For $p,q\in\mathbb{N}$ and $α,β\in\mathbb{R}$, we investigate the family of improper integrals \[\int_0^\infty\frac{(\cosαx-\cosβx)^p}{x^q}dx.\] We establish a complete classification of the parameter ranges $(p, q; α, β)$ for which the integrals converge or diverge, and we derive explicit closed-form evaluations in all convergent cases. The analysis also reveals a family of combinatorial identities arising naturally from coefficients in the trigonometric power expansions. As a further application of the same method, we study an analogous class of integrals involving powers of sine differences. This extends the work of Laoharenoo and Boonklurb in 2022.

Almost every known planet host will evolve into a white dwarf, and the surviving planetary material will continue to orbit this stellar remnant. Asteroids perturbed onto star-grazing orbits will become disrupted, forming an accretion disk which causes "enrichment" of the otherwise pure hydrogen or helium atmosphere. Measurements of these photospheric abundances give detailed insights into the interior compositions of exo-planetesimals with an accuracy not possible for intact exoplanets around main sequence stars. This method has revealed the diversity of rocky material in our solar neighborhood, including primitive, chondritic planetesimals, fragments of planetary cores, and even analogues of Kuiper belt objects. The planetesimal abundances can be used as an input to interior structure models. The far-ultraviolet is a key wavelength range for this field because it contains strong transitions for almost every element of interest, many of which are undetectable using ground-based optical spectroscopy. Without the FUV, we will no longer have access to the C, N, P, S content of exoplanetary bodies and thus will no longer be able to probe how volatiles interact with refractories, which is crucial to understanding planet formation-and even the origin of life. The medium resolution and high sensitivity of COS on HST has been indispensable in determining the compositions of dozens of exo-planetesimals. However, the only two medium resolution FUV-capable spectrographs are currently onboard HST, with no plans for replacements until the 2040s. An extension to the HST mission is critical for the field of white dwarf planetary systems, because the loss of FUV capability would leave us blind to volatiles. Boosting the orbit of HST would allow us to measure volatile abundances, determine the rocky planetary occurrence rate, investigate differentiation, and probe for photospheric abundance variability.

Young solar-like stars are efficient generators of magnetic activity, superflares, coronal mass ejections (CMEs), and stellar energetic particles. These phenomena drive the early evolution of stars and shape the habitability of exoplanets. The Hubble Space Telescope (HST), with its unmatched far-ultraviolet (FUV) and near-ultraviolet (NUV) sensitivity, provides a uniquely powerful window into these processes one that no current or near future facility can replicate. This white paper articulates four interconnected science questions that require Hubble continued operation and targeted observing programs over the next 10 to 15 years, enriched by new multi-wavelength insights from deep X-ray surveys of open clusters. We describe required instrument capabilities, critical synergies with contemporaneous missions (JWST, Chandra, XMM Newton, TESS, and the Nancy Grace Roman Space Telescope), and the fundamental role Hubble observations will play in calibrating and informing the design of the Habitable Worlds Observatory (HWO). We advocate for large scale coordinated campaigns targeting young solar-like stars as the highest priority science program for the coming decade.

A rainbow is a captivating natural phenomenon resulting from the refraction, dispersion, and reflection of sunlight within water droplets. Traditional classroom demonstrations often focus on qualitative explanations of the formation of rainbows using prisms or water bowls. This study presents a simple experimental approach to analysing the process of rainbow formation through quantitative analysis using a cylindrical glass filled with water, graph paper, and three semiconductor laser sources emitting red, green, and blue light. By measuring the angles of minimum deviation for different wavelengths, we have found that the experimental values closely match the theoretical predictions. This method offers a hands-on, cost-effective approach to enhance students' understanding of the physics behind rainbows.

This is an incomplete attempt to show that the upper bound of $\lesssim n^\frac{4}{3}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$ points and $n$ lines that attain the sharp bound of the Szemerédi-Trotter point-line incidence bound.

When does a crisis-induced surge in radical-party support fade away, and when does it become a durable realignment? We address this in a mathematical sociology threshold model on a conserved population. The baseline admits a global classification through a Perron--Frobenius threshold. Adding a crisis-induced disengagement compartment, we separate state shocks (which alter the current state) from structural shocks (which alter parameters). State shocks affect transients but cannot move the long-run attractor; durable realignment requires structural threshold-crossing. We derive a critical shock amplitude and a finite mobilisation-window bound, and show that cumulative structural shifts can produce staircase realignment. A stylised illustration uses German federal elections, 2013--2025.

