Papers

In 1878, Pafnuty Chebyshev presented to the Association fran\c caise pour l'avan\-cement des sciences [French Association for the Advancement of the Sciences] an article \cite{Chebyshev1878} dealing with garment cutting. According to Chebyshev himself, his interest was sparked by a lecture given by Édouard Lucas that he had attended in 1876 \cite{Lucas1876}. There is a second story on the origin of Chebyshev's interest in garment cutting according to which in the 1850s, being short of money, Chebyshev got himself a job as a consultant to a clothing factory. At the time of the Crimean War (1853-1856), there was a great demand for uniforms. Chebyshev was allegedly asked to optimize the use of fabric, and it was there that his interest in garment cutting was born. This second story appears to have its origin in a post by Clive J. Grant to MacTutor in 1996 \cite{Grant1996}. However, this contribution contains no references, and no other source of information that I have found offers any first-hand documentation to support this story. Our conclusion is that this second story is a fabrication, invented out of whole cloth.

This article presents a mathematical model of the Motivated Emotional Mind cognitive architecture developed for embodied intelligent systems. Such a system learns to maintain its homeostasis through a generalized form of reinforcement learning based on its internal motivations, termed motivated learning (ML). The principal contribution of this article is a rigorous formalization of the re-entrant loop integrating feedforward processing, lateral interactions, and feedback pathways, together with the representational selection mechanisms that govern adaptive system responses. The model specifies how ongoing exteroceptive and interoceptive signals, bodily-motivational context, and memory traces are bound into associative memory structures termed semblions, which compete for access to further processing and top-down reconstruction. The formalization encompasses secondary perception, representational competition, curiosity, procedural gaps, and action selection directed toward limiting allostatic violations. Within this framework, motivated learning is tailored to embodied systems whose dynamics are shaped by needs, affect, and the current regulatory state. Unlike standard reinforcement-learning models, the proposed approach incorporates need thresholds, goal generation and shifting goals, bodily state, resource constraints, and action uncertainty, thereby providing a more adequate account of response selection under regulatory pressure. Global affect functions as a central control signal, modulating the learning rate, representational valence, and the balance between exploration and exploitation. The model presented here is a step toward a more rigorous formalization of cognitive phenomena and may provide a basis for further theoretical analysis, computer simulation, and implementation in artificial-intelligence systems inspired by biological processes.

A coloring of $\R^n$ is proper for the forbidden distance segment $[1,\ell]$ if no two points of the same color are at a distance from $[1,\ell]$; the minimum number of colors is $\chi(\R^n,[1,\ell])$, and $\ell=1$ gives the classical chromatic number $\chi(\R^n)$ of the Nelson–Hadwiger problem. We prove the new upper bounds $\chi(\R^4)\le43$, $\chi(\R^5)\le132$, $\chi(\R^7)\le1029$, $\chi(\R^9)\le7203$, $\chi(\R^{10})\le45619$, improving the previously known $49$, $140$, $1372$, $17253$ and $3^{10}$; in particular, this refutes the conjecture of Arman, Bondarenko, Prymak and Radchenko that $49$ and $140$ are optimal among all lattice colorings of $\R^4$ and $\R^5$. The first four bounds come from explicit rational lattices — an Eisenstein lattice in $\R^4$, a lattice in general position in $\R^5$, and laminations of the Eisenstein colorings $E_6^*/343$ and $E_8/2401$ in $\R^7$ and $\R^9$ — and each is reduced, by one verification protocol, to a finite list of inequalities between explicitly written rational numbers checked in exact arithmetic. The fifth bound is analytic: we prove that for every Eisenstein lattice $\Lambda$ the distance between same-colored cells of $(3+\omega)\Lambda$ equals $\sqrt{7/3}\,\lambda_1(\Lambda)$, which gives the exact widths of all known colorings with $7^{n/2}$ colors, and a product rule $\sum_i1/d_i^2\le1$ for the widths of orthogonal products; together they yield $45619=2401\cdot19$, the first bound in $\R^{10}$ below $3^n$, as well as $\chi(\R^{25})\le4\cdot7^{12}$ and $\chi(\R^{26})\le19\cdot7^{12}$. We also show that no sublattice of $E_8$ of index below $2401$ defines a proper coloring. All code, exact certificates and data are open.

