We prove an Erdős--Szekeres type result for finite words over $\mathbb{N}$ with repeated values. Specifically, we define a \emph{repeat} in a word to be an occurrence of a value which is not its first occurrence. We define an occurrence of a \emph{pattern} $π$ in a word $w$ to be a (not necessarily consecutive) subword of $w$ that is order isomorphic to $π$. In this note, we show that every word with $kn^6+1$ repeats contains one of the following patterns: $0^{k+2}$, $0011\cdots nn$, $nn\cdots1100$, $012 \cdots n012 \cdots n$, $012 \cdots nn\cdots 210$, $n\cdots 210012\cdots n$, $n\cdots 210n\cdots 210$. Moreover, when $k=1$, we show that this is best possible by constructing a word with $n^6$ repeats that does not contain any of these patterns.

Finite-horizon probabilistic multiagent concurrent game systems, also known as finite multiplayer stochastic games, are a well-studied model in computer science due to their ability to represent a wide range of real-world scenarios involving strategic interactions among agents over a finite amount of iterations (given by the finite-horizon). The analysis of these games typically focuses on evaluating (verifying) and computing (synthesizing/realizing) which strategy profiles (functions that represent the behavior of each agent) qualify as equilibria. The two most prominent equilibrium concepts are the Nash equilibrium and the subgame perfect equilibrium, with the latter considered a conceptual refinement of the former. Computing these equilibria from scratch is, however, often computationally infeasible. Therefore, recent attention has shifted to the verification problem, where a given strategy profile must be evaluated to determine whether it satisfies equilibrium conditions. In this paper, we demonstrate that the verification problem for subgame perfect equilibria lies in PSPACE, while for Nash equilibria, it is EXPTIME-complete. This is a highly counterintuitive result since subgame perfect equilibria are often seen as a strict strengthening of Nash equilibria and are intuitively seen as more complicated.

We report on recent advances in rule-based graph programming, which allow us to match the time complexity of some fundamental imperative graph algorithms. In general, achieving the time complexity of graph algorithms implemented in conventional languages using a rule-based graph-transformation language is challenging due to the cost of graph matching. Previous work demonstrated that with rooted rules, certain algorithms can be implemented in the graph programming language GP 2 such that their runtime matches the time complexity of imperative implementations. However, this required input graphs to have a bounded node degree and (for some algorithms) to be connected. In this paper, we overcome these limitations by enhancing the graph data structure generated by the GP 2 compiler and exploiting the new structure in programs. We present three case studies: the first program checks whether input graphs are connected, the second program checks whether input graphs are acyclic, and the third program solves the single-source shortest-paths problem for graphs with integer edge-weights. The first two programs run in linear time on (possibly disconnected) input graphs with arbitrary node degrees. The third program runs in time $O(nm)$ on arbitrary input graphs, matching the time complexity of imperative implementations of the Bellman-Ford algorithm. For each program, we formally prove its correctness and time complexity, and provide runtime experiments on various graph classes.

The PXP chain was recently shown to exhibit superdiffusive energy transport with Kardar-Parisi-Zhang-like scaling, $z\approx3/2$, joining a growing number of spin chains with this exponent. An understanding of how this anomalous hydrodynamics emerges from microscopics is, however, still lacking. In this work, we show that finite-temperature energy transport in this model provides a window into the emergence of superdiffusion. At finite temperature, the energy autocorrelation function exhibits a crossover from short-time coherent dynamics to long-time hydrodynamics. The short-time behavior is dominated by a single magnon band and can be understood analytically. In momentum space, this regime is characterized by spectral weight near $q=π$. The damping time $τ$, which separates the short-time magnon-dominated behavior from the late-time hydrodynamics, grows rapidly upon cooling, consistent with an activated form $τ(β)\sim βe^{Δβ}$ with a gap scale set by the magnon band. At longer times, the spectral weight transfers to $q=0$ and the running decay exponent drifts toward the superdiffusive value $z=3/2$. Finite-temperature energy transport therefore provides a bridge between microscopic magnon physics and late-time superdiffusion in the PXP model.

