Papers

We compute the on-shell closed, generalized Komar charges associated to an arbitrary Killing vector of the two 10-dimensional $\mathcal{N}=2$ supergravity theories which are the low-energy effective field theories of the type~IIA and IIB superstrings. We study different rewritings of this charge in terms of Page charges and forms with special properties that are used in the derivation of thermodynamical relations in black $p$-brane spacetimes. The, we show that, when the Killing vector is the supersymmetric Killing vector built as a bilinear of a Killing spinor, the momentum maps of the different fields of the theory are

Layout generation for real-world facilities is a challenging problem, requiring reasoning over irregular site boundaries, heterogeneous orientations, access-aware placements, and motion-planning feasibility. Yet, most existing layout benchmarks in the generative AI space target simpler placements over rectangular domains and rely on distributional metrics such as FID and IoU that reward conformity to dataset priors, thus discounting design innovation. Motivated by these gaps, we introduce ALPS-Bench, a benchmark of $1,000$ professionally annotated real-world facility layouts paired with an instance-specific scoring protocol grounded in a structured design manual. As a strong baseline for ALPS-Bench, we propose CANDO, a training-free multi-agent framework in which specialized agents iteratively refine layouts through a verification-grounded loop, concentrating reasoning on strategic spatial decisions. We demonstrate that CANDO surpasses state-of-the-art trained and LLM-based baselines on the widely adopted PubLayNet, RICO, and PKU-PosterLayout benchmarks, establishing cooperative agentic design as a broadly effective recipe for constraint-aware layout synthesis.

What should a macroeconomically relevant climate index contain? The composition is inherently ambiguous, and aggregation choices affect structural inference. We introduce the Assemblage VAR, which jointly estimates nonnegative aggregation weights and VAR parameters by maximizing the system likelihood-gain criterion, effectively outsourcing aggregation to observed macroeconomic dynamics. Two variants operate in component-space and rank-space, reweighting named subcomponents or emphasizing regions of the cross-sectional distribution. Applied to disaggregated U.S. climate data from the Actuaries Climate Index and NOAA, VARs using the assembled climate measures yield contractionary impulse responses that are substantially larger than those estimated using fixed-weight benchmarks. Weights emphasize high-wind variables and distributional tails over slow-moving components such as sea level.

Large language model (LLM) agents can improve their performance by reusing knowledge distilled from past interactions. However, curating new experiences into a knowledge bank that becomes more useful as it grows remains challenging. Effective knowledge accumulation should limit redundant overlap among entries and ensure that new knowledge contributes beyond what the bank already provides. Yet training a curator with Group Relative Policy Optimization (GRPO) on standalone task success can reinforce general guidance even when it duplicates existing knowledge. Therefore, we propose Knowledge Weaver, a reinforcement learning framework that trains a language model to curate reusable knowledge from agent trajectories. We couple feedback inspired by token-wise mutual information (MI) with marginal success rewards to guide knowledge accumulation. Together, these signals encourage the curator to preserve distinct information from experience and produce entries that improve task success when added to existing knowledge. Standalone success rewards also favor entries that are useful on their own. On ALFWorld and WebShop, Knowledge Weaver achieves mean success rates of 54.0% and 42.0% with k=10 retrieved entries, exceeding GRPO by 16.9 and 18.7 percentage points, respectively. Its knowledge banks also outperform the evaluated prompt-based and established banks, including human-written banks, in overall ALFWorld success rate and WebShop score with the executor frozen. Our codebase is available at https://github.com/LaoKuiZe/Knowledge-Weaver.

On-shell forms are differential forms on the Grassmannian which arise in particle physics. They are defined using bipartite graphs with $n$ distinguished “boundary” vertices. Mathematical investigation of on-shell forms has largely focused on the case where the graph is planar, in which case one can utilize combinatorial tools pioneered by Postnikov in the study of the totally nonnegative Grassmannian. In this article, we investigate on-shell forms for arbitrary graphs. We first discuss how to extend various tools from the planar case to arbitrary graphs. We then prove a determinantal formula for a class of on-shell forms which first appeared in physics literature. We explain the relation between this class of forms and the hypertree divisors of $M_{0,n}$, introduced by Castravet–Tevelev.

