systems, where path loss, short channel coherence times and phase noise introduce severe fluctuations that degrade link spectral efficiency. We propose an extension to conventional Adaptive Modulation and Coding (AMC) framework that incorporates perceptual quality awareness into link adaptation. In this framework, the decision metric is a Perceptual Quality Indicator (PQI) derived from the Structural Similarity Index Measure (SSIM). The receiver employs a Denoising Convolutional Neural Network (DnCNN) denoiser to enhance post-decoding image quality before feedback estimation. The resulting perceptual metric replaces the standard Channel Quality Indicator (CQI) in the AMC loop, enabling adaptation to maximize spectral efficiency while satisfying a perceptual-fidelity constraint. Experiments on a 5G-compliant mmWave testbed demonstrate up to a twofold gain in spectral efficiency while maintaining perceptual fidelity, underscoring the potential of perception-optimized link adaptation.

We propose a new family of polynomial approximation kernels for approximating nonnegative polynomials on the hypercube $[-1,1]^n$. Our Kernels produce polynomial sums-of-squares of degree $r$, achieving an $O(\log^3 r/r^2)$ error in the $\ell_1$-norm of the coefficients. This improves on the known error bound $O(1/r)$ from the literature. As a corollary, we obtain an improved convergence rate for the Lasserre hierarchy for polynomial optimization on the hypercube, again improving a known rate by Baldi and Slot from $O(1/r)$ to $O(\log^3 r/r^2)$.

The targeted discovery of inorganic materials remains challenging due to the vastness of compositional design spaces and the high cost of exhaustive screening. Task-specific generative artificial intelligence represents a particularly efficient alternative to screening, yet demands tedious collection of training data before providing real benefit. General-purpose large language models (LLMs) have recently shown tremendous potential for the targeted generation of single, optimal materials compositions without the need for task-specific fine-tuning. However, it is unclear whether LLMs generally pose an advantage compared to specialized generative models, in particular in large design spaces. Here, we demonstrate that such models are capable of covering entire regions of the targeted property space effectively and systematically. Using Elpasolite materials as an established benchmark for generative tasks in large chemical spaces, we find that an iterative prompt-and-response framework is able to recover on average 96% of all low-energy Elpasolites in the target region. This performance, driven mainly by iterative in-context learning, surpasses the generative abilities of previous, task-specific models. Our results establish general-purpose LLMs as flexible and accessible components for inverse materials design workflows.

The distribution of entangled pairs of photons on the links composing a quantum network, combined with Bell state measurements and teleportation, is the basic apparatus to transfer quantum bits (qubits) over long distances. Entanglement distribution establishes an end-to-end entangled pair while consuming intermediate pairs on links and holding them for a certain time period. The technical literature identifies two main kinds of protocols, parallel and sequential ones, the latter having an advantage in resource consumption over the former. In this paper, we introduce an efficient swapping protocol called Progressive Swapping to the Middle (PSM) as it combines the existing Progressive Swapping (PS) protocol from both extremities of a path that meet in the middle where the received pairs are swapped. We compare PSM with two parallel protocols and PS; in our evaluation, we take into account imperfect memories and fidelity degradation. We demonstrate that PSM yields a much better link probability than PS while keeping a reasonable link fidelity, and shows an advantage in resource consumption over other protocols.

We study the space of conformally covariant $n$-point functions of spinning operators using methods from invariant theory, commutative algebra, and combinatorics. We show that the rational part of any such function can be expressed in terms of the basic building blocks introduced by Costa, Penedones, Poland, Rychkov, thereby providing a rigorous proof of a result that is widely used in the physics literature. We reformulate the problem of enumeration of $n$-point structures in terms of counting lattice points in fractional matching polytopes, and compute these counts using vector partition functions, Hilbert functions, and Kostka numbers. We show that all algebraic relations between the building blocks follow from Gram constraints and compute the number of algebraically independent building blocks. For three-point functions, we derive closed counting formulas for arbitrary integer spins, both with and without Bose symmetry, and discuss a necessary and sufficient condition for the partial conservation operator to lift to a differential operator written in terms of the building blocks. We provide code that generates a basis of three-point structures satisfying these constraints for given values of spins and scaling dimensions.