White dwarf stars, the endpoint of stellar evolution for 97% of stars in our Milky Way, offer a unique and powerful window into the bulk elemental composition of rocky exoplanetary bodies. Up to 50% of single white dwarfs are observed with photospheric metal lines from accreted exoplanetary bodies (called 'polluted' white dwarfs), and spectroscopic observations reveal the bulk composition of this material. High-resolution (R>15,000) UV spectra are essential for detecting many elements present in the material, such as the volatile elements imperative for habitability studies (C, N, O, P, S) and key rock-forming elements required to constrain interior structure (e.g. Fe, Si, Mg, Al, Ni). HST, through its COS and STIS spectrographs, remains the only facility capable of performing this science in the near future. Looking to the next decade, the scientific case for continued HST UV observations of polluted white dwarfs is compelling on three fronts (i) as a standalone to enable the bulk composition of exoplanetary material to be measured in a statistically significant sample, (ii) as essential groundwork for the Habitable Worlds Observatory (HWO), and (iii) in a powerful synergy with JWST, to enable characterization of the bulk mineralogy and bulk elemental composition of exoplanetary material. This white paper argues that continued UV spectroscopic capabilities with HST is a high-return investment for white dwarf and exoplanet science, and preserving and prioritizing HST's UV capabilities through at least 2035 is crucial to maximize the scientific return from HST, JWST, and HWO.

We prove sharp Heron-type majorization inequalities for two quadratic matrix expressions associated with the spectral and Kubo-Ando geometric means. For the spectral geometric mean cross term, we show that \[ λ\bigl(a^2A+b^2B+c(A\natural B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr), \qquad 0\le c\le 2ab, \] where $W_{a,b}(A,B)$ is the weighted Bures-Wasserstein expression. The coefficient $2ab$ is sharp, and at this endpoint the weak majorization becomes majorization. For the Kubo-Ando geometric mean, we prove the direct comparison \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr). \] This settles, in the two-variable setting, Bhatia's question of whether the Heron-type norm inequality of Bhatia-Lim-Yamazaki admits a weak-majorization refinement. More precisely, we prove \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl((aA^{1/2}+bB^{1/2})^2\bigr), \] and consequently obtain the corresponding inequality for all unitarily invariant norms.

We show explicitly that radiation from a uniformly accelerating charge escapes outside Rindler wedge, while within Rindler wedge there is no flux through infinity, neither in the Minkowski frame nor in the Rindler frame. This remains true despite the fact that the coordinate transformation between the Rindler and Minkowski frames is not trivial at infinity.

Mars provides a critical analog to once habitable exoplanets that have since lost their surface liquid water. The current atmospheric state of Mars retains the chemical fingerprints of that transition, including isotopic signatures of atmospheric escape and climate evolution. As the closest accessible example of a terrestrial world with definitive evidence for once supporting liquid water on its surface, Mars presents a unique opportunity to test hypotheses about planetary habitability and atmospheric evolution in a spatially and temporally resolved way.

We review recent results showing that, within the framework of quantum field theory in curved spacetime, the semiclassical energy-momentum tensor of the neutrino flavor vacuum fulfills the equation of state of dust and cold dark matter. By considering spherically symmetric spacetimes in the weak field approximation, the flavor vacuum is shown to contribute as a Yukawa correction to the Newtonian potential. We discuss how this modified potential provides a mechanism to account for the flat rotation curves of spiral galaxies. In this perspective, neutrino mixing is presented as a viable contributing factor to the dark matter content of the universe.

We investigate a ModMax black hole surrounded by perfect-fluid dark matter within the framework of Lorentz-violating Kalb-Ramond gravity. The model combines three physically distinct contributions: nonlinear electrodynamic corrections from the ModMax sector, Lorentz-symmetry-breaking effects induced by the background Kalb-Ramond field, and environmental modifications associated with the surrounding dark matter fluid. We obtain the corresponding static and spherically symmetric black hole geometry and analyze how the charge, ModMax parameter, Kalb-Ramond coupling, and dark matter parameter affect the horizon structure and thermodynamic behavior. In particular, we study the Hawking temperature, entropy, heat capacity, and Helmholtz free energy, showing that the combined effects of nonlinear electrodynamics and Lorentz violation may shift the extremal configuration, modify the thermal stability regions, and generate nontrivial phase behavior. The perfect fluid dark matter contribution introduces an additional logarithmic correction to the geometry, becoming especially relevant at intermediate radial scales. Our results indicate that ModMax electrodynamics can effectively screen the electric sector, while the Kalb-Ramond parameter amplifies the geometric deformation and changes the thermodynamic response of the system. These features suggest that black holes in Lorentz-violating backgrounds surrounded by dark matter provide a useful arena for probing deviations from standard charged black-hole thermodynamics.