Non-Hermitian operators are now routinely used to describe few-mode systems such as optical resonators and superconducting qubits, and exceptional points (EPs) are defective spectral singularities of such non-Hermitian operators. In contrast, the scattering-theoretic formulation of EP physics for unbounded Hamiltonians remains less settled. In this work, we formulate the geometric phase associated with encircling an EP when the underlying eigenstates are quantum resonances within a one-dimensional scattering model. To do this, we employ the complex-scaling method, where resonance poles of the S matrix are realized as discrete eigenvalues of the non-Hermitian dilated Hamiltonian, to construct situations in which resonant and scattering states coalesce into an EP in the complex energy plane, that is, the resonance pole is embedded into the continuum spectrum. We analyze the self-orthogonality in the vicinity of an EP, the Berry phase, and the Chern characteristic. Our results clarify how EP branch structure and geometric holonomy arise directly from resonance poles in scattering theory, thereby connecting non-Hermitian spectral topology with the traditional theory of quantum resonances.

We present a detailed analysis of slowly driven quantum thermal machines based on interacting qubits within the framework of the Lindblad master equation. By implementing a systematic expansion in the driving rate, we derive explicit expressions for the rate of work of the driving forces, the heat currents exchanged with the reservoirs, and the entropy production up to second order, ensuring full thermodynamic consistency in the linear-response regime. The formalism naturally separates geometric and dissipative contributions, identified by a Berry curvature and a metric in parameter space, respectively. Analytical results show that the geometric heat pumped per cycle is bounded by $k_B T N_q \ln 2$ for $N_q$ non-interacting qubits, in direct analogy with the Landauer limit for entropy change. This bound can be surpassed when qubit interactions and asymmetric couplings to the baths are introduced. Numerical results for the interacting two-qubit system reveal a non-trivial role of the interaction between qubits and the coupling between the qubits and the baths in the behavior of the dissipated power. The approach provides a general platform for studying dissipation, pumping, and performance optimization in driven quantum devices operating as heat engines.

While the auto-bidding literature predominantly considers independent bidding, we investigate the coordination problem among multiple auto-bidders in online advertising platforms. Two motivating scenarios are: collaborative bidding among multiple bidders managed by a third-party bidding agent, and strategic bid selection for multiple ad campaigns managed by a single advertiser. We formalize this coordination problem as a theoretical model and investigate the coordination mechanism where only the highest-value bidder competes with outside bidders, while other coordinated bidders refrain from competing. We demonstrate that such a coordination mechanism dominates independent bidding, improving both Return-on-Spend (RoS) compliance and the total value accrued for the participating auto-bidders or ad campaigns, for a broad class of auto-bidding algorithms. Additionally, our simulations on synthetic and real-world datasets support the theoretical result that coordination outperforms independent bidding. These findings highlight both the theoretical potential and the practical robustness of coordinated auto-bidding in online auctions.

We introduce Pudlak-Buss style Prover-Adversary games to characterise proof systems reasoning over deterministic branching programs (BPs) and non-deterministic branching programs (NBPs). Our starting points are the proof systems eLDT and eLNDT, for BPs and NBPs respectively, previously introduced by Buss, Das and Knop. We prove polynomial equivalences between these proof systems and the corresponding games we introduce. This crucially requires access to a form of negation of branching programs which, for NBPs, requires us to formalise a non-uniform version of the Immerman-Szelepcsenyi theorem that coNL = NL. Thanks to the techniques developed, we further obtain a proof complexity theoretic version of Immerman-Szelepcsenyi, showing that eLNDT is polynomially equivalent to systems over boundedly alternating branching programs.

We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs. Our framework is based on a new hybrid approximation algorithm, which uses a combination of the Gaussian elimination technique (i.e., solving a system of linear equations over an Abelian group) and the semidefinite programming relaxation. We complement our algorithm with a matching dictator vs. quasirandom test that has perfect completeness. The analysis of our dictator vs. quasirandom test is based on a novel invariance principle, which we call the mixed invariance principle. Our mixed invariance principle is an extension of the invariance principle of Mossel, O'Donnell and Oleszkiewicz [Annals of Mathematics, 2010] which plays a crucial role in Raghavendra's work. The mixed invariance principle allows one to relate 3-wise correlations over discrete probability spaces with expectations over spaces that are a mixture of Guassian spaces and Abelian groups, and may be of independent interest.