Many Quantum Field Theory models, either designed to improve the ultraviolet behavior via higher derivative terms or aimed at providing unified descriptions of Gravity, inevitably introduce negative-energy 'ghosts' states. These states raise fundamental questions regarding the stability of the theory. In this study, we investigate the time evolution of simplified quantum mechanical models consisting of two coupled normal and ghost subsystems. Although the full spectrum of these models is unbounded from below and, therefore, lacks a true ground state, we focus the time evolution of a 'would-be ground state': a zero-energy eigenstate of the decoupled system. By analyzing its metastability under interactions, we gain insights into the dynamics of ghost-like states. Contrary to the standard paradigm, our results demonstrate that, across a broad range of couplings, the would-be ground state does not decay and remains dynamically stable over long times. This provides evidence towards resolving the stability problems of Hamiltonians that are unbounded from below. We also discuss the connections with the corresponding classical and semiclassical dynamics.

This paper presents a video-to-model framework for automatically modeling the motion of a deformable suture thread from an input video. We utilize a recently developed CBF–CLF–QP numerical model that simplifies the characterization of deformable string motion through the selection of a small number of parameters. A perception module first localizes and tracks the thread in video, producing an ordered sequence of thread nodes. The observed thread motion is then processed by a spatio-temporal CNN network that estimates the effective parameters of a structured CBF–CLF–QP model. These parameters are used to simulate the thread under a user-defined needle velocity input. Experiments using unseen thread configurations and motion demonstrate that the framework can reliably reconstruct the thread behavior from video, automatically configure the structured model, and reproduce the expected thread motion with low tracking error. The proposed approach reduces the need for manual parameter tuning and provides a step toward automatic video-based modeling of deformable linear objects.

Accurate forecasting models are usually large, expensive to update online, and fixed in architecture once trained. We apply ONE-NAS, an online neuroevolutionary architecture search that evolves a population of small recurrent networks as each window of data arrives, to daily cross-sectional stock return prediction, and pilot it on a host and endpoint pipeline: the host runs the search and ships each generation's champion genomes over TCP/IP to a Raspberry Pi 4B, which predicts online. On the Pi a single champion predicts a 50-stock window in 24.6~ms and the ensemble of 40 island champions in 556~ms, far inside the daily decision cycle. On four panels of US mid-cap equities over 2022–2024, reading the population as a rank-mean ensemble of island champions returns $+27.5\%$ net of realised transaction costs, against $+11.3$ to $+14.8\%$ for online LSTM, online GRU and monthly-retrained LSTM baselines and $+4.5\%$ for the single best genome used in prior ONE-NAS work.

Physical reservoir computing exploits the nonlinear dynamics of physical systems to process time-dependent data with greater energy efficiency than conventional machine learning approaches. However, physical reservoirs have fixed intrinsic response timescales, whereas real-world signals combine deterministic and stochastic components across multiple timescales. Here we show, using a nanoporous niobium oxide reservoir, synthetic noisy signals and cryptocurrency-price volatility, that the relationship among noise correlation time, reservoir memory and forecast horizon determines whether correlated noise is filtered or predicted. Noise varying faster than the relevant reservoir memory and forecast horizon is averaged by the reservoir, whereas the temporal structure of slower-varying noise is sufficient for algorithmic forecasting. We introduce the reservoir memory horizon and forecasting regime index to distinguish these operating regimes. These contributions demonstrate that timescale matching can guide the encoding of input time series and development of physical reservoir architectures that filter, analyse and predict stochastic signal components across distinct temporal scales.

