Papers

The Solvability Complexity Index (SCI) provides an extensional limit-height formalism for recovering a target map $\Xi$ from finite samples of an evaluation interface $\Lambda$ by finite-height towers of pointwise limits. At first sight this sounds like a typical Type-2 question, so it seems natural to ask how deep and rich the connection of SCI approximation questions and Type-2 computability questions is. This note aims to explain this connection further with explicit examples mostly from the literature.

Autonomous driving involves coupled decision-making and scene evolution under multi-mode uncertainty. To capture this coupling and uncertainty, we introduce MM-Future, a world-action model that generates multiple paired scene-action hypotheses and models bidirectional interaction within each pair. Each hypothesis is initialized from a structured action prior and an independent future scene source, which are then co-evolved through a modality-aware diffusion Transformer. To support efficient multi-mode rollout, MM-Future compresses multi-view video into planning-oriented representations, dubbed MM-Tokens. Finally, a future-conditioned proposal scorer ranks trajectory candidates by shared history context and their paired predicted future. On NAVSIM navtest, MM-Future achieves 94.0 PDMS and 91.5 EPDMS, while attaining a 32.3 HD-Score in zero-shot closed-loop evaluation on HUGSIM. Ablations show consistent improvements over both single-mode and action-only variants, validating the benefit of multi-mode joint world-action modeling.

We investigated the pressure-induced evolution of the electronic and crystal structure of the noncentrosymmetric BaNiSn3-type antiferromagnet EuIrGe3 using x-ray absorption spectroscopy, synchrotron x-ray diffraction complemented by density functional theory calculations, and electrical resistivity measurements. The Eu L3-edge spectra reveal a continuous increase in the mean Eu valence under compression, accompanied by modifications of the Ge and Ir electronic states. High- pressure x-ray diffraction shows anisotropic lattice compression in the tetragonal (I4mm) phase and provides evidence for a structural phase transition above 38 GPa. The experimentally deter- mined lattice and equation-of-state parameters are in good agreement with the DFT calculations. Electrical resistivity measurements reveal a monotonic increase in the antiferromagnetic ordering temperatures up to 18 GPa, indicating that the antiferromagnetic ground state remains robust despite the increasing contribution of the nonmagnetic Eu3+ configuration to the intermediate va- lence state. These results demonstrate that EuIrGe3 exhibits a pressure response distinct from EuCoGe3 and EuRhGe3, highlighting the important role of the transition metal d-electron states in the pressure-induced electronic and structural evolution of the EuT Ge3 family

We study a variable-exponent double-phase Neumann problem on a domain in a metric measure space, where the boundary measure satisfies an upper codimension-\(\theta\) bound for \(0\le\theta<1\). Under a local comparability condition on the double-phase growth function, we establish a modular estimate for a fractional maximal operator and use it to prove the existence and boundedness of the associated trace operator. We then establish coercivity of the associated energy functional on the normalized double-phase Newtonian space and, under reflexivity, prove the existence of minimizers. We further show that the set of minimizers is closed and convex, and that any two minimizers have the same minimal weak upper gradient almost everywhere. Finally, we prove the stability of minimizers under perturbations of the Neumann data.

The HI Zeeman effect provides one of the few direct probes of magnetic field strengths in the diffuse atomic interstellar medium (ISM), yet the physical quantity recovered by the standard weak-field estimator remains uncertain in multiphase gas. This is especially relevant to the low-column-density branch of the observed magnetic field–column density relation, which is nearly flat below NH~1E22 cm-2 and remains difficult to explain theoretically. We investigate this issue using synthetic HI Zeeman observations of multiphase magneto-hydrodynamic simulations of the atomic ISM performed with RAMSES, complemented by analytical two-phase models of the thermally bistable atomic medium. We find that both emission and absorption Zeeman measurements recover the magnetic field weighted by optical depth along the line of sight, rather than the density-weighted field. The two converge only in limiting cases, such as isothermal gas or homogeneous magnetic fields across phases. In realistic multiphase media, the Zeeman-derived field statistically exceeds the density-weighted field by factors of a few, with the strongest departures when cold and warm atomic gas both contribute significantly to the column density. Our analytical models reproduce this behavior and show that thermal bistability generates both an excess and a large dispersion between Zeeman-derived and density-weighted fields, even when the intrinsic magnetic field increases monotonically with density. The effect is strongest below NH~1E22 cm-2, where the temperature contrast between cold and warm HI causes optical-depth and density weighting to diverge. We conclude that HI Zeeman observations primarily trace the optical-depth-weighted magnetic field. This intrinsic thermal weighting may contribute to the broad distribution, and possibly part of the apparent flattening, of the observed magnetic field-column density relation.