Autonomous mobile robots require timeefficient planning and safety-critical dynamic obstacle avoidance under constrained onboard computation. While Iterative Learning Planning (ILP) offers lightweight and efficient traversal planning, it lacks explicit mechanisms for dynamic obstacle perception and avoidance. This article extends ILP to safety-critical navigation in dynamic environments by integrating an anticipatory risk-blended control barrier function (ARB-CBF). The extended ILP learns traversal-speed and steering-bias profiles via a fractionalpower update based on local obstacle risk, generating nominal control commands that ARB-CBF modifies at runtime for real-time safety guarantees. Algorithmic analysis demonstrates that the ILP replanning stage scales at O(kN) for k iterations and N waypoints, while ARB-CBF executes with linear complexity. Comprehensive simulations and real-world experiments validate the framework, demonstrating superior temporal efficiency and safety with lower computational overhead compared to optimizationbased baselines, making it highly suitable for resourceconstrained platforms.

We study a Fermi-Hubbard model coupled to a open dissipative single cavity mode. Using Keldysh diagrammatics we derive and solve the quantum kinetic equations for the fermions, taking the bosonic cavity mode as a source of non-equilibrium noise and dissipation. In absence of Hubbard interactions we show that the fermions reach generically a non-equilibrium steady-state, characterized by a non-thermal distribution function. Quite interestingly we demonstrate that the latter exactly nullifies the heat-current between fermions and cavity mode. We discuss the regimes of parameters where a low-frequency effective temperature description emerges and how the fluctuations affect the mean-field phase diagram for the superradiance phase transition. Finally we include Hubbard interaction in the weak-coupling regime and show that it leads to a crossover towards a full equilibrium distribution.

The formulation of statistical mechanics in terms of ensembles, originally proposed by Gibbs, has proved to be extremely effective in describing the equilibrium state of many-particle systems, in the most diverse contexts (critical phenomena, quantum mechanics, biophysics, just to mention a few). The reasons of this success are still debated, and the connection between dynamics and probability in large systems remains somehow elusive. Indeed, in order to derive the main results of equilibrium statistical mechanics, strong assumptions on the ergodicity of the dynamics are usually required. Nonetheless, empirical observations seem to suggest that the predictions of the theory hold true even when such assumptions are not verified. In this paper we reconsider the point of view put forward by Khinchin, stating that the only relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom in the system and the choice of extensive observables, irrespectively of the details of the microscopic dynamics. In particular, the presence of dynamical chaos is not required. To this aim, we discuss some analytical and numerical results on a couple of classical integrable systems, the harmonic chain and the Toda model, showing that many important features predicted by equilibrium statistical mechanics, as for instance Maxwell-Boltzmann distribution, are found even in absence of chaos.

A central goal of modern condensed matter physics is to uncover new quantum states of matter arising from the intertwined effects of strong electron correlations, magnetism, and band topology. Heavy-fermion phases, generated by Kondo interactions, represent one of the most remarkable manifestations of electronic correlations, and topological heavy-fermion states have been identified in several non-magnetic materials. Yet, the consequences of their competition with magnetic order have remained largely unexplored. Here, we reveal a new phenomenon: the spontaneous spatial separation of correlated quantum phases. By showing that magnetism can drive distinct strongly correlated electronic states to coexist in different regions of a single material, our work establishes a previously unknown mechanism for organizing quantum matter and opens a new direction in the study of correlated topological systems. Using bulk-sensitive probes, we show that UAsS crystals are, in the bulk, metallic ferromagnets with only moderate correlation-driven band renormalizations. First-principles calculations reveal a topological electronic structure hosting both nodal lines and Weyl points, pointing to a rich underlying topology. Angle-resolved photoemission spectroscopy (ARPES) measurements are consistent with these predictions, resolving the nodal lines and Weyl crossings. In striking contrast, surface-sensitive ARPES and scanning tunneling microscopy/spectroscopy (STM/STS) measurements reveal a pronounced flat band pinned at the Fermi level, accompanied by a sharp resonance – hallmarks of an emergent, strongly correlated Kondo state not captured by first-principles calculations.