We show that every pair of 2-torsion elements in the Tate-Shafarevich group of an elliptic curve are visible in the same abelian surface. This was previously only known for a single 2-torsion element. Our result explains some of the original observations on visibility in the paper of Cremona and Mazur.

We address the effective nonequilibrium drag force field that emerges from the microscopic interparticle interactions in steady states of counterdriven binary fluid mixtures. Using power functional scaling arguments for adaptive Brownian dynamics computer simulation results, we establish quantitatively the crossover between near-equilibrium linear response and far-nonequilibrium square root asymptotics. An algebraic expression captures both limiting cases and remains applicable in the crossover regime. Using simulation results as benchmarks, we verify that a local power functional approximation based on the scaling law reproduces the spatial nonequilibrium structure formation in inhomogenously driven systems. The crossover scenario transcends dynamical density functional theory and it sheds light on general nonequilibrium scaling of driven fluids.

We confirm a compactness conjecture of M. Li. If a complete Riemannian manifold has nonnegative Ricci curvature and uniformly convex boundary in the sense that the second fundamental form satisfies $h\ge1$. Then we prove it is compact, and consequently has finite fundamental group. The proof uses monotone quantities constructed via positive proper harmonic functions with Neumann condition.

We establish intrinsic area--length--topology inequalities for compact free boundary constant mean curvature (CMC) surfaces in three-manifolds with Ricci curvature bounded from below. Our main result is obtained from a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator, derived via a balancing argument. This yields a quantitative inequality that does not require stability and captures both interior and boundary contributions. As an application, we obtain explicit topological restrictions for stable free boundary CMC surfaces under a natural curvature pinching condition. In particular, in weakly convex domains, stability forces low topological complexity, with genus at most three and a small number of boundary components. These results show that effective topological control persists even in negatively curved settings, where classical rigidity phenomena are no longer available.

A multitude of findings and theories from cognitive, behavioural and computational neuroscience show that neural activity unfolds in a variety of meaningful temporal units. Behavioural research on event segmentation (ES) has shown that continuous experience is segmented into discrete events and sub-events, which aid real-time comprehension, memory, and decision-making. Computational neuroscience research observes and models ongoing brain activity as a series of stable population activity that occur across wide spatial and temporal scales, referred to as metastable neural activity (MNA). Through this review, we show that these isolated branches of literature, the cognitive theory of Event Segmentation (ES) and the mechanistic approach of metastability (MNA), actually study the same metastable neural states from different perspectives. While the behavioural branch offers a theory for the cognitive and behavioural utility of segmentation, the metastability literature provides the mechanistic account at the implementational level. We describe how metastable neural states act as the fundamental computational units of cognition and identify a number of core principles of how they operate. One is the spatio-temporally nested hierarchy of states, where longer-duration states in higher-order regions both constrain and are shaped by states in faster-operating regions. Another is that neural states are a reflection of underlying predictive models which shape perception, decision making, memory encoding and recall. And finally that neural states are periods of more modular processing, which are interspersed by boundaries where there is a reconfiguration of connectivity. Understanding how neural states emerge, interact, and shape cognition brings us closer to understanding the brain in its natural mode of operation.

Quantum phase transitions manifest as an abrupt change in the ground state of a many-body system; yet it is an open question whether this sudden change necessarily precludes a continuous dynamical connection between the two phases. We introduce a classification of quantum phase transitions based on this geometric aspect of the ground-state manifold, that differs from known classifications. By leveraging the framework of counterdiabatic driving, we explicitly construct schedules that dynamically connect one phase to another. This strategy allows us to uncover a large class of quantum phase transitions, where the states on both sides are separated only by a finite geometric distance in the thermodynamic limit. We term such transitions traversable, since exact counterdiabatic driving links the two phases via a finite dynamical protocol in the thermodynamic limit. We show that multiple known transitions fall into this class -- e.g., symmetry-breaking transitions obeying hyperscaling and discontinuous transitions with an enhanced continuous symmetry. We further show the existence of quantum phase transitions that cannot be crossed dynamically even with the help of nonlocal counterdiabatic driving, as they would require divergent amplitudes and frequencies. Geometrically, these nontraversable transitions correspond to an infinite distance separating the two phases of matter; we show that the class comprises continuous transitions exhibiting mean-field universality, and discontinuous transitions arising from the competition between metastable minima. Our geometric classification goes beyond the known taxonomy, is independent of local order parameters and renormalization group fixed points, and has direct implications for the complexity of state preparation and adiabatic quantum computation.