A novel class of polycatenated architected materials (PAMs) has recently emerged, exhibiting distinctive mechanical properties such as stress-strain hysteresis and a geometry-driven transition between solid-like and fluid-like behavior. Although finite element models exist for conventional PAMs, none currently account for electrostatic effects. Here, we propose and qualitatively validate a computational framework to simulate the structure, dynamics, and mechanical response of electrostatically charged PAMs. Our approach successfully reproduces electrostatic actuation in close agreement with experiments, and reveals that electrostatic charges enable a reversible fluid-to-solid transition while also providing a means to estimate electrostatic stiffness. Notably, stiffness enhancements of up to sevenfold are achieved upon charging. This work expands the design space of PAMs and offers a predictive tool for actively controllable mechanical metamaterials.

Patient-specific biomechanical models of the left ventricle can relate cardiac magnetic resonance imaging to regional myocardial material properties, but existing emulator-based studies typically treat the myocardium as mechanically homogeneous, limiting representation of localised dysfunction. We propose a Bayesian surrogate-modelling framework for inferring regional Holzapfel-Ogden material parameters in a left ventricle partitioned into five physiological zones derived from the American Heart Association 17-segment model. Eight emulator strategies spanning single- versus multi-output, local versus global, and Gaussian-process- versus neural-network-based architectures were screened using parameter point-estimation accuracy; the three retained models were evaluated using empirical marginal credible-interval coverage and posterior contraction. We found that models with comparable point accuracy nevertheless differed markedly in uncertainty. A multi-output variational Gaussian process provided the most favourable balance across these criteria and was retained for the subsequent analyses. In synthetic local and global stiffening scenarios, maximum a posteriori estimates generally distinguished stiffened from baseline zones, but the nonlinear-stiffening parameters were more difficult to identify from end-diastolic observations than the stiffness-magnitude parameters. A healthy-volunteer analysis demonstrates feasibility with an incomplete observation vector and jointly inferred strain-noise scales. These results suggest that the proposed framework provides a computationally feasible, uncertainty-aware approach to regional left ventricle parameter inference.

Quantifying the interactions between active Janus particles remains a challenge that needs to be addressed to describe their collective behaviour. Inspired by a photonic force microscopy setup where two active Janus particles are confined in separate optical tweezers, we derive the steady-state probability distribution of each particle bearing signatures of the pair phoretic interaction, which can be generalised to a broader range of interactions. This approach allows us to infer interaction forces and torques from positional configurations without any knowledge of the particle orientations. Unraveling the distance-dependent interaction between active particles is essential to harnessing them as building blocks for advanced materials with dynamic properties.

In pay-per-token LLM services, the more a model says, the more users pay. Dishonest providers can covertly manipulate generation to inflate output tokens while largely preserving task utility. We define such manipulation as a Provider-Side Token Inflation Attack (PTIA) and instantiate five representative attacks at the query, prompt, representation, and model levels of the provider-controlled pipeline. Our experiments show that each attack increases mean output length to more than 10.2x the clean baseline, demonstrating PTIA's financial appeal and feasibility at multiple stages of generation. Yet auditing PTIA from black-box responses is difficult for users. Our key observation is PTIA saturation: an initial attack sharply lengthens output, but further strengthening or composition has much less effect. We trace this saturation to stopping behavior: an initial PTIA sharply lowers the end-of-sequence token probability, whereas further intervention lowers it only marginally. Building on this insight, we design a lightweight single-probe audit that applies a controlled lengthening intervention. Under PTIA, the probe induces far fewer additional tokens than under normal service. The audit requires neither a trusted local reference model nor historical clean responses, and its separately issued original and probed requests resemble ordinary traffic, making evasion difficult. Across four open-weight models, it achieves an average detection rate of 85.1% with false-positive rates below 2%. Across 15 real LLM API services, the audit flags 7 for PTIA-consistent behavior.