Non-Gaussian entanglement is a promising resource in various quantum tasks. A recently defined class identifies entanglement that cannot be generated by applying Gaussian operations to separable inputs. To further explore the entanglement in this context, we introduce a quantitative witness $E_{\rm NG}$ in bipartite bosonic systems, which satisfies $E_{\rm NG}=1$ for all Gaussian-entanglable states, while $E_{\rm NG}>1$ certifies non-Gaussian entanglement. Its ceiling $d=\lceil E_{\rm NG}\rceil$ provides a lower bound on the Schmidt number irreducible by Gaussian transformations, thereby defining a natural hierarchy of non-Gaussian entanglement. For pure states, the condition is sharp and the hierarchy reflects the complexity of state learning. We benchmark the framework with some paradigmatic non-Gaussian states, such as NOON states and squeezed Kerr states, and analyze its robustness against loss. Moreover, we construct an experimentally economical NOON-type witness requiring only four density-matrix element measurements. These results establish an operationally meaningful and experimentally accessible framework for identifying non-Gaussian entanglement resources in continuous-variable quantum platforms.

Stellar mergers produce explosive outflows that serve as transient sources of IR line luminosity and inject mechanical energy early into the natal molecular cloud. We present the first velocity-resolved maps of the [O I] 63 and 145 um fine-structure line emission from the wide-angle outflow in Orion BN/KL, the nearest explosive outflow. The data were obtained with SOFIA and include sensitive [C II] 158 um and OH maps. They allowed us to disentangle the quiescent cloud gas from the outflow, traced by a broader [O I] component with a line FWHM of about 20-30 km/s and exhibiting a spatial distribution similar to that of the shock-excited H2 emission seen with JWST. The OH 119 um line shows a prominent P-Cygni profile covering 160 km/s, similar to the very broad CO lines. The total [O I] 63 and 145 line luminosity is remarkably high, 86.5 L_sun, comparable to the H2 and CO line luminosities, implying an outflow mass-loss rate of (9.1+/-2.6)x10^-3 M_sun/yr and a mass of 3.3-5.9 M_sun. The [O I] 63 / 145 and [O I] 63 / [C II] 158 intensity ratios reach very high values in the line wings (20-30 and 40-60, respectively), exceeding those found in PDRs. These ratios are consistent with the presence of dense (10^5 to 10^6 cm^-3 ) and warm (~500 K) postshock gas. We analyzed the fine-structure line-wing intensities using magnetized shock models that include UV irradiation, to which the [C II] 158 line intensity is particularly sensitive. We find that the [O I] and [C II] intensities are consistent with emission from dissociative J-type shocks with velocities of 30-40 km/s and preshock gas densities of a few 10^4 cm^-3, illuminated by external UV radiation generated by surrounding fast shocks and possibly by massive (proto)stars in the region. We also report a broad [O I] emission feature around the BN star, which we attribute to an unresolved outflow or wind bow shock.

Fractional Chern insulators (FCIs) are lattice analogs of fractional quantum Hall systems, where the interplay of strong interactions with a frustrated tunnelling kinetics leads to the emergence of a gapped ground state with long-range entanglement and anyonic excitations. The highly correlated nature of such systems makes their adiabatic preparation challenging already beyond the minimal system size of two particles. Considering Floquet implementations of the bosonic Harper-Hofstadter-Hubbard model of few photons in superconducting circuits, we design protocols for the driven-dissipative stabilization of its FCI ground state at half filling via quantum bath engineering. Dissipation control is achieved through the coupling to driven leaky cavity modes, which realize a tuneable artificial environment having the Floquet-FCI as its approximate fixed point. For systems of two, three and six particles, we show numerically how the flexibility of the control scheme further allows for the detection of fractional quantum Hall signatures in the stabilized steady states, including bulk incompressibility, Hall response and the trapping of fractional charges. Our results provide a concrete pathway to dissipation-assisted preparation of strongly correlated states in quantum simulators.

