AI for Science

We present a method to compute Ewald summation for the irreducible components of flow around active particles to study hydrodynamic interactions in active colloidal suspensions. An active particle is modeled as a colloidal sphere with a surface slip velocity. Using this model, we obtain an irreducible representation of the fluid flow produced by an active particle in periodic geometry of Stokes flow for an arbitrary surface slip. The solution of the active flow is obtained in terms of lattice sum of the Oseen tensor and their derivatives. The lattice sum is accelerated using the Ewald summation technique. We apply the method to compute explicit expression for rigid body motion of hydrodynamically interacting active particles. Our method presents a way for dynamic simulation of active particles due to arbitrary mode of active slip in periodic geometry of Stokes flow.

We study Josephson dynamics in a long atomic Bose-Josephson junction formed by two tunnel-coupled coplanar Bose-Einstein-condensate rings. An in-plane linear acceleration breaks the axial symmetry of the trap and transforms a single Josephson plasma oscillation into a multimode population-imbalance response. Gross-Pitaevskii simulations and Bogoliubov-de Gennes analysis show that the additional spectral components arise from collective modes that acquire finite overlap with the population-imbalance operator under symmetry breaking, with their activation governed by reflection symmetry about the acceleration direction. We also propose a mode-resolved Josephson-spectroscopy protocol based on a weak localized periodic perturbation. Frequency scans reveal resonant amplitude peaks and phase shifts at the eigenfrequencies of active Bogoliubov modes, while angular scans of the drive position provide access to the angular structure of the corresponding mode density perturbations. A dissipative time-dependent Bogoliubov theory yields analytical response functions in quantitative agreement with full Gross-Pitaevskii simulations in the linear regime. Our results demonstrate that accelerated dual-ring condensates provide a controllable platform for symmetry-selected Josephson dynamics and spectroscopic probing of collective modes.

Antiferromagnetic skyrmions are swirling topological spin textures with rich dynamics and intriguing transport properties, yet their bidirectional dynamics remain largely unexplored. Here, we investigate the dynamics of antiferromagnetic skyrmions driven by current-induced spin-transfer and spin-orbit torques. We computationally demonstrate that antiferromagnetic skyrmions moving in one direction at low current densities can reverse their motion direction when the driving current is above a threshold. Based on the Thiele approach analysis, we show that this bidirectional motion originates from a change in the relative strengths of two effective forces arising from spin-transfer and spin-orbit torques. Furthermore, exploiting this bidirectional motion on a single racetrack, we design programmable logic gates. Our results not only uncover a hidden mechanism for bidirectional skyrmion motion but also facilitate the development of antiferromagnet-based logic devices.

Understanding strongly correlated quantum materials remains a central challenge in condensed matter physics and materials science. While angle-resolved photoemission spectroscopy (ARPES) has become an indispensable probe of single-quasiparticle excitations, it accesses electronic correlations only indirectly. Here we show that unlike one-photon in, two-electrons out coincidence ARPES ($γ\!\rightarrow\!2e$ 2eARPES), the two-photon in, two-electron out $2γ\!\rightarrow\!2e$ 2eARPES provides a direct and unambiguous probe of electronic pairing. We establish this on general theoretical grounds and substantiate it through large-scale numerical simulations of strongly correlated models with both paired and unpaired ground states. The key result is a model-independent separation in the $(ω_1,ω_2)$ plane of the two photoelectrons' energies, between signal from electrons emitted from the \emph{same} pair and signal from electrons emitted from \emph{different} pairs; this follows from energy conservation alone and is independent of any material-specific assumptions. Our findings demonstrate that $2γ\!\rightarrow\!2e$ 2eARPES can identify pairing and extract the pair binding energy as well as the energy of the 'glue' boson without any sophisticated data analysis or complementary measurements.

Terahertz radiation from $\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_{8+δ}$ intrinsic Josephson junctions (IJJs) provides an ideal platform to study the synchronization of a macroscopic quantum system. Here, we present a spectral analysis of a frequency-modulated Josephson plasma emitter coupled with patch antennas. In the unmodulated intensity distribution as a function of radiation frequency $I_{\mathrm{UM}}(ω)$, we observe a double Gaussian peak structure. Crucially, double-peak spectra obtained at a constant bias voltage imply either a rapid temporal distribution of resonances or their simultaneous excitation, driven by the mutual electromagnetic coupling between the IJJ mesa and the antennas. At low modulation frequencies $f_m$, the spectra are well reproduced by the products of $I_{\mathrm{UM}}(ω)$ and frequency combs, yielding a synchronized relaxation time $τ_s \simeq 0.28\text{ ns}$. Incorporating $τ_s$ quantitatively reproduces a drastic spectral transformation observed around $f_m \sim 1\text{ GHz}$, unveiling the sub-nanosecond non-equilibrium dynamics of coupled Josephson plasma.