The present paper reports a study of a unified cosmological scenario in the framework of holographic f(Q) gravity, where, in a single theoretical setup, both the early inflationary epoch and the late-time accelerated epoch are studied. Considering f(Q) = $ζQ^n$, we reconstruct the Hubble parameter in the presence of Barrow holographic fluid and study the inflationary behaviour through the slow-roll parameters, scalar spectral index $n_s$, and tensor-to-scalar ratio r. The obtained inflationary predictions are found to be consistent with the latest Planck 2018 observational constraints, with a very small value of the tensor-to-scalar ratio. In the next phase, we extend the study by including the matter sector. The Chevallier-Polarski-Linder (CPL) parametrization is used to connect the theoretical model with observational cosmology. Using combined Cosmic Chronometer (CC) and Baryon Acoustic Oscillation (BAO) datasets, the study constrains the model parameters through Markov Chain Monte Carlo (MCMC) analysis. From the observational results obtained this way, the study concludes that at low redshifts, the holographic f(Q) model considered here remains compatible with the standard LambdaCDM model, while mild deviations are observed at higher redshift. We have also performed the AIC and BIC analysis and commented on the goodness of fit in comparison with the LambdaCDM model. Hence, the present framework provides a viable unified description of inflation and late-time cosmic acceleration within holographic f(Q) gravity.

Future missions to Mars and Venus will make use of aerobraking and aerocapture in order to gain mass through the saving of fuel at planetary arrival. So far only aerobraking has been demonstrated, if the Mars Premier project has paved the way for aerocapture, no demonstration was performed due to the project interruption. The use of these techniques induces additional constraints for planetary probes, since additional heating and mechanical loads have to be carefully managed. Moreover, aerocapture requires a high level of accuracy for the Guidance Navigation and Control aspects, since a pass at an altitude of the atmosphere with a different density could lead to the vehicle destruction. This document surveys the existing state-of-the-art on inflatable devices (including ballutes, sails, or inflatable heat-shield capsule) for orbital manoeuvres at planetary arrival.

We investigate the impact of braneworld theory on the astrophysical environment surrounding a black hole. The black hole is sourced by localized matter from the bulk, which could describe both regular and singular (Schwarzschild) black hole. Employing an Einstein cluster description for the environment, we find that the anisotropic nature of the cluster, coupled with finite brane tension, leads to a weakening of gravity due to the quadratic and nonlocal corrections to the effective four-dimensional field equations. Consequently, this effect prevents horizon formation within the environment. Applying current constraints on the brane tension derived from neutron star observations, we demonstrate that this effect is particularly relevant for sub-stellar mass black holes embedded in compact environments. Furthermore, we investigate the optical signatures of finite brane tension in this scenario, specifically focusing on the black hole shadow and Einstein ring radii. We show that the Einstein ring radius decreases with a smaller brane tension, whereas the black hole shadow radius increases--somewhat contradicts the weakening gravity effects. Ultimately, these two observables may jointly serve to constrain the value of the brane tension in a very specific astrophysical scenarios.

The operational reliability of a high performance marine vessel depends critically on the health of its marine propulsion systems, which are increasingly subjected to diverse operational loads and environmental stressors. This paper proposes a robust mathematical framework for non-linear state-space forecasting of marine engine parameters using adaptive-window multi-particle stochastic differential equations. Traditional time-series models such as Vector Autoregressive Integrated Moving Average, often fail to capture the inherent stochasticity and transient dynamics of complex systems due to their reliance on fixed-window linear assumptions. To address this, we develop a dual-layered estimation approach: first, an adaptive lookback mechanism dynamically adjusts the learning window size based on the instantaneous drift magnitude, ensuring responsiveness during non-stationary regimes. Second, a Multi-Particle ensemble is evolved via Euler-Maruyama discretization, where each particle trajectory represents a stochastic realization of the system state. To refine the ensemble mean and mitigate the "noise-chasing" behavior of raw estimators, a Girsanov transform induced change of probability measure is implemented, assigning higher probabilistic weights to particles that align with the physical drift. Theoretical evaluation and empirical benchmarking demonstrate that the proposed adaptive SDE framework significantly outperforms classical statistical baselines in multi-step prediction stability and computational efficiency. The model provides a scalable, "grey-box" solution for real-time risk quantification in systems characterized by high-frequency volatility and non-linear transitions.