We consider debiased inference on finite-dimensional functionals of infinite-dimensional least-squares solutions to inverse problems as a way to avoid having to assume exact solutions exist. Such assumptions are substantive and not innocuous, and their failure may imperil inference when we impose them on the statistical model. Our approach instead allows us to conduct inference on a quantity that is defined regardless of solutions existing and coincides with the usual estimands when they do. For the case of instrumental variables, this means we can motivate the analysis with structural models but these do not need to hold exactly for the semiparametric inferential procedure to remain valid.

Heterogeneous reconfigurable platforms with tensor cores, such as AMD ACAP, are increasingly adopted for deep neural network (DNN) inference due to their high throughput and flexibility. However, their suitability for microsecond-scale inference on small problem sizes remains underexplored. In jet-tagging applications in high-energy physics, inefficient on-chip communication and large inter-layer latency prevent existing frameworks from meeting the 1-μs latency budget. Moreover, hardware overheads such as synchronization and VLIW processor prologue are often overlooked, making it infeasible to optimize accelerators correctly. To address these problems, we propose μ-ORCA, a customized heterogeneous accelerator framework for ultra-low-latency model inference. μ-ORCA enables direct inter-layer communication between DNN layers on the AIE array, instead of using shared memory tiles or FPGA fabric. Moreover, a 512-bit/cycle cascade connection is applied instead of a 32-bit/cycle DMA connection. μ-ORCA also provides an overhead-aware performance model that adapts to different NN layer sizes, and conducts design space exploration to optimize end-to-end latency. μ-ORCA supports MLP and DeepSets models with non-MM kernels, including bias, ReLU, and global aggregation on AIE. We evaluate μ-ORCA on the AMD ACAP VEK280 platform. Experimental results show that μ-ORCA achieves average latency reduction of >1.70$\times$ and >1.83$\times$ compared with different state-of-the-art ACAP frameworks, and achieves 0.93 μs latency for a 6-layer real-world DeepSets model, satisfying the latency budget. We open source μ-ORCA at https://github.com/arc-research-lab/u-ORCA.

We introduce action-driven flows for causal variational principles, being a class of non-convex variational problems emanating from applications in fundamental physics. In the compact setting, Hölder continuous curves of measures are constructed by using the method of minimizing movements. As is illustrated in examples, these curves will in general not have a limit point, due to the non-convexity of the action. This leads us to introducing a novel penalization which ensures the existence of a limit point, giving rise to approximate solutions of the Euler-Lagrange equations. The methods and results are adapted and generalized to the causal action principle in the finite-dimensional case. As an application, we construct a flow of measures for causal fermion systems in the infinite-dimensional situation.

In an oriented graph $\overrightarrow{G}$, the inversion of a subset $X$ of vertices is the operation that reverses the orientation of all arcs with both end-vertices in $X$. The inversion graph of a graph $G$, denoted by $\mathcal{I}(G)$, is the graph whose vertices are orientations of $G$ in which two orientations $\overrightarrow{G_1}$ and $\overrightarrow{G_2}$ are adjacent if and only if there is an inversion transforming $\overrightarrow{G_1}$ into $\overrightarrow{G_2}$.The inversion diameter of a graph $G$ is the diameter of its inversion graph $\mathcal{I}(G)$, denoted by $\mathrm{diam}(\mathcal{I}(G))$.Havet, Hörsch, and Rambaud~(2024) first proved that for $G$ of treewidth $k$, $\mathrm{diam}(\mathcal{I}(G)) \le 2k$, and that there are graphs of treewidth $k$ with inversion diameter $k+2$.In this paper, we construct graphs of treewidth $k$ with inversion diameter $2k$, which implies that the previous upper bound $\mathrm{diam}(\mathcal{I}(G)) \le 2k$ is tight.Moreover, for graphs with maximum degree $Δ$, Havet, Hörsch, and Rambaud~(2024) proved $\mathrm{diam}(\mathcal{I}(G)) \le 2Δ-1$ and conjectured that $\mathrm{diam}(\mathcal{I}(G)) \le Δ$. We prove the conjecture when $Δ=3$ with the help of computer calculations.