The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent $λ_{\mathrm{L}}=2πT$ at temperature $T$), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, $g\equiv J_{0}/δJ$ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing $g$ yields the following effects: (i) The system remains a fast scrambler with $λ_{\mathrm{L}}=A~T$, but with a suppressed coefficient $A<2π$. (ii) In the limit $g\to \infty$, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with $λ_{\mathrm{L}}\simeq 0$. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of $g$. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all $g$ values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.

Spin dynamics in ordered magnets with topological lattice defects is investigated. Using fracton--elasticity duality, we develop an effective field theory of magnons coupled to quantized lattice dislocations (dislons) in magnetic insulators. Within this framework, an elastic gauge field mediates a nonlocal interaction between dislocations and magnetization gradients. The resulting magnetoelastic coupling gives rise to coherent magnon-dislon hybridization whose properties are dictated by dislocation topology. Screw dislocations exhibit helicity-selective hybridization and symmetry-protected dark dislon sectors, while edge dislocations generate anisotropic hybrid excitations with finite spin-precession ellipticity through the glide constraint. Our results establish dislocations as dynamical topological defects with directly observable polarization fingerprints in magnon spectra, and reveal magnon-dislon hybridization as a new route to control spin dynamics.

We study the non-equilibrium dynamics of emergent dynamical axion quasiparticles (DAQ) coupled to a photon bath in equilibrium via a Chern-Simons term as a quantum open system. A quantum master equation (QME) is derived up to second order in this coupling implementing only a \emph{partial} Markov approximation, allowing time dependent rates in the Lindblad (QME). These are determined by the equilibrium correlation functions of the Chern-Simons density, and their time dependence allows us to explore transient dynamics in coherences and population: the formation of the quasiparticle on short time scales and its decay, and the build-up of population with an effective time dependent rate. Early time evolution features quantum \emph{anti} Zeno dynamics with enhanced quasiparticle decay and population growth. These phenomena describe transient violations of Fermi's Golden rule and of \emph{detailed balance}, and are distinct \emph{non-Markovian} effects directly related to the spectral density of the Chern-Simons correlators. We obtain the equation of motion of coherent (DAQ) condensates both with the (QME) and with quantum many body linear response establishing a direct bridge between both methods. As a corollary we obtain the expectation value of the Chern-Simons density \emph{induced} by a (DAQ) condensate in linear response, the topological susceptibility is shown to be proportional to the (DAQ) many body self-energy. We provide a Feynman diagram-based interpretation of approximations invoked in the (QME) and corrections from system-bath correlations in higher order.

In a magnetic field bilayer graphene hosts an octet of pseudo-zero-mode electron levels nearly degenerate in orbitals $n=(0,1)$, valleys and spins. They split in valleys by electrostatic gating. In gated bilayer graphene, in which the interlayer bias is set up to flip sign across a line, one has a line junction that traps a portion of pseudo-zero-mode electrons inside the insulating bulk band gap, giving rise to electron states localized along the junction, known as kink states. A close look is made into the spectra and electromagnetic response of such kink states. There are two species of valley current associated with them, a drift current driven by the bias gradient and a circulating current coming from cyclotron motion. It turns out that they both flow in essentially the same direction, with the circulating current exhibiting a magnetic character distinct from those of other higher levels. In equilibrium they spatially circulate within the kink states, creating a quasi-one-dimensional channel of orbital magnetization. The electric control of the orbital magnetization and valley currents via a network of gated junctions will find useful applications in valley electronics.

In this article, we develop a multi-dimensional two-layer thin film model extending the thin film model proposed in \cite{barthwal2025hyperbolic}. The model considered in \cite{barthwal2025hyperbolic} considered a very specific Marangoni scale by choosing Marangoni numbers in both layers to be $1$. We relax this condition here and prove that the obtained system possesses a full set of Riemann invariants. Based on these findings, we develop a Riemann Invariant-based Local Characteristic Decomposition WENO (RI-WENO) method for the two-layer thin film model in one and two dimensions. The method is built upon a specially designed variable transformation constructed from the derived Riemann invariants of the system. This transformation partially diagonalizes the governing equations and yields a sparse structure in the transformed eigenvector matrices. As a result, the proposed RI-WENO framework significantly reduces the computational cost of the standard Local Characteristic Decomposition WENO approach while retaining its strong capability to suppress spurious oscillations. Numerical experiments, including new benchmark test cases, demonstrate that the RI-WENO method achieves an effective balance between accuracy and computational efficiency, making it a promising and practical choice for solving the two-layer thin film model.