Quantum $p$-local spin glass Hamiltonians are natural quantum analogues of the classical spin glass models. We provide an asymptotic characterization of the maximal energy achievable by the product states. We prove that for every $p \ge 2$, the limit of ground state energy exists and is given by a Parisi-type variational formula. This settles a question left open by Anschuetz et. al. (2025). The variational formula also recovers the known large-$p$ asymptotics as a consequence. Finally, we prove that the limiting product state energy is universal for a broad class of non-Gaussian interactions.

The great transoceanic voyages of the Age of Discovery imposed severe human costs on their crews. Long travel durations, deteriorating sanitary conditions, and finite food and water supplies made disease, exhaustion, and mortality central features of maritime expansion. We propose a minimal compartmental model describing the voyage itself as a dynamical system coupling crew health and resource depletion. The population is divided into healthy, ill, and dead individuals, while an additional variable represents the effective level of provisions on board. Illness incidence depends nonlinearly on resource availability, capturing the increasing health risk associated with deteriorating food, water, and sanitary conditions. The coupled ordinary differential equations are analyzed in the transient regime relevant to finite-duration voyages. Rather than reconstructing specific historical expeditions, the model identifies generic mechanisms linking provisioning, health deterioration, mortality, and voyage duration. Numerical results show that resource limitations can substantially reduce the healthy and operational fraction of the crew even when overall survival remains relatively high, revealing a distinction between demographic survival and operational viability. Increasing initial provisioning delays the crossover at which ill individuals become as numerous as healthy individuals and extends the period over which the crew remains predominantly operational. We also introduce a critical provisioning time, defined as the time required for resources to reach a prescribed stress threshold, and find an approximately linear dependence on the initial resource buffer. The framework provides a quantitative perspective on the internal dynamics of early transoceanic navigation and illustrates how dynamical-systems approaches can contribute to the study of historical processes.

Our first main result is a rigidity theorem for complete self-expanders in the Euclidean space with higher codimension, assuming an integral cur More precisely, any smooth complete self-expander $x:M\to \mathbb R^{n+p}$($n\geq3$) that satisfies both $(\int_{M}|A|^{n}d\mu)^{\frac{1}{n}}<K(n)$ and $\int_{M}|A|^{n}e^{\frac{|x|^{2}}{2}}d\mu<\infty$ for a positive constant $K(n)$ depending only o isometric to $\mathbb R^{n}$. Moreover, we show that the rigidity result persists when the pinching condition is expressed in terms of the trace-free second fundamental form.

In order to understand which initial states retain memory under periodic driving, it is necessary to depart from the ideal limit giving rise to invariant structures and work inside near-degenerate manifolds. We show that degeneracy itself is not conclusive and what actually matters is how perturbations, such as structural and driving imperfections, acts on the projected multiplet. For systems decoupling into the local clusters in the ideal limit, the projections can be computed within degenerate perturbation theory. There the stability is decided by two criteria: the coupling can not connect the state with other degenerate partners in the manifold, either by kinetic blocking, or more generally, by diagonalizing the projected coupling, and the drive's local selection rule must admit only transfer energies detuned from Floquet sidebands, in which case the projected drive generator exactly vanishes. We formalize these understandings in a periodically driven clean, short-ranged alternating XXZ chain and demonstrate a set of results that include, long-lived states, both of product form and entangled across the clusters, a periodic family of sideband resonances, and an explicit example showing opposite fates for states with identical energy and charges.

Automated pain recognition from facial expression could make continuous welfare assessment practical in sheep, but adoption depends on trust: a stockperson cannot act on a score that arrives without justification. We ground a model in the Sheep Pain Facial Expression Scale (SPFES) by letting each detected facial region attend over text embeddings of the clinical descriptors and then test whether the resulting explanations mean anything. They do not. Ablating an entire descriptor changes the predicted logit by about $10^{-4}$, and the most-attended cue agrees with the predicted pain level in only $32.6\%$ of regions, although the attention maps, the learned gate, and the generated text all proposed otherwise. We therefore remove the appearance bypass with a concept bottleneck whose classifier reads only SPFES concept scores, supervised by per-region state annotations that image-level pipelines discard. This costs $0.05$–$0.10$ in Cohen's $\kappa$ but yields concepts that are demonstrably learned: minority pain-indicating states are recovered at $3.5$–$8.3\times$ their base rates, and the ear and eye severity orderings emerge without severity supervision. Removing the supervision alone leaves $\kappa$ unchanged while concept accuracy falls to $0.109$, showing that architectural necessity does not imply semantic validity. We also show that pooled concept accuracy is misleading under clinical imbalance and provide a cross-validated, protocol-matched benchmark of seven methods on this dataset.