Escardó's dialogue interpretation assigns to each closed term $t:(\iota\to\iota)\to\iota$ of Gödel's System~T a well-founded, countably branching tree $D(t)$, where $\iota$ is the natural-number type. We give a direct proof that its classical ordinal height is below $\epsilon_0$. More precisely, we compute a natural number $K(t)\ge2$ from the type levels occurring in the source term and prove $h(D(t))<\theta_{K(t)}$, where $\theta_0=\omega$ and $\theta_{n+1}=\omega^{\theta_n}$. Our proof translates recursors into closed infinitary templates and eliminates $\beta$-redexes by a finite sequence of passes indexed by ordinary type level. The translation and every pass preserve the dialogue denotation exactly. An auxiliary rank $\rho$ satisfies an additive substitution bound; each pass sends rank $\alpha$ to at most $2^\alpha$. Combining these estimates with a computable initial bound $\omega+m(t)$ and a dialogue-height bound $2^{\rho(N)}$ for closed ground normal forms $N$ yields the stated tower bound. A semantics-preserving translation transfers the result to Escardó's original combinatory interpretation. We formalise the proof in Agda over classical ordinals under explicit foundational assumptions.

Mathematical Programs with Blocks of Vanishing Constraints (MP-BVCs) are a challenging class of optimization problems due to the loss of standard constraint qualifications. Based on existing theory for classical MPVCs, i.e., MPBVC with block size one, we extend the constraint qualification theory for MPBVCs and show that the known LICQ-type condition for MPBVCs implies the Guignard Constraint Qualification (GCQ). A novel piecewise gradient flow approach converges under GCQ. It directly computes strongly stationary points and is an extension of the piecewise gradient flow approach for classic MPVC problems. Numerical experiments on several truss topology design instances with two and even three applied load forces demonstrate the robustness and effectiveness of the method.

We present a localized rooted-operator argument for Weil positivity in the real odd logarithmic channel, retaining the polar rank-one term throughout. The three gates are structural levels rather than a serial dependency: Gate I is an independent strict cellular/shell positivity theorem and diagnostic; Gate II carries the load-bearing operator argument; Gate III uses compression, closure, and the restricted odd Weil criterion. In Gate II, the inherited parent response is placed in the same post-old-core/common-cut primal form in which the arithmetic mismatch is charged. The audited MASTER-P2 same-block splice combines the aligned reference contribution with the actual Schur response before the target-ground coefficient is used, while the literal-slot/true-ground mismatch is paid explicitly by a leakage term. The source elimination is written as one exact common-source Schur completion on the same harmonic vector, and the endpoint fold is used only as unitary coordinate transport, so it creates no second source debit. An outward-rounded finite computation and analytic tail give a strictly positive MASTER margin for k >= 7. Starting from the rigorously certified physical coefficient-2 endpoint Y=7, Schur induction yields positivity at the integer endpoints; zero-extension gives non-negativity at every finite support radius. Closure and the restricted odd Weil criterion then give the conclusion stated in the main theorem.

We consider infinite walks in $\mathbb{N}^k$ with standard unit basis vector steps that avoid $t$ collinear points, and show that these walks exist for $(k,t) \in \{(6,3), (4,4), (3,7)\}$. In particular, our construction for $k = 3$ improves the previous bound $189$, obtained by Lidbetter, to $7$. Our results also imply the existence of infinite words over small finite alphabets that are weakly abelian squarefree (resp., weakly abelian cubefree, weakly abelian 6th-power-free).

Surface reflectance and roughness are key to understanding Mercury's geologic evolution. The BepiColombo Laser Altimeter (BELA) will measure these properties alongside topography, but predicting its performance requires realistic modeling of laser return pulse shapes, unlike a simplified Gaussian approximation in earlier models. We develop a comprehensive return-pulse simulation framework for BELA that includes non-Gaussian transmitted pulses, analog filtering in the receiver chain, and pulse-shape modification by footprint-scale topography. Using high-resolution lunar digital terrain models and synthesized fractal topography as Mercury analogs, we develop calibration and correction methods and estimate measurement errors expected in future BELA observations. Range errors average 1.5 m or less below 1000 km altitude, and energy errors are small enough to distinguish dark deposits, regolith, and exposed ice below 900 km. Pulse width proves unreliable as a roughness proxy under realistic conditions; we instead constrain roughness from digitized pulse shapes, underscoring the value of time-resolved data for future planetary altimeters.