In this paper, we investigate the stability of minimizing Yamabe metrics on compact manifolds with boundary, in the sense introduced by Escobar. We show that if a function nearly minimizes the Yamabe energy, then the associated conformal metric is quantitatively close to a minimizing Yamabe metric within its conformal class. Moreover, this closeness is controlled by an appropriate power of the Yamabe energy deficit.

We consider neutral-current parity-violating interactions in an atom mediated by the exchange of a neutrino-antineutrino pair. We explicitly account for the nuclear finite size encoded in the nuclear form factor. Based on its general properties, we derive an effective neutrino-mediated potential and determine its properties at short and long distances. We demonstrate that, once the form factor properties are correctly accounted for, the range of such an effective potential corresponds to the nuclear radius, removing any sensitivity to shorter-distance contributions. This potential changes sign over the atom's volume, so that the correction to the effective nuclear weak charge induced by this interaction is tiny and does not alter the interpretation of atomic parity violation experiments.

The Frauchiger--Renner paradox derives an inconsistency when quantum theory is used to describe the use of itself, by means of a scenario where agents model other agents quantumly and reason about each other's knowledge. We observe that logical contextuality (à la Hardy) is the key ingredient of the FR paradox, and we provide a stronger paradox based on the strongly contextual GHZ--Mermin scenario. In contrast to the FR paradox, this GHZ--FR paradox neither requires post-selection nor any reasoning by observers who are modelled quantumly. If one accepts the universality of quantum theory including superobservers, we propose a natural extension of Peres's dictum to resolve these extended Wigner's friend paradoxes.

We examine the correspondence principle for an equiprobable superposition of high-energy eigenstates of the infinite square well using a fully analytical Fourier-based approach. We derive a closed-form asymptotic expression for the interference terms $ρ_α^{\text{a}}(x)$ by expanding them into a geometric series of quantum Fourier coefficients. We show these terms act as functional envelopes that do not vanish individually but become asymptotically equivalent in the large-$n$ limit. Furthermore, we prove the total probability density for a superposition of $2Δ+1$ states converges exactly to the uniform classical distribution as $Δ\to \infty$. Dynamically, the expectation value of position reproduces the classical triangular trajectory asymptotically. Residual quantum deviations remain confined to boundary layers whose relative width vanishes under macroscopic resolution. These results establish a rigorous asymptotic realization of the classical limit for isolated bound systems in both static and dynamical contexts.

We derive the canonical-ensemble scaling of Tan's contact for $N$ harmonically trapped Tonks--Girardeau bosons at finite temperature in the large-$N$ limit. The leading scaling coefficient reproduces the local-density-approximation result and is obtained from a contour-integral representation of the canonical partition function followed by a saddle-point reduction to a phase-space integral with a self-consistent scaled chemical potential. The subleading coefficient is the central new object of this work: it admits an explicit representation in terms of universal phase-space integrals of the Fermi factor, has closed-form Sommerfeld and virial limits, and is identified with the canonical-versus-grand-canonical ensemble difference at fixed mean particle number. In the high-temperature Boltzmann regime the ratio of subleading to leading coefficients collapses to a universal value, traceable to the Poissonian particle-number statistics of the dilute grand-canonical gas. We construct Padé approximants for both scaling functions that interpolate uniformly between the low-temperature Sommerfeld and high-temperature virial regimes; for the subleading coefficient we report a form that is uniformly accurate on our working range of temperatures and asymptotically correct beyond. The scaling law is verified against canonical contour-integration data across the full temperature range.