Atomically thin transition-metal dichalcogenides, such as MoS2, offer a promising route to surpass the scaling limits of silicon owing to their excellent electrostatic control and resilience to short-channel effects. Realizing this potential depends critically on gate-stack engineering, where strong gate coupling must be achieved without compromising the pristine two-dimensional interface. Here, we demonstrate a coplanar lateral-gating architecture for MoS2 field-effect transistors fabricated directly on single-crystal SrTiO3. The exceptionally high permittivity of SrTiO3 enhances gate-channel coupling. The coplanar geometry eliminates the need for a separate gate insulator, reducing interface disorder. Owing to the quantum paraelectric nature, the SrTiO3 dielectric constant increases upon cooling, enhancing the gate coupling, leading to a decrease in threshold voltage and subthreshold swing an effect that contrasts with conventional FET architectures. The dielectric-free MoS2 surface, combined with enhanced electrostatic control, makes this architecture a promising platform for low-power two-dimensional electronics and cryogenic quantum devices.

We study income taxation when an AI agent implements economically relevant choices through a rule hidden from the government. Alongside unobserved productive ability, this hidden preference-to-execution mapping creates double unobservability: the same observable tax-base response can carry different welfare consequences. We call the resulting income welfare-opaque. Our constructions show that tax-base statistics can coincide while reform welfare effects differ, even when mechanical welfare weights are identical. We derive an optimal-tax condition that adds a response-weighted execution wedge to the familiar sufficient statistics. A higher marginal rate gains a corrective benefit under local over-execution and an additional cost under local under-execution. Observing the wedge identifies the welfare effect of a marginal reform at the prevailing schedule; bounds on it deliver bounds on that effect. A controlled laboratory compares 4,500 model runs across five AI engines. Faithful delegation selects the score maximizer in essentially all runs. Conflicted objectives produce heterogeneous responses: Claude largely preserves the score maximizer, GLM moves predominantly downward, and GPT-mini and Qwen show concentrated lower-tail increases. Qwen also makes substantial downward adjustments. Different engines locate their departures at different points and in different directions of the designed distribution. Explicit scores align model rankings; formula-based objective instructions yield more uneven agreement. Qwen shows a clear positive tax-by-objective interaction, but its direction does not generalize across engines and the pooled sign depends on its inclusion. The analysis identifies execution information as a complement to conventional tax-base statistics.

Structural similarity, the extent to which economic mechanisms respond alike to common shocks, matters for economic forecasting, policy transfer, and other decisions that rely on evidence from comparable settings. For mechanisms approximated by linear models, this idea has a simple geometric representation: The smaller the angle between their coefficient vectors, the more similarly they respond to the same shock. This paper introduces the cosine of this angle as a measure of structural similarity and develops the corresponding inference procedure. Applied to New Jersey's wage mechanism around the 1992 minimum-wage reform, the approach yields interesting evidence: Although the reform clearly affected the low-wage part of the wage distribution, the underlying mechanism linking worker characteristics to wages was no different from what would have prevailed without it.

Document parsing converts document images into structured content and requires reliable performance across diverse layouts and acquisition conditions. Yet training corpora are biased toward common document types and clean digital pages, while expanding coverage alone does not specify how to address a parser's remaining weaknesses. We present WeVisDoc, a two-stage data-centric framework for robust end-to-end document parsing. Stage I broadens semantic, structural, and appearance coverage through heterogeneous data and structure-preserving degradation synthesis. Stage II uses a held-out probe to measure the Stage I parser's residual errors within fixed visual-structural clusters. These diagnostics guide targeted data construction and reallocation of the target-token budget. WeVisDoc-4B achieves an Overall score of 95.38 on OmniDocBench v1.6 and a mean Overall score of 75.54 across the three PureDocBench tracks, ranking first among the compared end-to-end parsers in all four settings. Compared with Stage I, Stage II improves Overall scores for the 2B and 4B models on both benchmarks, with larger gains on the degraded PureDocBench tracks, including a 4.03-point gain for the 4B model on the Real Degraded track.