Deploying large Transformer-based vision models on resource-limited mobile devices at network edge is severely constrained by hardware limitations and dynamic wireless environments. While federated learning (FL) enables collaborative training without sharing raw data, strictly local fine-tuning of such massive models remains computationally prohibitive for edge devices. Split federated learning (SFL) alleviates this burden by offloading deep layers to the edge server, yet it suffers from heavy communication overhead when transmitting high-dimensional activation tokens. To address this bottleneck, we propose ST-SFLora, a semantic token-based split federated LoRA fine-tuning framework. We introduce a new metric, \emph{Semantic Transmission Efficiency} (STE), to balance semantic retention and transmission cost. Based on STE, we formulate a joint resource optimization problem that dynamically determines token selection, uplink bandwidth allocation, and transmit power under latency and energy constraints. The resulting mixed-integer nonconvex problem is efficiently solved via an alternating algorithm. Experiments on multiple benchmarks demonstrate that ST-SFLora achieves the lowest client-side resource consumption among baselines while delivering a favorable trade-off between communication efficiency and model performance.

For business process modeling, we can choose between graph-oriented and block-oriented languages. Block-oriented languages are more structured and therefore better understandable for domain experts, while graph-oriented languages allow more modeling freedom and technical versatility for process designers. To bridge this gap between understandability and technical versatility, we propose a participative forward engineering approach. It uses our block-oriented CUTA4BPM language to support high level process modeling together with domain experts, while graph-oriented BPMN is used for further detailed process design and automation. To support smooth transition from the high level modeling step to the detailed one, CUTA4BPM process models are automatically transformed to BPMN models using model driven engineering techniques.

In order to meet the high-precision measurement of the current baryon spectroscopy, for the first time, we rigorously analyze the spin-dependent interactions and the fine structure of the negative-parity singly heavy baryons in the relativized quark model, which was previously unfeasible in the three-quark system. This gains access to the exploration of the strong interactions dominated by the non-perturbative QCD, and reveals how the various forms of strong interactions in a baryon compete with each other, affect the evolution of the energy levels, cause the energy level splitting and contribute to the mixing effect responsible for the formation of the fine structures. It is shown that the rigorous calculation can perfectly reproduce the data, the averaged deviation between the calculated and experimental energy levels is less than 5 MeV for the negative-parity singly heavy baryons. Therefore, the theoretical precision has reached the experimental high precision. This confirms the reliability of the calculation and also helps make reasonable assignments for the observed negative-parity baryons. The large amount of data obtained by the rigorous calculations contains a wealth of interaction information and is helpful for both of the theoretical and experimental studies. The key to the rigorous calculation in this work is the proposal of a new method, namely the two-step Gaussian expansion method. This new method not only overcomes the long-standing unresolved problem in the relativized quark model, but also provides an effective approach for the high-precision calculation of other few-body systems such as the compact tetraquarks and pentaquarks, especially for the treatment of spin-orbit interactions and tensor interactions which actually appear ubiquitously in all of quantum many-body systems.

We investigate the electromagnetic form factors of the nucleon within an effective chiral theory derived from the QCD instanton vacuum, taking into account the finite current quark mass. The momentum-dependent dynamical quark mass, generated by the instanton-antiinstanton medium, naturally plays the role of a regulator, so that no additional regularization is required to tame the divergences arising from quark loops. The instanton parameters, the average instanton size $\barρ=0.35$ fm and the average interdistance $\bar{R}=0.86$ fm, together with the dynamical quark mass at zero virtuality $M_0=385$ MeV, are all fixed by the saddle-point equation beyond the chiral limit, leaving no adjustable free parameter in the present calculation. We compute the Sachs electric and magnetic form factors of the proton and neutron, the nucleon charge and magnetization radii, the magnetic moments, and the ratios $μ_{p,n} G_E^{p,n}(Q^2)/G_M^{p,n}(Q^2)$. The present results are compared with the experimental data, the chiral quark-soliton model ($χ$QSM), and the Kelly parametrization. The proton charge radius, $\sqrt{\langle r^2 \rangle_\mathrm{ch}^p}=0.841$ fm, is in remarkable agreement with the recent muonic-hydrogen value, and the $Q^2$ dependence of the proton form-factor ratio $μ_p G_E^p/G_M^p$ is reproduced very well, in clear contrast to the $χ$QSM. The overall agreement with the experimental data confirms that the effective chiral theory derived from the QCD instanton vacuum provides a consistent and predictive framework for describing the electromagnetic structure of the nucleon.