We investigate uniform weak convergence rates for probabilistic numerical methods applied to backward time-fractional diffusion equations whose dynamics are driven by diffusions with possibly unbounded coefficients, such as the Geometric Brownian Motion. The fractional structure is represented through a random time-change by the inverse of a stable subordinator. To approximate the underlying fractional dynamics, we combine discrete Markov chain schemes for the diffusion component with heavy-tailed random walk approximations of the time change. Our analysis builds on Feller semigroup techniques and a high-order sensitivity framework for diffusion semigroups based on the Kunita stochastic flows and tensor fields. We derive uniform bounds for all orders of sensitivities, establish a quasi-contraction property for the associated semigroup, and transfer these estimates to the fractional setting via the convolution representation with the inverse subordinator. As a result, under killing conditions which dominate at least the base-space semigroup growth, we obtain weak convergence rates for the combined continuous-time-random-walk scheme to the time-fractional diffusion, with a logarithmic regime before the discount dominates the stronger smooth-space growth.

What made useful knowledge cumulative was not discovery alone but the institutions that transmitted it. We provide the first exhaustive structural measurement of the network through which upper-tail human capital passed from master to student across a millennium. Using 470,000 mentor-student records from Wikidata (which integrates the Mathematics Genealogy Project and MacTutor Archive), and all 64 historical Fields Medalists as a fixed, ex ante tracer set, backward traversal yields a directed acyclic graph of 25.5 million paths reaching 57 generations. We document two institutional transitions. First, a 17th-century watershed concentrates lineage traffic on Leibniz: 47 of 64 lineages pass through him with a 10:1 downstream-to-upstream ratio, and seven independent attributes -- learned-society membership (a 46-fold rise per scholar), field, language, employer, institutional diversification, student production, and diffusion entropy -- re-organize coherently across the same window. This is the network signature of Mokyr's Republic of Letters, and it reframes the Newton-Leibniz priority dispute as a distinction between the possession and the transmission of upper-tail human capital: it is transmission that generates the spillovers on which growth depends. Second, 84% of lineages converge upstream on five 12th-13th-century Islamic and Byzantine scholars before terminating at an 11th-century boundary -- the ``Monastery Wall'' -- at which personal academic mentorship first becomes record-generating in Europe. Our claims are descriptive-structural, not causal. Because exhaustive traversal at this scale defeats standard tools, we also contribute a deterministic, algebraic graph-traversal instrument whose measurement bias we characterize in closed form, and report one emergent property of independent methodological interest.

This work introduces a theoretical framework based on the Kadanoff-Baym equations in spinor space to study ultrafast magnetization dynamics in itinerant ferromagnetic systems from first principles. By incorporating spin-orbit coupling into the ab initio Hamiltonian and generalizing the self-energies to include terms beyond the charge sector, I derive scattering integrals within the Markov approximation and quasiparticle renormalizations for electrons and phonons in the presence of spin-dependent effective interactions within a many-body perturbation theory approach. I explicitly discuss how magnons emerge in this framework, and derive, through suitable approximations, a close and tractable set of equations for coupled electron,phonon and magnon dynamics that can be solved from first principles. This approach allows to treat coherent and incoherent magnetic dynamics on an equal footing, paving the way for a first principles understanding of ultrafast magnetic dynamics and demagnetization, and opening to a truly predictive theory of femtomagnetism in periodic systems.