We study inhomogeneous long-range first-passage percolation on $\mathbb{Z}^d$ with edge passage times $\lVert x-y\rVert^\alpha\omega_{xy}/(V_xV_y)$, where vertex weights have polynomial upper-tail exponent $\gamma$ and edge noises have polynomial lower-tail exponent $\theta$ at zero. The model interpolates between long-range first-passage percolation and scale-free percolation and exhibits competition between reusable heavy-vertex hubs and pair-specific small-noise bridges. We conjecture an eight-regime phase diagram, organized into five growth phases governed by $q_{\rm hub}=d/\gamma$ and $q_{\rm edge}=d/\theta$. For $T_n=T(0,\lceil nx\rceil)$, we prove upper bounds of the conjectured order in every regime and matching lower bounds in phases I and II. Specifically, $T_n=0$ a.s. when $\alpha<q_{\rm hub}\vee q_{\rm edge}$, while $T_n=\Theta_{\mathbb{P}}(1)$ when $q_{\rm hub}\vee q_{\rm edge}<\alpha<2q_{\rm hub}$. In the edge-dominated intermediate regime, $T_n=O_{\mathbb{P}}((\log n)^{\Delta_{\rm III}+\varepsilon})$, where $\Delta_{\rm III}=\log 2/\log(2q_{\rm edge}/\alpha)$. The two power-law regimes satisfy $T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm hub}+\varepsilon})$ and $T_n=O_{\mathbb{P}}(n^{\alpha-2q_{\rm edge}+\varepsilon})$, and the linear regime satisfies $T_n=O_{\mathbb{P}}(n)$. All upper bounds are constructive, based on hub-chain and binary edge-bridge multiscale constructions.

The random-order interpretation of the Shapley value specifies both a terminal allocation and a payment path: each entrant is paid her marginal contribution at the time of entry. We embed this path in a finite dynamic entry problem in which a selected player may accept immediately or defer and allow another outsider to enter first. A flow is fully voluntary when immediate acceptance is optimal at every decision state. Although this criterion is defined through optimal continuation values, backward induction yields an equivalent system of linear inequalities in full-consent continuation values. The distinction between allocation and path is sharp in convex games: the Shapley allocation lies in the core, but the Shapley flow is fully voluntary only in additive games. Nevertheless, every efficient allocation admits a budget-balanced fully voluntary implementation when signed payments are available. We select a canonical implementation by minimizing distance from the Shapley flow, derive a reduced quadratic program for symmetric cardinality games, and study equal residual sharing, which selects the terminal allocation endogenously. With two player types, the selected allocation is an affine combination of the Shapley and equal-division allocations. Examples illustrate the effects of payment restrictions and a possible conflict between full voluntariness and coalitional stability.

Water masers provided powerful probes of shocked gas and protostellar activity during the early stages of star formation. Using VLA K-band observations of the 22 GHz water line, we report the first detections of water masers toward dense cores in five 70 ${\mu}$m infrared dark clumps, providing evidence for active star formation at an early evolutionary stage. We detected ten unresolved maser spots comprising thirteen velocity components, with peak flux densities of 13–499 mJy and deconvolved full widths at half maximum (FWHMs) of 0.51–3.54 km s-1. Eleven components had FWHMs below 1 km s-1, while their velocity offsets from the systemic velocities traced by the NH3(1,1) main line ranged from 0.83 to 50.73 km s-1. Four water maser spots were associated with 1.3 mm continuum emission, three of which were also associated with molecular outflows traced by CO 2-1 emission. No CO outflow signature was detected toward the remaining spots within the observed field of view, suggesting that water masers could trace deeply embedded protostellar activity whose outflows were not yet detectable in CO emission. For AGAL031, the non-detection of StokesV emission toward the strongest maser component yields an upper limit on the line of sight magnetic field strength on the order of a few hundred mG. Our results showed that star formation was already active in 70 ${\mu}$m infrared dark clumps and that 22 GHz water masers provided insights into early protostellar environments that remained difficult to probe with common star formation tracers.