We give a deterministic polynomial-time $1.6908$-approximation for Maximum Weighted $3$-Set Packing, breaking the $\sqrt3$ locality-gap barrier of squared-weight local search. The approximation ratio for this problem progressed from Berman's $2$ [Ber00] to Neuwohner's $2-\frac{1}{63{,}700{,}992}+\epsilon$ [Neu21]. Thiery and Ward then obtained $1.786$ [TW23], while Thiery subsequently improved the bound to $1.761+\epsilon$ and finally to $\sqrt3 \approx 1.732051$ through a layered exchange analysis [Thi23]. Thiery also proved that $\sqrt3$ is a locality-gap lower bound for the squared-weight objective even with exchanges of arbitrary size. Our algorithm performs in two phases and combines two objectives. Phase~I computes a bounded-exchange local optimum for the squared-weight potential and analyzes it through Thiery's layered framework, while strengthening the terminal analysis by preserving internal tree-edge slack for nonsingleton components and exploiting the incidence structure of $3$-sets for final singletons. This yields an augmented structural inequality with residual positive claw gain under the original objective. Phase~II switches to the original objective and recovers sufficient residual gain through an auxiliary weighted $9$-Set Packing instance. A covering argument transfers the structural bound through the high-girth lift used only in the analysis.

We study the tableaux monoid from the perspective of rewriting theory. We first construct a finite quadratic convergent presentation based on column generators and prove that it yields a biautomatic structure. We then extend this presentation to a finite coherent presentation in the sense of polygraphic rewriting. These results provide canonical normal forms, yield homological and algorithmic consequences, and produce a higher-dimensional presentation that can be used to describe actions of the tableaux monoid on categories.

Ferromagnetic spin models on random regular graphs, such as the Ising and Potts models, have been studied extensively. Several equivalent variational and recursive representations of their limiting pressure are known, and these representations can be used to establish agreement between the quenched and annealed pressures. We give a concise overview of these representations and highlight the connections among the different formulations. We then derive analogous representations for the exponential growth rate of the second moment of the partition function, and prove that it has the same order as the square of the annealed pressure, up to a subexponential error. Combined with concentration of the quenched pressure, this identity yields a transparent second-moment proof of quenched–annealed agreement on random regular graphs. By bringing these representations and arguments together in a unified presentation, we aim to provide an accessible entry point to the literature on ferromagnetic spin models on random regular graphs.

A theoretical understanding of multi-head self-attention is fundamental to the study of modern neural networks. Using random matrix theory, we establish Gaussian equivalence for multi-head self-attention: replacing softmax attention with rescaled scores plus Gaussian noise preserves the limiting spectral law of the centered output. This equivalence also covers value and output projections that depend on the keys. The resulting laws separate the effects of head allocation and projection widths, and distinguish spectrum-preserving across-head sharing from within-head key–value dependence.

Massive multiple-input multiple-output (MIMO) achieves a high spectral efficiency (SE) by employing a large number of receive antennas in the uplink. Employing many receive antennas requires a large number of analog-to-digital converters (ADCs) and a high fronthaul data rate. Low-resolution ADCs can reduce power consumption and cost, while fronthaul quantization can reduce the required fronthaul data rate, at the expense of quantization distortion at both stages. In wideband orthogonal frequency-division multiplexing systems, ADC quantization is performed in the time domain, while fronthaul quantization may be applied after the discrete Fourier transform to transmit only active subcarriers. In this study, we investigate the achievable uplink SE over fronthaul links under double quantization and imperfect channel estimation. We derive a linear minimum mean-squared error channel estimator from the double-quantized pilot observations and obtain an achievable SE using the use-and-then-forget bound. Numerical results demonstrate that the ADC and fronthaul quantization resolutions have comparable impacts on the achievable SE. When the two resolutions differ, the lower resolution becomes the dominant factor limiting the SE. Furthermore, our numerical results show that six-bit ADC and fronthaul quantization achieve more than 90% of the SE attained in the ideal case.