A nonparametric variant of the Kiefer-Weiss problem is proposed and solved. The objective is to minimize a weighted sum of the error probabilities of a binary sequential test subject to a constraint on its maximum expected sample size. This maximum is taken over all possible probability distributions on the given sequence space. First, it is shown that the nonparametric Kiefer-Weiss problem can be reduced to an optimal stopping problem. Then, the optimal stopping policy is derived under the assumption that at most k uses of randomization are permitted during any run of the test. The solution to the original problem is then obtained by letting k go to infinity. The optimal cost function is shown to be the solution of a nonlinear Bellman equation. The corresponding optimal stopping policy is shown to be based on a two-dimensional test statistic, with one component tracking the likelihood ratio and the other one tracking the expected remaining sample size. Critically, the stopping policy uses randomization to increase the remaining expected sample size for some runs, while stopping early for others. The optimal randomization rule is shown to be determined by a function that maps the likelihood ratio to an integer-valued sample size. Two approximations of this function are proposed that can be evaluated easily in practice. The results are illustrated with two numerical examples of nonparametric Kiefer-Weiss tests, one for a shift in the success probability of a Bernoulli distribution, and one for a shift in the mean of a normal distribution.

The stretching behavior of linear macromolecules undergoing conformational transitions is investigated. An exact solution is provided for a two-state system within the elastic freely jointed chain model. This minimal framework contains the smallest set of parameters required to describe such transitions: two Kuhn lengths, two elastic force constants, a free energy difference between both states and a nearest-neighbor interaction energy accounting for cooperativity. Explicit analytical expressions are derived for the chain extension and the probabilities of each state as functions of the applied force.The approach accurately reproduces the experimental force-extension curves of poly(ethylene-glycol) (PEG) and hyaluronic acid (HA), revealing no cooperativity for PEG and negative cooperativity for HA. It also describes the B-DNA to S-DNA conformational transition, a process that exhibits positive cooperativity.We analyze the mathematical conditions required for a transition and identify two fundamental driving mechanisms: differences in Kuhn lengths and differences in force constants.Extensions of the model to systems with more than two conformational states per Kuhn segment are also discussed. The results presented here apply equally to transitions that are intrinsic to the macromolecular structure or induced by ligand-receptor interactions, unifying both cases within a single thermodynamically consistent framework.

This paper studies the local well-posedness and the behavior at potential finite blow-up times of mild solutions to the three-dimensional fractional Boussinesq equations in Lei-Lin and Lei-Lin-Gevrey spaces $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$. By combining fixed point arguments with Fourier estimates adapted to these spaces, we obtain local existence and uniqueness results for data in $\mathcal{X}_{a,σ}^s(\mathbb{R}^3)$, including the usual Lei-Lin case $a=0$. We also establish several blow-up criteria for maximal mild solutions and derive lower bounds for the growth of the corresponding norms as the maximal time is approached. In the strict Lei-Lin-Gevrey regime, and when the dissipative exponents coincide, these estimates yield an exponential type blow-up criterion.

Superconducting domes are conspicuous features of the phase diagrams of most unconventional and high-temperature superconductors. The superconducting transition temperature ($T_{c}$) of FeSe can be dramatically enhanced with electron doping, but unlike all other high-temperature and unconventional superconductors, its full phase diagram and superconducting dome have yet to be fully explored. Here, we employ a combination of molecular beam epitaxy synthesis, alkali surface doping, in-vacuum electrical transport, and angle-resolved photoemission spectroscopy to investigate the entire superconducting dome of electron-doped FeSe, achieving a fully metallic state where superconductivity is suppressed in the heavily overdoped regime. We discover a robust scaling between $T_{c}$ and the residual resistivity ($ρ_{0}$) which holds across the entire superconducting dome, suggesting that the evolution of $T_{c}$ is heavily influenced by the evolution of the elastic scattering rate in the high-$T_{c}$ electron-doped phase. This in turn suggests that the superconducting dome in electron-doped FeSe appears to be fundamentally different than that of other unconventional superconductors where doping plays the primary role, and may be driven primarily by the sensitivity of the superconductivity to disorder.