Chemical substitution provides an effective route to tune the structural and superconducting properties and to explore the evolution of superconductivity across related materials. In this work, we investigate the effect of Pb/Bi mixing on superconductivity in possible topological superconductors \ch{Pb2Pd} and $\beta$-\ch{Bi2Pd}, by synthesizing and characterizing single crystals of \ch{Pb_{2-x}Bi_{x}Pd} ($0.2\le x\le1.8$). Increasing Bi content triggers a structural phase transition from non-symmorphic ($I4/mcm$, $x<1$) to layered ($I4/mmm$, $x\ge1$) and induces a monotonic expansion of the lattice parameters. Across the doped range, all superconducting samples behave as weakly-coupled type-II superconductors exhibiting isotropic s-wave superconducting gaps.

Let $\mathbb{D}$ be a division ring with an involution $x \mapsto x^\star$, and $n \geq 2$ be an integer. Denote by $\mathcal{H}_n(\mathbb{D})$ the set of all $n$-by-$n$ Hermitian matrices with entries in $\mathbb{D}$, and by $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ the set of all matrices $A-A^\star$ with $A \in \mathcal{M}_n(\mathbb{D})$. Here, we give a complete solution to the following problem: Determine all group homomorphisms from $\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ (respectively, from $\mathcal{A}\mathcal{H}_n(\mathbb{D})$ to $\mathbb{D}^n$ unless $(-)^\star$ is the identity) that take every matrix to a right linear combination of its columns. The solution to this problem was already known when $(-)^\star$ is the identity, and the novelty here lies in the generalization to arbitrary involutions, and in particular in the noncommutative case. These results are to be used in a subsequent article on subspaces of Hermitian matrices of bounded rank, and on large spaces of diagonalisable matrices.

A quantitative comparison of the predictions of three theoretical descriptions (TMD, pQCD and geometrical) of the angular coefficients in Drell-Yan Z-boson production with new experimental data on angular coefficients obtained by the ATLAS, CMS and LHCb collaborations has been performed. In the central rapidity region at low-$Q_T$, the TMD calculation provides a better description of the $A_0$ and $A_2$ coefficients among the descriptions considered. The differences between the approaches are less pronounced for $A_1$,$A_3$ and $A_4$ coefficients. The results identify $A_0$,$A_2$ and the Lam–Tung relation as the most sensitive observables at low-QT and illustrate the domains of applicability of the three descriptions.

We prove a generalization of the version, due to A. Gabrielov, of the Łojasiewicz inequality for not necessarily restricted semi-Pfaffian sets. Unlike the most variants of the Łojasiewicz inequality, it describes the rate of growth of a Pfaffian function on a semi-Pfaffian set $X$ not only in a neighbourhood of its zero set in the domain of $X$ but also in a neighbourhood of zeroes on the boundary of the domain. Gabrielov's version is essentially one-dimensional, we generalize it to a multidimensional case.

In this paper, we prove the long-time existence of the heterotic $G_2$ flow with bounded Ricci curvature, torsion tensor and dilaton. Besides, we also get the forward and backward uniqueness results for the solution.

The symbiotic scaling of artificial intelligence models and high-performance computing systems continually creates algorithmic challenges in their convergence. Foundation models (FMs) are a crucial example, requiring months-long training on thousands of cutting-edge GPUs. Sharded data parallelism (DP) is the dominant strategy to accelerate such computations by splitting data and models across multiple GPUs. However, it incurs prohibitive communication overhead when deployed at scale, particularly on multi-tier interconnects with heterogeneous performance. Inspired by the efficient communication principles of federated learning (FL), this work introduces two hybrid algorithms - FL+FSDP and FL+HSDP - interleaving sharded DP with FedAvg-style aggregations. Such approaches decouple large DP deployments into smaller, loosely-coupled federation groups, requiring minimal inter-group traffic while keeping the global batch size bounded by the groups' size. Formal analysis of communication costs and experimental validation prove their scalability and flexibility. A Llama3.1 8B pre-training on 512 A100 GPUs shows that, under identical hyperparameters, FL+FSDP and FL+HSDP achieve up to 8.04 faster data processing and 4.48 lower evaluation perplexity than their counterparts, demonstrating superior computational efficiency and improved model quality. These properties stem from reduced communication overhead and the bounded growth of the global batch size relative to the federation group size.