We study the new eigenvalues of a fixed point interaction on the cubic three-torus. Their mean-one consecutive spacings converge to a probability law independent of the interaction parameter. We characterize this law by the consecutive zeros of a random meromorphic function built from three-squares congruence densities. Its small-gap distribution has the form $s^5\Psi(\log_2(s^{-2}))+o(s^5)$, where $\Psi$ is continuous, positive, and one-periodic. The proof combines periodic mean-square approximation, a weighted Hilbert-transform estimate, and Fourier estimates that retain the interactions between primes.

We calculate the Rayleigh-Schrodinger perturbation series for the ground-state energy and the effective mass of the three-dimensional Frohlich polaron through fourth order in the coupling constant alpha. The third- and fourth-order coefficients of the effective mass are new. Because the Frohlich interaction is a form-bounded perturbation of an isolated nondegenerate ground state at fixed total momentum, the series has a nonzero radius of convergence, and we analyze it with the ratio and Pade methods appropriate to convergent series. The coefficients are strikingly regular and are described by a simple power-law singularity at alpha near 8.5 with an exponent near 1.4. The couplings at which successive truncations of the inverse mass vanish, the first of which is the textbook breakdown value alpha = 6, are slowly converging estimates of this radius. Pade approximants built from the weak-coupling coefficients alone agree with diagrammatic Monte Carlo results for the mass to better than one percent for alpha up to 5. Exact results on the analyticity of the ground state and on the strong-coupling asymptotics imply that the coefficients cannot keep a constant sign, although all known coefficients do, so that the apparent singularity must be a complex-conjugate pair close to the positive real axis. It marks the crossover to strong coupling: the weak-coupling series locates this crossover accurately but cannot be continued through it.

We study a nonlocal stochastic optimal control problem in which the control acts through the elliptic smoothing operator $\mathcal{S}=(I-\Delta)^{-1}$ on $\mathbb{R}^d$. The state is described by the law of a controlled diffusion, equivalently by a controlled Fokker-Planck equation with coefficients depending on $\mathcal{S}u$. We prove the existence of an optimal control by the direct method in the calculus of variations. We then derive a Pontryagin-type maximum principle by spike variations, relying on sharp properties of the operator $\mathcal{S}$. The result yields a pointwise minimisation rule for the optimal control. We discuss the meaning of the optimal control problem in the case of population dynamics.

In the development of photovoltaic materials, computational simulation has become a important tool for reducing the time and cost of exploring novel materials, as well as providing insights that help drive improvements in efficiency through increasing fundamental understanding. We present Solphin, a Python package for the generation, post-processing and analysis of photovoltaic calculations from periodic solid state Density Functional Theory. We include the ability to calculate the photovoltaic Figure of Merit, detailed balance and spectroscopic limited maximum efficiency, alongside other analysis methods. Solphin has been built to improve the access, reproducibility and ease of evaluating the photovoltaic potential of a material in an efficient and user-friendly manner.

Tensor-ring (TR) decompositions provide a natural representation of periodic systems but are difficult to optimize because the closed geometry prevents a global canonical form and leads to costly, ill conditioned environments. Existing periodic DMRG methods alleviate this difficulty by compressing long environments to a low-rank representation, reducing local operations to $\mathcal{O}(p\chi^3)$, where $p$ is the retained environment rank. Here, we extend this approach to an efficient two-site Ring-DMRG algorithm and improve its numerical robustness through appropriate gauge transformations and a generalized Davidson solver. A central challenge in the two-site formulation is the truncation step, which must account for the surrounding environment. When this environment is sufficiently separable, a suitable change of frame reduces the truncation to an ordinary SVD. When it is not separable, we instead use an alternating optimization that retains the full environment. The computational advantage of tensor rings over tensor trains depends on the scaling of the required environment rank $p$ with bond dimension and system size. For critical systems, the divergent correlation length makes long-range observables remain sensitive to the periodic geometry as the thermodynamic limit is approached. In this regime, the environment rank required to reach a fixed accuracy in the observable does not grow with the system size. Ring-DMRG therefore retains the same cubic scaling with bond dimension as standard DMRG, but at a substantially smaller bond dimension. The low-rank environment construction also provides a systematic generalization of belief propagation (BP), with standard BP recovered in the rank-one limit.