Ladder logic translation is an important problem in industrial automation because without it, it is difficult to switch Programmable Logic Controller (PLC) vendors. The prevailing translation problem highlights mismatched programming environments, incompatible ladder logic constructs, limitations in terms of differences in the semantic expressiveness of the vendor formalisms and integrated black-box proprietary engineering tools which are exemplified in our example case; Rockwell to Siemens PLC code translation. This work presents a mathematical formulation of the problem, the detailed architecture of a solution which supports XML extraction, structural normalization, constrained generative function (LLM), and system integration via the TIA Portal Openness API as rigorously engineered pipeline for automated translation of Rockwell Ladder Programs to Siemens S7 ladder programs. Finally, we present results that show that the translations retain high semantic consistency across instruction categories.

We study the low-scale type-I seesaw with three right-handed neutrinos (i.e. heavy Majorana neutrinos) when the CP-violation arises solely from the low-energy Dirac phase $δ$ of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix and the heavy neutrinos have testable mixings. We derive a CP-conserving and non-real structure of the $3\times 3$ orthogonal matrix entering the Casas-Ibarra parametrisation in terms of two real angles and one single imaginary parameter, ensuring that the only CP-violating phases in the neutrino Yukawa couplings are those of the PMNS matrix. We then focus on the case of CP-violation from $δ$ alone and discuss the phenomenological implications of this hypothesis. We concentrate on quasi-degenerate heavy Majorana neutrinos with masses within $\sim (0.1-100)\,\text{GeV}$, as relevant for low-scale leptogenesis. Only certain subregions of the full ternary space defined by the ratios $Θ^2_e:Θ^2_μ:Θ^2_τ$ -- where $Θ^2_α$ denotes the squared coupling of the heavy neutrinos to leptons of flavour $α= e,\,μ,\,τ$ -- are compatible with Dirac-phase CP-violation while being testable at collider experiments. Our assumption also implies specific forms of the effective Majorana mass parameter that can be tested at neutrinoless double-beta decay searches. Finally, low-scale leptogenesis under this restrictive scenario can still reproduce the observed baryon asymmetry of the Universe (BAU) in the entire testable region of the parameter space. The BAU vanishes in the exact limit of CP-conserving values of the Dirac phase $δ= 0,\,π,\,2π$, but the observed BAU can be reproduced within the testable region even if $δ$ deviates from these values by a factor as small as $\mathcal{O}(10^{-5})$, with important implications for ultraviolet completions with approximate CP-symmetry.

Free-fermion solvability in quantum spin systems is increasingly understood to be governed by a graph Clifford algebra defined from the frustration graph of the Hamiltonian. When the frustration graph belongs to certain classes, such as the even-hole-free and claw-free (ECF) class, the Hamiltonian is solvable by hidden free fermions: it admits a free-fermion solution although it does not reduce to a Majorana bilinear under the Jordan-Wigner transformation. However, unlike in the Jordan-Wigner case, where each mode is a linear combination of single Majorana fermions, the explicit operator structure of the hidden free-fermion modes -- and that of the local conserved charges -- has remained obscure. In this work, we derive a path-product expansion that expresses each free-fermion mode as a linear combination of path products along induced paths in the extended frustration graph. The expansion is obtained from the generating function of the Krylov basis and yields the modes directly, without using the transfer matrix or the nonlocal conserved charges as input. As an application, the mode decomposition computes infinite-temperature dynamical correlation functions for arbitrary ECF frustration graphs. We further obtain explicit expressions for local conserved charges as linear combinations of path products along induced paths; these charges apply beyond the free-fermion (ECF) class to more general claw-free frustration graphs. We also identify a unified family of generalized conserved charges that contains both the previously known nonlocal conserved charges and these local conserved charges as special cases. For Fendley's original FFD chain with homogeneous couplings and periodic boundary conditions, in a suitable basis, the structure of these local conserved charges exhibits the same Catalan-tree pattern as in the spin-$1/2$ XXX chain.