Graph foundation models (GFMs) aim to learn transferable representations across severely heterogeneous graph domains. However, severe domain shifts in topology, graph scale, and feature semantics impede the construction of a unified, domain-agnostic representation space. To address this, we propose SCGFM-ART, a structure-centric GFM framework that aligns arbitrary graphs onto a shared relational atlas via Amortized Relational Transport (ART). The relational atlas serves as a universal coordinate system defined by a finite set of relational landmarks (bases), while ART directly predicts reusable, end-to-end graph-to-base transport plans, bypassing costly runtime Gromov-Wasserstein optimizations. Under this formulation, SCGFM-ART decomposes a graph into a unified representation: globally via its relational response coordinates relative to the atlas, and locally via its node-to-role structural correspondences. These correspondences project disparate node attributes into a canonical role space, resolving structural and semantic heterogeneity within a singular alignment interface. Rigorously modeling graphs and atlas bases as finite measured relational spaces, we establish coordinate fidelity bounds, prove stability under predicted transport plans, and derive an amortized coverage bound that guarantees our learning objective tightly surrogates ideal relational coverage. Benchmarked across 14 cross-domain graph- and node-level classification tasks, SCGFM-ART achieves state-of-the-art transferability, securing superior average ranks of 2.29 and 1.14, respectively. Topological perturbation analyses demonstrate that node-role transport retains fine-grained structural nuances beyond global coordinates. On real-world benchmarks, the amortized formulation yields 44.2 to 85.1 times faster frozen target-domain inference by avoiding iterative alignment at test time.

Group testing has been widely used to identify infected individuals with a limited number of tests, typically under the assumption of independent infections. Recent studies have exploited correlations among individuals, but often require additional information beyond the contact graph, such as community structures, interaction strengths, or detailed infection dynamics. Such information may be unavailable, incomplete or unreliable in practice. In this work, we assume that only the contact graph is known and develop a graph-aware group testing framework that exploits localized infection clustering in pooling design, fundamental limits of recovery, and decoding. Specifically, we propose an optimal transport-based pooling design that incorporates graph proximity and pooling constraints into a unified optimization framework. We prove that, under mild conditions, the proposed design eliminates uninfected individuals with higher probability than the Bernoulli pooling design, reducing the feasible search space for decoding. Then, we characterize the family of possible infected sets induced by localized infection clustering and derive necessary conditions on the number of tests required for exact recovery, revealing a lower testing requirement than that under the combinatorial prior. For decoding, we model the infection states of the population as a piecewise-constant graph signal and propose a graph total variation regularized decoder. We establish sufficient conditions for exact recovery under the Bernoulli pooling design in both noiseless and noisy settings, and show that O(Klog(n/K)) tests are sufficient under mild conditions in the noiseless case. Extensive simulations on synthetic and real-world networks demonstrate the effectiveness of the proposed framework and the benefit of exploiting graph-induced correlations in group testing.

The third Gaia data release (GDR3) includes the gdr3_rotmod catalogue of 474026 stars with variability that is attributed to magnetic activity analogous to that observed in the Sun, arising from rotational modulation induced by spots and faculae. These stars are therefore expected to exhibit flaring activity driven by magnetic reconnection processes. We aim to demonstrate that stellar flares can be reliably identified and characterised in the sparsely sampled \gaia\ photometric time series. We developed and validated a dedicated flare-detection pipeline that exploits the simultaneous multi-band photometry in the $G$, $G_{\rm BP}$, and $G_{\rm RP}$ bands. The algorithm identifies outliers associated with a concurrent increase in brightness and blueing of the stellar colour, and applies consistency criteria to distinguish genuine flare events from instrumental or calibration artefacts. Applying this method to the gdr3_rotmod catalogue, we detect 3217 flares occurring on 2818 stars out of 474026 analysed sources. All events are provided in a dedicated catalogue together with their photometric amplitudes. For a subset of 348 flares, we estimate an effective temperature using a black-body approximation, and we identify 29 hyper-flares with amplitudes $A(G)\ge0.75$ mag, which occur preferentially in M dwarfs. Despite the sparse temporal sampling of Gaia, our results demonstrate that its multi-band photometry and spectrophotometric information enable the robust detection of flares and the basic characterisation of their properties across an all-sky stellar sample. This work establishes the methodological foundation for future Gaia flare catalogues and provides a homogeneous, complementary view of stellar flaring activity alongside high-cadence missions such as Kepler and TESS.