Characterizing the physical properties of clay and cementitious materials matters across many fields, from materials science to geological waste disposal. Property simulation typically calls for 3D imaging, which is expensive, not always accessible, and technically limited for certain materials. Recent progress in deep generative models offers a way around this, reconstructing 3D volumes from the more easily acquired 2D images. Among GAN-based methods for 3D microstructure generation, SliceGAN has shown strong results for homogeneous isotropic and anisotropic systems. It struggles, however, to capture the finer detail of more complex heterogeneous microstructures, which motivates alternative generative frameworks. We introduce a hybrid approach that draws on the stability and generation quality of denoising diffusion models. Since no 3D ground truth is available, we replace the standard denoising loss with an adversarial loss, which yields a stable training process in our experiments. We show that the resulting model generates microstructures of varying complexity with minimal slice artefacts and close agreement with ground-truth phase fractions and structural descriptors.

The behavior of wave propagation and scattering in time-variant materials has been studied by many authors, for electromagnetic as well as for mechanical waves. Most studies consider two-dimensional waves, governed by scalar wave equations. Here we introduce a unified wave equation for three-dimensional electromagnetic and elastodynamic vector-wave propagation and scattering in homogeneous, time-variant materials. Using the symmetry properties of this equation, we establish the conservation of net field-momentum density, and a reciprocity relation between causal and acausal Green's functions. The latter relation is exploited in the derivation of the counterparts of classical time-reversed focusing and Green's function retrieval for time-variant materials.

This paper develops and analyzes an optimal-order fractional backward collocation method for adjoint Volterra integro-differential equations with weakly singular kernels. The backward Volterra structure, together with the weakly singular kernel, induces fractional-power singularities at the terminal endpoint, thereby reducing the classical regularity of the exact solution and causing order deterioration in standard polynomial collocation methods. We first establish a regularity result that characterizes the terminal singular behavior of the solution, showing that the solution is continuously differentiable, whereas its second derivative may exhibit a weak singularity at the terminal endpoint. Motivated by this regularity structure, we introduce a terminally graded mesh and construct a fractional backward collocation scheme whose local approximation space is adapted to the endpoint singularity. Rigorous convergence and superconvergence estimates are derived for both the solution and its derivative. With suitable choices of the fractional parameter and the mesh-grading exponent, the proposed method attains the optimal convergence orders dictated by the local approximation degree. Numerical experiments confirm the theoretical predictions and demonstrate the accuracy and effectiveness of the method for adjoint weakly singular Volterra integro-differential equations.

Since the detection of the first extrasolar planets thirty-five years ago, the dawn in detected signals has been driven primarily by radial velocity and transit techniques. However, other methodologies have gained significant interest due to their ability to explore specific niches that are challenging to access with traditional approaches. In this article, we focus on the photometric orbital phase modulations induced by planets, describe the methodologies and equations and highlight their developments and prospects for the future.

We classify, for every prime power $q$ and every $e\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\mathbb F_{q^e}$. Writing $\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\gcd(r,q-1)=1$, $(-a)^{\ell_e(q)}\ne1$, and $r\ell_h(q)\equiv1\pmod{\ell_e(q)}$ for some $1\leqslant h<e$ coprime to $e$. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from $(X^{q^h}+aX)\circ X^r$ for a suitable $h$. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family $X^r(X^{d(q-1)}+a)$ in the coprime-index case $\gcd(d,\ell_e(q))=1$. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas' theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-$q$ digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.

We study repeated contract design when a principal observes outcomes but not the actions that generate them. The principal may use any bounded outcome-contingent payment vector, and the agent's best response can make expected profit discontinuous in those payments. For every fixed number $m\ge2$ of outcomes, the minimax regret over $T$ rounds is of order $T^{m/(m+1)}$, up to logarithmic factors. The upper bound allows arbitrary action spaces and agent heterogeneity, without smoothness or monotone-surplus assumptions. Its key is an effective-dimension reduction that the benchmark can be normalized even when fixed tie-breaking is not shift invariant, after which revealed preference yields a monotone response map in payment-difference coordinates. A learning policy built on a Lipschitz parametrization of this map attains the rate using only observed outcome categories. The lower-bound construction accounts for how incentive losses accumulate across outcome dimensions. It shows that each additional contractible outcome creates a precise and unavoidable increase in the worst-case cost of learning.