This note establishes a connection between topological dynamics and statistical properties in ergodic theory. We begin by demonstrating that systems exhibiting the minimal oscillation property with respect to the Lebesgue measure m (that is, where the sequence of averages of the pullback of m accumulates on at least two distinct measures) are non-statistical. Consequently, this yields a sharp dichotomy: any minimal system on the circle is either uniquely ergodic or non-statistical with respect to the Lebesgue measure. As a result, we conclude that the set of minimal non-statistical systems with respect to Lebesgue measure is non-empty.

Solar radio noise storms are common phenomena, composed of broadband continuum emission embedded with diverse fine structures, yet their unusually compact apparent sizes remain unexplained. We present frequency-time-resolved imaging spectroscopy of a near-disk-center noise storm observed by LOFAR between 30--40 MHz, together with anisotropic radio-wave scattering simulations. The continuum forms a bright, spatially compact source that drifts across the solar disk over tens of minutes. Across the band, the measured apparent major axis is $\sim8.0^\prime$ to $\sim4.3^\prime$ between 31.3 and 38.4~MHz, less than half the size of typical type III burst sources at comparable frequencies. Embedded type I bursts, S-bursts, and spikes exhibit a range of dynamic spectra appearances, yet share nearly identical apparent sizes within uncertainties, suggesting a common size-determining mechanism. Using anisotropic scattering simulations, we show that compact apparent source sizes naturally arise for emission embedded within closed magnetic field structures, where anisotropic turbulence directs radiation away from the observer's line of sight. Additional modifications arise from enhanced coronal densities, steeper density gradients, reduced turbulence levels, and strong fluctuation anisotropy, but these exert secondary influence. Our results provide a unified explanation for the similar apparent sizes of diverse fine structures in noise storms, and demonstrate that the compactness of type I storm sources is governed primarily by the large-scale coronal environment rather than intrinsic differences in emission processes, where the magnetic topology plays a crucial role in determining the observed source size.

Quantum support vector machines are classification algorithms that rely on quantum-generated kernels. The fidelity quantum kernel commonly used in quantum support vector machines suffers from exponential concentration as system size increases, preventing an efficient scaling beyond fewqubit systems. We introduce the Hamming quantum kernel, a classical post-processing method that is based on the same measurement outcomes as the fidelity quantum kernel. However, it avoids the exponential concentration problem by using the full measurement statistics rather than a single fidelity value. We evaluate the approach on both classical data (MNIST) and synthetic data generated from quantum circuits, using systems ranging from 2 to 27 qubits. Throughout the simulations, the Hamming quantum kernel outperforms the fidelity quantum kernel whenever 15 or more qubits are used. Furthermore, for synthetic quantum data, our method consistently outperforms the classical Gaussian kernel. This demonstrates that the Hamming quantum kernel improves the expressivity and robustness at larger qubit scales without requiring any additional quantum ressources.

We construct a family of 2D-local constant-depth quantum circuits that output states whose entanglement entropy across a specified cut cannot be estimated in quantum polynomial time. As constant-depth quantum circuits can be learned from polynomially many quantum samples, our resulting pseudoentangled states are implicitly public-key and not pseudorandom. This separates pseudoentanglement from pseudorandomness in the shallow-circuit regime: the former is possible, while the latter is not. The construction is based on the quantum intractability of the Dense-Sparse Learning Parity with Noise problem introduced in [DJ25] and uses a bounded-fan-in, bounded-fan-out classical randomized encoding for linear maps $\mathbf{x} \mapsto \mathbf{Mx},$ which could be of independent interest. As applications, we obtain quantum hardness for the problem of learning the entanglement structure (across a fixed cut) of the ground-state of 1D and 2D local Hamiltonians. The 1D Hamiltonian has an inverse polynomial gap, whereas the 2D one has a constant gap. This complements the result of [BZZ24] that showed only factoring-based hardness for the 1D case, though achieving a volume versus area entanglement difference.