Score-driven models update time-varying parameters using conditional likelihood scores. This paper develops a Bayesian interpretation of such updates through Tweedie's formula, which connects posterior mean corrections with marginal scores. In Gaussian signal extraction, this gives an exact posterior-correction identity. For natural exponential families, related identities characterize posterior means in natural- and expectation-parameter spaces. Building on these identities, we show that conjugate Bayesian filtering in expectation space coincides exactly with an inverse-Fisher-scaled conditional score update under local precision discounting. For general conditional densities, the exact Bayesian correction involves a generally unavailable predictive-marginal score. A local Gaussian approximation shows that the conditional likelihood score provides the leading approximation to this posterior correction; under local precision discounting, the predictive covariance becomes proportional to inverse Fisher information, yielding the familiar inverse-Fisher-scaled score recursion. The results clarify when score-driven updates are exact Bayesian filters and when they should instead be viewed as tractable local approximations.

Neutron monitors (NMs) are ground-based devices designed to measure cosmic-ray count rates by monitoring atmospheric neutrons from cosmic-ray showers. We present results from new electronics that have recorded cross-counter time delay histograms for the {\it Princess Sirindhorn Neutron Monitor} (PSNM) at the summit of Doi Inthanon, Thailand. From these histograms, we have extracted the cross-counter leader fraction ($L$) and corrected it for atmospheric effects. For large counter separation, we measure nearly constant $L\approx0.997$, implying that 0.3\% of counts in one counter are temporally associated with later counts on a given distant counter. Monte Carlo simulations confirm that individual secondary particles cannot account for the associated counts at large counter separation, which instead requires a contribution from multiple secondary particles in the same cosmic ray shower that is apparently independent of distance over 3 to 7.5 m. We infer that $\approx$4.5\% of PSNM counts are associated with a later count in at least one of its 18 counters from a different secondary particle in the same shower. Monte Carlo simulations of atmospheric showers and NM yield functions can be validated using our measurements of neutron multiplicity across counters and the contributions of single and multiple secondary particles. These measurements also improve understanding of the single-counter $L$, which has been used for precise tracking of cosmic-ray spectral variations and extending the range of NM observations to higher energies.

In this work, we investigate the Dymnikova regular black hole within the framework of unimodular gravity, emphasizing the role of the effective vacuum sector in the regularization of the geometry. By allowing a controlled violation of the covariant conservation of the energy--momentum tensor, the cosmological contribution emerges dynamically as a radial-dependent function, $Λ=Λ(r)$. We first reinterpret the Dymnikova spacetime as a charged configuration supported by nonlinear electrodynamics and derive the corresponding electric and magnetic sources. Subsequently, we demonstrate that the same geometry can be consistently generated by standard Maxwell electrodynamics in unimodular gravity. In this construction, the resulting electric field is everywhere regular and corresponds to a localized charge distribution with vanishing asymptotic charge, indicating that the spacetime does not behave as an asymptotically charged object.

In the first part of this series, the authors introduced the quantum wreath product, providing a unified framework that encompasses numerous results previously addressed only through case-by-case analysis. This paper shifts focus to the fundamental construction of modules over these products, termed wreath modules. Our approach utilizes parabolic induction on tensor products combined with a sophisticated labeling scheme based on multipartitions. While the underlying constructions are technically involved, they offer a transparent realization of several prominent module families. Specifically, these wreath modules recover and unify: Simple modules over the Ariki-Koike algebra; Specht and simple modules over the Hu algebra; (anti)spherical modules and Kashiwara-Miwa-Stern modules over the affine Hecke algebra and its pro-p Iwahori variants. Finally, we demonstrate that these wreath modules for the Hu algebra serve as a critical component in solving the Ginzburg-Guay-Opdam-Rouquier problem. This solution enables a concrete realization of Category O for the rational Cherednik algebra in Type D.

Topological phenomena in quantum critical systems have recently attracted growing attention, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, a general framework for identifying such nontrivial phenomena, particularly in higher-dimensional systems, remains insufficiently explored. In this work, we propose a universal fingerprint for detecting nontrivial topology in critical free-fermion systems protected by global on-site symmetries. Specifically, we analytically establish an exact relation between the bulk entanglement spectrum and the boundary energy spectrum at topological criticality in arbitrary dimensions, demonstrating that the degeneracy of edge modes can be extracted from the bulk entanglement spectrum. These findings, further supported by numerical simulations of lattice models, provide a universal fingerprint for identifying nontrivial topology in critical free-fermion systems.