Understanding the scrambling of quantum information is central to many areas of quantum physics, including quantum thermalization, entanglement growth, and quantum information processing. Insights from these studies have, in turn, inspired the development of novel quantum protocols and algorithms. Recently, a bidirectional teleportation protocol was proposed to implement a digital SWAP operation between qubits by leveraging chaotic Hamiltonian evolution combined with measurement and post-selection. In this work, we provide a comprehensive study of two central quantities that characterize the protocol, the post-selection probability and the fidelity, taking into account possible errors in time-reversed dynamics. We show that these quantities can be expressed in terms of standard diagnostics in quantum dynamics, including the Loschmidt echo and its subsystem variant. The results unveil (1) the initial-state dependence of the fidelity and (2) the stability of the post-selection probability in integrable models. Our findings offer practical guidance for the implementation of the protocol on realistic quantum devices.

In this work, we formulate self-similarity from the perspective of scale invariance, where a self-similar form is understood as the transformation of a function into a form invariant under scale transformations. By applying this formulation to physical parameters, which consist of numerical values and units, it is demonstrated that dimensional analysis works for physical problems because scale invariance is partially shared between units and physical parameters. This naturally leads to the distinction between similarity of the first kind and similarity of the second kind according to whether the scale functions induced by units and those associated with physical parameters are equivalent or not. Self-similar solutions of the second kind can be further classified according to whether the power exponents of the similarity parameters include functions of dimensionless numbers. This leads to the conclusion that there are three kinds of self-similar solutions. The present work provides a unified framework for understanding dimensional analysis and a universal classification of self-similarity in physical problems.

We study the moduli space of non-Hermitian Yang--Mills connections over a compact Kähler manifold. Using normalized harmonic metrics, we construct a natural Hermitian metric on the unobstructed locus and show that, near the Hermitian locus, the unobstructed locus carries an almost hypercomplex structure which is compatible with the associated Riemannian metric.

We give a dynamical proof of Benoist's non-arithmeticity theorem for Jordan spectra of Zariski dense subgroups of connected semisimple real algebraic groups. After passing to a Zariski dense Schottky subgroup, we use the coding of the limit set to realize Jordan projections as periods of a vector-valued Busemann return map for an expanding map on the Furstenberg boundary. The key step is to prove that a suitable two-branch asymptotic discrepancy is not locally constant on the limit set. We also show that the same criterion applies beyond Lie groups; in particular, it yields a direct density result for multiplier spectra of hyperbolic rational maps whose Julia set is not contained in a circle.

We prove, to all orders, maximal transcendentality of the strongly coupled large-N Principal Chiral Model in the double-scaling regime introduced in our earlier work. We also prove that, after a natural shift of the coupling constant, the coefficients of the vacuum-energy expansion are expressed purely as polynomials in odd zeta values with rational coefficients. The first 35 explicitly computed orders reveal further number-theoretic regularities, pointing to hidden structure beyond maximal transcendentality.

In this paper, we prove the local boundedness of solutions to systems of partial differential equations in divergence form. More specifically, we consider systems that include the first variations of functionals depending on the spatial variable and exhibiting nonstandard growth with respect to the gradient, such as $$\int_Ω \left( 1+ h(|Du|)\right) ^{α(x)} \, d x,$$ where the convex function $h=h(t)$ does not satisfy the so-called $Δ_2$ property and does not exhibit the conventional polynomial growth behavior.

This paper studies the problem of detecting a change in the distribution of a sequence of independent observations when the post-change distribution is unknown. We propose a novel change detection algorithm, termed Predictive-Mixture CuSum (PM-CuSum), which combines predictive distributions constructed from sliding windows of different lengths within a CuSum recursion. The predictive distributions are aggregated using adaptive weights based on their recent predictive performance. We show that PM-CuSum achieves first-order asymptotic optimality under mild conditions, and that its asymptotic delay bound has a smaller remainder order than what is achieved procedures using a single fixed (even oracle) window. Numerical simulations demonstrate that PM-CuSum performs well compared to existing methods. Moreover, it is demonstrated that forming likelihood ratios using full predictive distributions can substantially improve performance compared to plug-in likelihoods.

The realization of Coulomb coupled electron-hole double layers in 2D semiconductor heterostructures has enabled the thermodynamic and transport studies of equilibrium exciton fluids without a magnetic field. By doping the exciton fluid with additional electrons/holes, an equilibrium fluid of trions - three particle bound states of electrons and holes - further emerge, providing the platform to explore new transport phenomena associated with such composite particles. Here we report the observation of a Hall effect for trions in MoSe2/WSe2 heterobilayers, which support Coulomb-coupled electron and hole fluids with tunable densities. The Hall effect arises from a Lorentz force on trions under a perpendicular magnetic field. It is manifested in both Hall drag measurements and standard Hall effect measurements on just one of the semiconductor layers. For negatively charged trions, an electron Hall effect is observed even in a hole doped WSe2 monolayer due to the presence of trion drags. The trion Hall effect also disappears when the trions are ionized at elevated temperatures and/or high trion densities. Our work opens the door for realizing quantum oscillations and the quantum Hall effect for trions.