The exponential growth of many-body Hilbert space presents a fundamental barrier to quantum technology, obscuring the search for physically significant states within an astronomically vast landscape. Consequently, resources for quantum-enhanced metrology have been largely confined to the symmetric subspace whose dimensionality scales only polynomially with the particle number-leaving the vast majority of the Hilbert space largely unexplored and poorly understood. Here we challenge this paradigm by demonstrating that metrological advantage can arise as a ubiquitous feature across exponential-dimensional manifolds. By tailoring the first-moment structure of random unitaries, we uncover dense manifolds of engineered random states (ERSs) where Heisenberg-limited scaling emerges as a statistically generic property. This ubiquity endows these resource states with inherent resilience against parameter disorder. We experimentally validate this framework on a trapped-ion processor, achieving a metrological enhancement of $6.98 \pm 0.38$ dB beyond the standard quantum limit. Potential applications extend to diverse platforms, ranging from superconducting circuits and waveguide QED to solid-state spins and polar molecules. Our results establish a powerful paradigm where quantum-enhanced precision can be harvested from the exponential vastness of the Hilbert space.

Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes that is independent of any background geometry and naturally incorporates flexible architectures and accommodates noninteger values, drawing on mathematical machinery from fractal geometry and geometric measure theory. Important scenarios include topological codes and algebraic codes such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--König bounds for regular topological codes, respectively. Furthermore, we discuss implications on thermal properties, presenting a conditional no-go result for self-correcting quantum memories in dimension $3-ε$ for any $ε>0$. Our theory lays a versatile and unifying mathematical foundation for studying the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.

We propose a Bayesian framework for data synthesis and imputation in complex survey settings with informative sampling. To address variance underestimation in existing Bayesian approaches and to accommodate the missing data encountered in survey data, we introduce an adaptive weighting scheme for parameter estimation. We show that the proposed weighting yields consistent estimators with an asymptotically valid Godambe information matrix. The framework is flexible, accommodating a broad class of Bayesian models and facilitating practical implementation. Simulation studies demonstrate that the proposed method provides accurate uncertainty quantification for both model parameters and synthetic population inference.

Open clusters gradually dissolve, and their stars disperse into the Galactic field. Lost stars form tidal tails-elongated streams that trace the cluster orbit ahead of and behind its core. From the shape and orientation of the tidal tails, it is possible to infer the shape of the gravitational potential governing the cluster's motion. The orbits of open clusters, including those in the Solar neighbourhood, are sensitive to the gravitational potential of the inner Galaxy, which is dominated by the Galactic bar. Using n-body simulations of synthetic and real open clusters, we investigate how sensitive the shapes and orientations of tidal tails are to variations of the gravitational potential of the Milky Way. We consider the effects of the bar as well as spiral arms, giant molecular clouds (GMCs) and satellite galaxies. We analyse the stellar distributions within tidal tails using statistical metrics that quantify the differences between tail morphologies. Such non-parametric approach enables us to efficiently explore tidal tails across a large parameter space of gravitational potential models. We find that the Galactic bar-particularly its pattern speed-has a strong influence on the orbits of open clusters and the shape of their tails. Spiral arms have a limited effect, and satellite galaxies do not disturb the tidal tails of nearby open clusters. Perturbations by GMCs affect most clusters, with distortions stronger than those by the bar observed in old and in-plane clusters. We identify nearby open clusters that are most sensitive to the pattern speed of the bar. By observing the tidal tails of a handful of well-selected nearby clusters, we should be able to measure the pattern speed of the bar with a precision in the order of $1\ \mathrm{km\,s^{-1}\,kpc^{-1}}$ independently from length and orientation of the bar. We will present the observability of tidal tails in paper II.

We show that a spin-1/2 particle can behave as if it were spin-$s$, and generate $2s+1$ spots in a Stern Gerlach measurement (albeit with a smaller gyromagnetic ratio). This arises from some subtle properties of effective Hamiltonians and Hamiltonians with constraints. Examples of implications of the effect in condensed matter are discussed. We also give some simple non-perturbative bounds for a system subjected to strong constraints.