In this paper, we propose an Anderson-accelerated stochastic extragradient algorithm for solving a class of stochastic variational inequalities, by incorporating Anderson acceleration into the stochastic extragradient method under a stochastic approximation framework. A key challenge in our setting is that the pseudomonotonicity assumption is only imposed on the expectation of the stochastic operator, rather than on the individual stochastic operator itself and the sample averages utilized in the algorithm. We prove that, despite the lack of pseudomonotonicity in the sampled operators, the sequence generated by the proposed algorithm converges almost surely to a solution of the stochastic variational inequality problem. Additionally, we establish the sublinear convergence rate of the proposed algorithm in terms of the mean residual function, along with its optimal oracle complexity. Finally, we validate the effectiveness of the proposed algorithm through numerical experiments.

Collagen remodeling in living tissues exhibits anisotropic orientation patterns commonly described by Von Mises distributions, yet the physical origin of such nonequilibrium organization remains unresolved. In the present work, we demonstrate analytically that the combined action of Malthusian growth dynamics and the introduction of linear relations governing mechanical remodeling naturally gives rise to generalized bimodal Von Mises distributions as emergent states of living matter. The theory reveals a {\it rotation without rotation} mechanism, in which fibers progressively reorient in the absence of angular mechanical coupling via selective deposition and removal along preferred directions. The resulting analytical solutions quantitatively reproduce experimentally observed distributions and establish a direct mechanobiological origin for directional statistics in biological tissues. By interpreting the evolving normalized fiber density as a probability distribution function, we formulate a dynamical Shannon entropy framework that captures the temporal emergence of microstructural organization. The theory further yields closed-form expressions for the drift of the associated Fokker--Planck equation, enabling the corresponding stochastic differential equation to be derived, thus revealing that tissue remodeling is the collective outcome of noisy single-fiber dynamics. These results establish a minimal theoretical framework that connects biomechanics, stochastic processes, and nonequilibrium statistical organization in living matter.

Boundary time crystals exhibit spontaneous breaking of continuous time-translation symmetry through persistent periodic oscillations in driven-dissipative many-body systems. Here, we show that multilevel interference provides a natural route beyond periodic order, enabling dissipative time quasicrystals without externally imposed quasiperiodic driving. We consider a collectively driven-dissipative four-level ensemble with two degenerate excited states and two degenerate ground states. In the thermodynamic limit, the exact mean-field dynamics reduces to an irrational flow on a two-dimensional torus, yielding quasiperiodic order parameters with discrete spectra generated by two incommensurate fundamental frequencies. Vanishing maximal Lyapunov exponents demonstrate that the nonlinear self-consistent dynamics remains nonchaotic. Our results establish a minimal interference-induced mechanism for time-quasiperiodic order and open a route toward higher-dimensional quasiperiodic dynamics in multilevel systems.

Mining anomalies from unmanned aerial vehicle (UAV) state-estimation logs is challenging because failures are sparse, temporally structured, and distributed across heterogeneous PX4 telemetry streams with variable sensor availability and missing values. We present AeroTSBoost, a temporal-statistical boosting framework for real-world UAV telemetry anomaly mining. AeroTSBoost aligns multivariate flight logs, converts each window into deterministic descriptors that capture distributional shifts, quantile structure, endpoint drift, local dynamics, and lag correlation, and trains a class-balanced LightGBM detector. On UAV-SEAD, AeroTSBoost achieves the strongest AUPRC among evaluated classical, supervised tabular, neural reconstruction, recurrent, Granger-causality-based, and frequency-domain baselines. Across five seeds, it reaches $0.7516\pm0.0043$ AUPRC and $0.5342\pm0.0108$ threshold-swept event F1, improving AUPRC by 5.79 absolute points over the strongest non-AeroTSBoost baseline. Under purged chronological and leave-log-out protocols, it remains the best AUPRC method, reaching $0.6066\pm0.0193$ and $0.6388\pm0.0315$, respectively. On related ALFA fixed-wing UAV fault logs, AeroTSBoost reaches $0.9259\pm0.0076$ leave-sequence-out AUPRC, ahead of RandomForest ($0.8835\pm0.0797$) and moments-only ($0.8700\pm0.0481$). These results show that deterministic temporal-statistical representations remain highly competitive for sparse anomaly mining in operational cyber-physical telemetry.