We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems. If the dominant root $\varphi$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\varphi^{-m}$. In the Zeckendorf case this resolves a recent conjecture. The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus. For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words. The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency. This proves the Tribonacci prime-number theorem suggested by Drmota–Müllner–Spiegelhofer. The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.

Language-queried audio source separation (LASS) enables extracting any sound source using natural language. However, adapting LASS models to application-specific sound classes is challenging due to noisy training data and limited semantic coverage of the CLAP-based control signals. We propose a framework comprised of an automated data engine for training-data curation and a two-stage optimization process for class-specific CLAP control signals. Our objective evaluations across seven sound classes show that data refinement and control signal optimization consistently improve source separation performance. Subjective evaluation with 17 participants further demonstrates perceptual improvements of the model trained with optimized control signals over baseline and similar commercial models.

Understanding the symmetry requirements that permit specific phenomena or physical effects is crucial in condensed matter physics for identifying candidate materials that exhibit one or more of these effects. To this end, screening tensor properties based on (magnetic) point-group symmetries is of particular interest. Here, we present the PythMPG code package, a tool designed for this purpose. Inspired by the MTENSOR utility of the Bilbao Crystallographic Server, PythMPG performs symmetry analysis in a self-contained way to determine which tensors, characterized by their Jahn symbols, are allowed for each magnetic point group and how many symmetry-allowed independent components each tensor possesses. We describe the structure of the code and present some illustrative examples.

Understanding the extent to which measured synaptic wiring determines computation remains a central challenge. Here, we couple the proofread adult Drosophila melanogaster connectome to an anatomically faithful model of its eye. Visual information is inputted in the eye model, then passed to the connectome, and finally read from a Kenyon-cell-centered linear decoder. This creates a connectome-only model in which the anatomical graph and eye geometry are fixed and only scalar synaptic gains and neuronal thresholds may be learned. The model supports multitask vision, including color discrimination, shape classification, and numerical discrimination that follows a ratio-dependent scaling characteristic of approximate number perception. To test whether precise connectivity is consequential under wiring economy, we compare the biological graph to randomized ensembles that increasingly preserve biological synaptic constraints. At matched wiring cost, the biological network consistently yields higher accuracy, whereas less constrained rewiring surpasses it at the cost of inflated wiring. These findings indicate that the measured connectivity and eye geometry jointly set efficient operating points for visual computation.

In this paper, we establish two nonasymptotic Berry–Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains the classical factor $d^{1/4}$ in the contribution of the independent sum, where $d$ is the dimension, while controlling the remainder through its size and its sensitivity to replacing a single observation. The second bound expresses the contribution of the independent sum in terms of fourth moments and can allow the dimension to grow faster with the sample size. For sums of independent random vectors, it removes the logarithmic factor from an existing fourth moment bound without imposing additional moment assumptions. As applications, we apply these results to Polyak–Ruppert averaging for nonsmooth stochastic approximation, temporal difference learning with linear function approximation, and multivariate $U$-statistics. The resulting bounds provide explicit Gaussian approximation errors and sufficient conditions under which these errors converge to zero as the dimension grows with the sample size.

Mutual information (MI) provides an objective for suppressing or encouraging statistical dependence in implicit generative models. However, direct MI evaluation is challenging in implicit models due to typically intractable densities. A remedy is estimating the generator gradient from the difference between conditional and marginal scores. This score difference can, in turn, be estimated by differentiating a log density ratio learned through classification. This construction nevertheless faces two difficulties: (i) singular distributions need not admit the required score functions, and (ii) poor overlap can hinder density-ratio estimation. We therefore introduce Spread Mutual Information (SMI), a weighted integral of MI across noise levels obtained by applying a common spreading kernel to the generated variable. Gaussian spreading yields smooth, strictly positive conditional and marginal densities, extending the gradient construction to distributions that may originally be singular. Across a variaty of experiments, SMI consistently achieves effective dependence control among MI-based methods and remains competitive with established task-specific approaches.