Modern machine learning is dominated by complex, overparameterized architectures capable of interpolating data and achieving zero training loss. For such models, we investigate the convergence properties of two popular modifications to standard SGD: clipped SGD and normalized SGD. We show that under overparameterization and a mild assumption on batch size, both clipped and normalized SGD do not suffer from the bias typically introduced by clipping, converging effectively at the same rate as their deterministic counterparts. This provides a rigorous theoretical justification for the empirical success of gradient clipping methods. In our analysis, we employ the $(L_0,L_1)$-smoothness condition, under which we obtain convergence rates that improve upon the best known results in prior work. Furthermore, we extend our analysis to specific challenging regimes, including heavy-tailed noise, $(H_0,H_1)$-smoothness (which is strictly weaker than standard assumptions in optimization literature) and the deterministic regime.

We study optimal portfolio choice models in markets with partial information about the stock's drift. We solve the single agent problem for general utilities using a new approach that yields regularity of the value function and closed form expressions for the optimal processes. We consider a N player game in which players interact through the law of peer's wealth and study its mean field limit. This leads to a a game with field equilibrium. We analyze the cases of separable couplings and general utilities, and represent the value of the game as a compilation of the single player problem and a function solving a non local quasilinear pde in the space of measures. We interpret the findings using elements from indifference valuation and arbitrage free pricing, Finally, when the couplings depend only on the average of peer's wealth, we derive explicit solutions and various regularity results for representative cases.

Recent works of El Hamam described explicit fundamental systems of units for several families of multiquadratic fields of degrees 8 and 16. In the degree-8 field $L^+ = \mathbb{Q}(\sqrt{2}, \sqrt{pq}, \sqrt{ps}),$ the corrected classification still leaves a residual binary indeterminacy: one must decide which of two explicitly constructed squareclasses gives the final unit generator. In this paper, we make this remaining bit explicit. First, we give an explicit local criterion deciding the parameter $μ\in \{1, ε_{pq}\}$ left open in recent literature. The criterion is first expressed in terms of Hilbert symbols at a single finite place, and is then sharpened to a residue criterion at a chosen split auxiliary rational prime. Second, we show that the standard residue datum $D(p,q,s) = \left( p \bmod 8,\,\, q \bmod 8,\,\, s \bmod 8,\,\, \biggl(\dfrac{q}{p}\biggr),\,\, \biggl(\dfrac{s}{p}\biggr),\,\, \biggl(\dfrac{q}{s}\biggr) \right)$ does not determine the final generator: we compute explicit triples with the same $D(p,q,s)$ but opposite values of the residual bit. Third, we place the one-bit problem inside a hierarchy of local-certification results in $K^\times/K^{\times2}$: besides the linear residual-choice statement, we prove an affine local-certification theorem for residual-choice cosets and a finite-test-set separation theorem for arbitrary finite candidate families.

We analyze the decoherence of a particle's spatial superposition moving along a stationary worldline through the Minkowski vacuum. The particle is modeled via an internal degree of freedom that couples to a scalar field, and an external degree of freedom, i.e., its quantized center-of-mass motion around the stationary worldline. Assuming a separation of time scales between the particle's internal and external dynamics, we first obtain an effective red-shifted polarizability of the particle, characterizing the trajectory-dependent linear response of the internal oscillator to the field. We then derive a quantum Brownian motion master equation for the particle's center of mass, under the Born-Markov approximation, which describes its decoherence in the position basis, as well as, Hamiltonian modifications corresponding to a dispersive potential. The resulting decoherence has two components: (1) arising from a modified field spectrum observed by the particle; and (2) due to a differential time-dilation over the particle's extended spatial wavefunction. For stationary trajectories, both contributions take an effectively thermal form. We evaluate the decoherence rates for two specific cases of hyperbolic and uniform circular motion.