Zero-knowledge virtual machines (zkVMs) are a key technology for driving the large-scale adoption of zero-knowledge proofs (ZKP), but their performance bottlenecks severely limit their practicality. While current hardware acceleration research has exclusively focused on backend proving, we identify that the frontend execution and trace generation phase is rapidly emerging as the new system bottleneck. To address this challenge, we propose ZK-Tracer, the first hardware accelerator architecture specifically designed for the zkVM frontend. ZK-Tracer features a novel heterogeneous design comprising a Main Trace Unit and parallel Permutation Trace Units. It exposes a fine-grained interface to the host software through a lightweight instruction set extension, enabling efficient task offloading. Our ASIC implementation results demonstrate that ZK-Tracer achieves up to 1829x speedup in trace generation over a high-performance multi-core CPU. When integrated with existing backend proving accelerators, it delivers a remarkable 963x end-to-end performance improvement for the entire ZKP system.

The shortest path multiplicity is an important metric of complex networks. The shortest path multiplicity of real networks is high and it correlates with their community structure. Since local network evolution induces network communities, it is possible that a high shortest path multiplicity is the natural expectation of local evolution rules. Here I demonstrate, by means of numerical simulations, that this is indeed the case.

With coupled detuned oscillators, either individual or collective oscillations are observable. The latter is used in quantum information processing in linear Paul traps. Here, we study the transition from collective radial modes at stronger axial confinements into individual radial oscillations at low axial confinements in a tapered Paul trap. The eigenmodes are experimentally studied in detail in the transition regime and compared with theoretical predictions. The features studied will enable investigation of modified heat transport phenomena and defect formation in trapped ions.

This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy's bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials.

We study a coevolutionary public goods game on a dynamic hypergraph, where an individual's payoff directly determines the number of hyperedges it can join. In the proposed mechanism, nodes adjust their participation according to the group payoffs of hyperedges, and hyperedges that remain occupied only by defectors for a sufficiently long time collapse and are rebuilt by selecting new members based on the current payoffs of nodes. This adaptive rule captures the performance-driven reorganization of group interactions in evolving collective systems. Using Monte Carlo simulations, we show that the cooperation fraction and average hyperdegree may converge to steady states with stochastic fluctuations or exhibit persistent oscillations, depending on the parameter regime. The steady-state outcomes are strongly nonmonotonic with respect to the structural adaptation parameters: cooperation is sustained only when the rate of link formation is properly balanced. If structural adaptation is too fast, frequent contacts between cooperators and defectors destroy cooperative clusters; if it is too slow, cooperators lack sufficient structural support to expand. This differs from the conventional expectation in static settings that larger benefit parameters always facilitate cooperation. We further introduce spectral entropy to quantify the regularity of the oscillatory dynamics and identify limit-cycle behavior in the phase space in certain regimes. These results suggest that adaptive higher-order restructuring can both promote and destabilize cooperation, offering insight into oscillatory cooperation and recurrent prosperity-decline cycles in real group-structured systems.

Memory-backed agents need provenance that can survive leaked or migrated snapshots, where logs, visible outputs, and trusted metadata may be absent. We propose MemMark, a state-evolution attribution watermark that embeds an owner-controlled signal into latent memory-write decisions. At each internal LLM call, MemMark samples among admissible candidates using keyed, distribution-preserving selection, and records cryptographic commitments with signed session anchors and reveal evidence. This makes attribution depend on reproducible backend behavior rather than mutable provenance fields. Across A-Mem and Graphiti on LoCoMo, with three LLM backbones, MemMark preserves memory utility: Overall F1 retains 99.6% of the unwatermarked baseline, while BLEU-1 changes by +0.2%. It also provides usable carrier capacity, with 1.16, 1.14, and 1.26 bits of mean entropy for update-target, link-target, and semantic-realization decisions. In the snapshot-only R3 setting, MemMark recovers the full 40-bit payload from final snapshots, while wrong-key verification remains near chance. Under nine memory-lifecycle attacks, verification distinguishes tampering, evidence deletion, and partial payload recovery. These results show that robust snapshot-only attribution is feasible for long-term agent memory without surviving traces, trusted metadata, or utility-degrading.