Binary modular Laplacian dynamics is organized by repeated replication, growth, and collapse on dyadic scales. We study how isolated and periodically repeated non-binary updates modify this organization and whether they can sustain densely occupied geometric structures. Computational experiments across multiple finite seeds, neighborhood masks, inserted moduli, and periodic schedules show that constant prime moduli retain the expected \(p\)-adic replication hierarchy. A single non-binary insertion mainly acts as an effective-seed replacement: subsequent evolution remains binary-like, but may resume at an integer phase shift of the collapse–recovery cycle, with only modest finite-time densification. A qualitatively different response occurs for periodic schedules \([2,k,2^s]^\infty\), where \(2^s\) denotes \(s\) consecutive binary updates. Odd-modulus insertions can sustain long-lived, spatially coherent carpet-like regimes with comparatively high and stable occupation, whereas even-modulus insertions remain binary-like. Ternary insertion gives the broadest and most robust high-density response. For \(k=3\), density depends non-monotonically on the binary-tail length, with troughs near \(s=7,15,23\) that reveal phase-sensitive interaction with the binary epoch structure. Cross-modulus synchronization also occurs: for degree-four diagonal Neumann and von Neumann masks, \(k=5,7,9\) repeatedly collapse onto the same complete configuration at subsequent binary phases. A parity argument explains this loss of memory. Increasing seed extent generally raises occupation while reducing tail-length sensitivity. Overall, periodic modular insertions act as phase-selective, geometry-dependent mechanisms transforming \(p\)-adically organized replication into long-lived carpet-like dynamics.

We develop a new framework for proving the undecidability of first-order theories of structures of the form $\langle \mathbb{N}; +, P \rangle$, $\langle \mathbb{N}; <, f \rangle$, and $\langle \mathbb{N}; +, f\rangle$, where $P \subseteq \mathbb{N}$ and $f \colon \mathbb{N} \to \mathbb{N}$. It is based on the recent proof of Hieronymi and Schulz that the first-order theory of $\langle \mathbb{N}; +, \{2^n \colon n \in \mathbb{N}\}, \{3^n \colon n \in \mathbb{N}\}\rangle$ is undecidable, and capable of transforming various randomness results about integer sequences into undecidability proofs. We apply our method to a large class of integer linear recurrence sequences, as well as various special functions, in particular showing that the first-order theories of $\langle \mathbb{N}; +, \{u_n \colon n \in \mathbb{N}\} \cap \mathbb{N}\rangle$, $\langle\mathbb{N}; <, n \mapsto \max\{0,u_n\}\rangle$, and $\langle \mathbb{N}; <, \phi\rangle$ are undecidable, where $(u_n)_{n\in\mathbb{N}}$ is any integer LRS with exactly two non-repeated dominant roots satisfying a non-degeneracy assumption, and $\phi$ is Euler's totient function.

Tactile signals provide direct contact and force measurements that are essential for understanding physical interactions and enabling dexterous robotic manipulation. However, tactile sensing requires direct measurement at contact interfaces, making large-scale data collection reliant on intrusive, costly, and restrictive instrumentation. We present TouchSight, a monocular egocentric vision framework for dense full-hand contact force prediction that leverages 500 hours of pressure-glove recordings and extensive hand-object interaction (HOI) data. To address the appearance gap between gloved training data and bare-hand real-world scenarios, we construct TwinTouch-20H: 20 hours of paired visual data in which generative video models re-render gloved recordings as bare-hand observations against new backgrounds while preserving the original measured tactile labels. TouchSight predicts dense force from both gloved and generated bare-hand videos, outperforms prior contact prediction methods on OakInk2, qualitatively generalizes to natural bare-hand egocentric videos from unseen datasets, and improves consistently as glove supervision scales. These results demonstrate that dense tactile signals can be recovered from egocentric vision alone, without tactile instrumentation at capture time.

The linking number and coefficients of the Conway polynomial are two examples of finite type invariants. Goussarov-Polyak-Viro proved that any finite type knot invariant can be computed in a two step process, where the second step is referred to as the GPV map for the invariant. This paper computes the GPV maps for the linking number and low-degree coefficients of the Conway Polynomial. Chmutov-Khoury-Rossi gave one description of the GPV maps for coefficients of the Conway polynomial in terms of arrow diagrams and state-sum calculations. We take a much more grounded approach using fundamental linear algebra to describe the GPV maps in terms of a basis for the vector space of Gauss diagrams.