Hybrid force/motion control requires knowledge of the interaction force and of a task frame defining the force- and motion-controlled directions. These quantities are usually obtained from force/torque sensing or model-based residuals, often assuming also a nominal environment model. This work addresses the online estimation of the contact force and a possibly time-varying task frame using only soft optical tactile sensing, under the assumption of locally planar contact with a negligible contact moment. The proposed method maps a single image of the deformed elastomer of a soft optical tactile sensor to observable contact variables: indentation depth, two surface-to-sensor tilt angles, and 3D contact force, each with a per-sample uncertainty estimate. The mapping is learned through a self-labeling acquisition procedure, in which a manipulator imposes controlled contacts while an auxiliary Force/Torque sensor is used offline to provide ground-truth labels. The tactile measurement is then fused with robot proprioceptive data in an Extended Kalman Filter, producing a continuously updated estimate of the contact task frame and of the interaction force. Control experiments with a DigiTac sensor mounted on a UR10 manipulator demonstrate closed-loop contact force regulation against a flat rigid board in linear and angular motion by a human operator, with touch as the only exteroceptive feedback.

We show that a knot in the 3-sphere with a nontrivial Jones polynomial of span at most 14 admits no purely cosmetic surgery. This extends a result of the first author for span at most 11. We determine all Laurent polynomials of span 15 satisfying the conditions on the Jones polynomial used in the proof. These polynomials form an infinite family. We also show that a knot of braid index at most 4 whose Jones polynomial has odd span admits no purely cosmetic surgery.

Effective long-range propagation remains a central challenge in graph neural networks, as increasing a model's propagation depth does not guarantee that distant nodes effectively influence each other. Sheaf neural networks enrich graph propagation through matrix-valued transport between stalks; still, this expressivity alone does not automatically imply effective long-range communication. We introduce ONDA, a long-range graph learning framework based on operator-valued information waves. Stalk-valued representations evolve through second-order dynamics governed by learned sheaf transport operators, combining wave-like propagation with expressive local geometry. We characterize long-range influence through a stalk-wise sensitivity analysis and show that the cross-influence never vanishes. Across long-range propagation, severe graph bottlenecks, graph transfer, and heterophilic benchmarks, ONDA consistently improves over scalar wave propagation, diffusive sheaf baselines, and state-of-the-art models, demonstrating the benefit of coupling wave dynamics with matrix-valued transport.

We study continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals $1/n$, where $n$ is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved $5$-cube, which mixes uniformly at time $\pi/4$. With the order $4\theta^2$ read literally, its parity condition also excludes the Clebsch graph and includes the parameters $(36,14,4,6)$, for which no time law gives uniform average mixing.

Motivated by the recent HAL QCD lattice results on the $D$ - $D^*$ potential at nearly physical point $m_\pi=146.4$ MeV, we theoretically examine effects from the two-pion exchange in (semi-)long-range parts of the $DD^*$ system. We employ the framework of heavy-meson chiral perturbation theory with heavy-quark spin symmetry. The pion exchanges up to next-to-next-to-leading order (N$^2$LO) are taken into account. In order to separate the (semi-)long-range contributions, we apply the dispersion-relation method to one-loop diagrams. As a result, we find that the lattice data on the $D$ - $D^*$ potential in the regime of $0.5\, {\rm fm} \lesssim r$ is reproduced by adjusting the unknown $O(p^2)$ couplings. In particular, it turns out that contributions from isospin-independent N$^2$LO triangle diagrams play a central role in reproducing the lattice data of the form $V^{DD^*}\sim \left({\rm e}^{-m_\pi r}/r\right)^2$ in $1.0\, {\rm fm} \lesssim r\lesssim 2.0\, {\rm fm}$. We also present predictions of other $D^{(*)}$ - $D^{(*)}$ potentials, where the similar two-pion exchange tail is predicted. Our findings provide useful information on the (semi-)long-range regime of the $D$ - $D^*$ potential focusing on two-pion exchanges.