Large-scale routing often requires visiting clusters of nodes in a prescribed order, giving rise to the Ordered Clustered Traveling Salesman Problem (OCTSP). Optimizing each cluster independently seems natural, but misses non-local dependencies. We introduce the COMPASS algorithm for OCTSP, which combines search with learning-accelerated routing by orchestrating parallel sub-solvers. COMPASS has no quality ceiling and its solutions keep improving with compute. It exploits the clustered structure, and can reach exact solutions in time exponential in cluster size rather than instance size. Empirically, COMPASS consistently outperforms alternative methods. Unlike common large-scale routing solvers, COMPASS consumes general distance matrices and is not limited to coordinate inputs. We demonstrate scaling to 100K synthetic nodes and to 28.5K real e-commerce nodes. To our knowledge, the latter is the largest reported routing solution over asymmetric distances, 9x beyond established ATSP benchmarks.

We study random linear projections of a finite-field subset for which every fiber has cardinality close to its mean. We bound the mean fiber size needed to ensure that all fibers satisfy a prescribed relative discrepancy, with a prescribed failure probability. For $S\subseteq\mathbb F_q^n$ projected to $\mathbb F_q^b$, one theorem gives three regimes: at fixed discrepancy and failure probability, sufficient mean fiber sizes are $O(q2^b)$ for arbitrary $q$, $O(q^2)$ when $q$ is at least a suitable constant multiple of $b$, and $O_q(b)$ for fixed $q$. The resulting entropy loss over fixed fields is $h-b=\log_q h+O(1)$, where $h=\log_q|S|$ is the input entropy. This matches the order of the binary obstruction of Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos (1999); we give a quantitative random-source refinement over every fixed field. Our proof combines a quotient-and-average counting lemma with the local balanced/unbalanced argument of Dhar and Dvir (arXiv:2204.01665) and Furstenberg estimates of Dhar and Dvir and Kumar and Mon (arXiv:2609.17020). We apply these bounds in the reduction of Ordentlich, Regev, and Weiss (arXiv:2311.04644) to improve their $O(n^3)$ bound for smooth lattice coverings to $O(n^2)$. For each fixed convex body $K\subseteq\mathbb R^n$, a Haar-Siegel random lattice of covolume one has the number of lattice points in every translate of $K$ within a prescribed relative error of $\operatorname{vol}(K)$, with prescribed high probability, once $\operatorname{vol}(K)\ge Cn^2$ and $n$ is sufficiently large. The constant and dimension cutoff depend only on the error and failure probability. Complements of higher-rank Kakeya sets of Kopparty, Lev, Saraf, and Sudan (arXiv:1003.3736) show that no hashing guarantee for arbitrary subsets can yield a smaller order in the same reduction.

Confidence intervals of location (CIL) of calling marine mammals, derived from time-differences-of-arrival (TDOA) between receivers, depend on errors of TDOAs, receiver location, clocks, sound speeds, and location method. When these errors are minuscule, simulations yield small errors of PAMGuard's 3D simplex localizer when click sounds of beaked and sperm whales originate in a 1000 x 1000 x 1000 $\mbox{m}^3$ region using five receivers having horizontal and vertical separations of 1000 m and 150 m respectively. Realistic uncertainties of sound speed up to $\pm 10$ m/s lead to errors up to $10^{14}$ m. With clocks maintained by atomic standards and common practice of correcting TDOA from synchronization measurements at the start and end of an experiment, errors of location are up to $10^{4}$ m. Errors up to $10^2$ and $10^3$ m are found when the receiver's locations are uncertain within 10 and 40 m respectively. Errors of PAMGuard's 3D hyperbolic localizer are almost independent of the above uncertainties, yielding errors of location up to about $10^4$ m even when simulated errors are minuscule. Causes of PAMGuard's 3D location errors are unknown. These algorithms are briefly compared to another method designed to yield a reliable CIL.

Spatial reasoning abilities correlate strongly with performance in STEM fields. Games offer a compelling medium for training these critical skills in developing children who have a natural proclivity for play. However, to facilitate human-like tutoring and player guidance, these games require an AI agent capable of making commonsense inferences from spatial events. Qualitative reasoning (QR) models appear to be a suitable framework for these application domains. As these models reason in symbolic representations, they can seamlessly translate game states into interpretable feedback for human-like player guidance. This paper introduces a hybrid qualitative model designed for Camelot Jr., a block-puzzle game that requires constructing multi-level bridges to connect two avatars stationed on separate towers. The game poses a challenge for the player, who must make platforms stable, plan their path, and ensure they use all the provided blocks. To handle the precise physics required by the domain, we integrate a mathematical center-of-mass stability logic to guide our qualitative solver. Our work facilitates spatial skill training in Camelot Jr. and contributes to the development of human-centric, explainable game-playing agents.