When aligning frontier models through post-training techniques, it is not possible to directly demonstrate all of the behaviours we want a model to exhibit in all possible deployment environments; our model must generalise outside of the post-training distribution. One proposed solution is alignment midtraining (AMT), which continues pretraining on large volumes of alignment-relevant documents to encourage generalisation in later stages of training. Despite the prominence of AMT as an alignment approach, there is limited public evidence for its effectiveness. To resolve this, we identify several assumptions around midtraining and evaluate them across scale: up to 110 billion-parameter models and 1 billion midtraining tokens. For instance, we study a scenario where post-training data is ambiguous between two possible motivations. We find that midtraining can steer the model's motivation in simple versions of this setting. However, the presence of a tiny fraction of finetuning data which suggests a competing motivation erases the effects of AMT. We also study scenarios in which we want an AI to follow a number of rules, but only demonstrate a subset of them. We find that demonstrations must be present either in midtraining or post-training datasets for these rules to be robustly learned. Based on these and other findings, we do not believe that there is sufficient public evidence for us to confidently state that midtraining can address the core difficulties inherent in aligning powerful AI systems.

Background: Code quality metrics are intended to measure latent properties of software source code. Although numerous code metrics have been proposed and used, their construct validity is rarely evaluated. Thus, the extent to which code metrics actually measure what they claim to measure is often unclear. Aim: Drawing from modern measurement theory, we investigate the construct validity of common class-level, object-oriented code quality metrics by identifying their factor structure using Exploratory Factor Analysis (EFA). The metrics were extracted from the Apache Maven project by three software tools: Designite, JHawk, and Understand. The factor structure was later verified using Confirmatory Factor Analysis (CFA) on 22 randomly selected open source projects meeting a predetermined eligibility criteria. Results: 24 code quality metrics that correspond to six constructs: Cohesion, In-Coupling, Out-Coupling, Size, Sub-Inheritance (related to subclasses), and Sup-Inheritance (related to superclasses) were revealed in the underlying factor structure. Ten metrics did not correspond to any known dimension of software quality and were removed in the EFA. Ten additional metrics exhibited low loadings in the CFA, suggesting their removal from the final measurement model. Size, Cohesion, Inheritance, and Coupling were the constructs retained, with subcategories identified for Inheritance and Coupling. Conclusions: Our results strongly support the construct validity of 24 code quality metrics. Coupling and Inheritance are revealed as multidimensional constructs, since they require measuring two different concepts, revealed as sub-categories in our analysis, and Complexity may be better explored in a multilevel model. Overall, our study demonstrates the value of applying modern measurement theory and latent variable modeling in validating software code quality metrics.

In the first half of the paper we revisit the question regarding MaxEnt probability distributions on spheres. We derive families of MaxEnt distributions on spheres in real and complex vector spaces with fixed expected Busemann function (energy). As a particular case, we deduce sub-families on canonical energy levels where inverse temperature equals the dimension of the sphere. In the second part we focus on the information manifold of canonical MaxEnt distributions on the sphere in the complex vector space. This manifold is isomorphic to the Bergman ball. We introduce the reproducing kernel on this manifold and use the RKHS theory to elaborate the model of cognition. In particular, we state the principle of minimal cognitive effort in RKHS and demonstrate its dual relationship with the MaxEnt principle for probability distributions on the boundary sphere.

Two-timescale stochastic approximation (TTSA) is a fundamental tool for analyzing coupled iterative algorithms in reinforcement learning, optimization, and stochastic control. However, finite-time guarantees for nonlinear two-timescale schemes remain difficult to obtain, especially under constant step-sizes. In this paper, we study nonlinear TTSA with step-sizes $\alpha\gg\beta$. Under standard stability, regularity, and Markovian noise assumptions, we upper bound the mean-squared error and the bias of both iterates around their limiting equilibria. Our bounds scale as $O(\alpha+\beta^2/\alpha^2)$, which we prove to be tight when $\beta\le\alpha^{3/2}$. The analysis separates the contributions of initial conditions, fast-timescale tracking error, Markovian dependence, and timescale coupling, thereby clarifying the origin of the $\beta^2/\alpha^2$ term. Our results reveal qualitative differences from the linear TTSA setting previously studied, showing that nonlinear dynamics introduce additional finite-time effects that are absent in the linear case.