By assuming reversible H-bonding association and dissociation, this work provides a description of the supramolecular structure and dynamics of monohydroxy alcohols (MAs) within the framework of a recently proposed living chain model (LCM). Structurally, reversible H-bonding leads to a single exponential distribution of the molar concentration of the supramolecular chain with length N. Dynamically, reversible H-bonding enables supramolecular chain breakage and recombination, which modifies the relaxation time of the supramolecular chains. In addition to the structural relaxation, tau_a, and the Debye relaxation, tau_D, two other relaxation times are revealed: the chain breakage time, tau_B, and the H-bonding lifetime, tau_H. The interplay among these four-time scales defines five distinct dynamics regimes. In Regimes I and V, no supramolecular chains form. In Regimes II and IV, supramolecular chains form and give a Debye relaxation. The characteristic chain length scales as Nc~tau_D/tau_a. In these two regimes, the H-bonding lifetime controls the Debye process. In Regime III, large supramolecular chains form. In all regimes with supramolecular chain formation, the Debye relaxation comes from the overall chain end-to-end dipole reorientation and scales with Nc. Excellent agreements between experiments and LCM have been observed, leading to quantitative descriptions of the dielectric and linear viscoelastic properties of MAs. These results thus establish a theoretical framework linking reversible H-bonding interactions to supramolecular structures, dynamics, and macroscopic properties of MAs.

Fluid and charge flow through interstitial volumes among impermeable randomly placed grains in porous materials ceases to occur at a critical concentration where networks of void volumes are disrupted at macroscopic scales. This critical density for void percolation can be difficult to calculate due to the irregular shape of the void regions. We develop and implement a geometrically exact method, scaling only linearly in the system volume, for identifying the shape and size of contiguous voids. In this manner, we calculate percolation thresholds for both grain cluster percolation (where system spanning networks of overlapping grains begin to appear with increasing density) and void percolation at much higher grain concentrations where networks of interstitial volumes no longer exist on macroscopic scales. For both the former and the latter, we calculate critical concentrations for inclusions in the shape of the Platonic solids (as well as truncated icosahedra) for both aligned and randomly oriented grains. In the case of critical densities for void percolation, the accuracy of our results is significantly improved relative to prior benchmarks. We also incorporate structural disorder of inclusions by considering impermeable grains in the form of cubes subject to a series of randomly placed and oriented fracture planes to mimic aggressively fractured inclusions found in nature. As the number of sustained slices becomes large, we find that the critical porosity for void percolation tends to 5%

We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $ω=2$ integer case of a problem of Erdős and Graham, stated only as a conjecture by Butler, Erdős and Graham (Integers 15 (2015), A51), who proved the $ω=3$ analogue. Counterintuitively the problem hardens as $ω$ decreases -- the induction's feed thins -- so $ω=2$ is the hard case; our proof adapts the Butler-Erdős-Graham induction to this thin-feed regime, where the entire content of the induction step reduces to an explicit onset inequality $Y_0(N)\le\min\{β(N),β'(N)\}$, proved for all $N\ge10$ by Olson's addition theorem and elementary Chebyshev bounds above a finite, machine-checked base range. The same engine extends to the rationals: for every squarefree $b$, every $a/b$ above an explicit threshold $\min\{B_{N_b}/6,\,1/5\}$ is $ω=2$ representable, unconditionally. As an application we give the first complete proof of the rational $ω=3$ statement -- every $a/b$ with squarefree $b$ is a sum of distinct sphenic unit fractions -- that Butler, Erdős and Graham conjectured but left unpublished; a descent settles every $ω\ge3$. What remains open is the $ω=2$ regime below this threshold, which we reduce to a single explicit conjecture -- that the gap-free floor of a semiprime subset-sum set tends to zero. This work is a human-AI collaboration: AI tools (notably Anthropic's Claude, used through Claude Code) contributed substantially to the Lean formalisation, the experiments, and the writing; correspondingly, every result is machine-checked in Lean 4 / Mathlib (no sorry; two cited classical axioms, plus the native_decide compiler-trust base for the finite computations), so its correctness is independent of the tools used.

We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If \(G\) is a finitely generated hyperbolic group and \(A\subseteq G\) is infinite with \[ A^2\subseteq AX \] for some finite \(X\subseteq G\), then either \(\langle A\rangle\) is virtually cyclic, or \(A\) has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if \(G\) is hyperbolic with finite generating set \(S\), then there is a constant \(c_{G,S}>0\) such that \[ |UV| \geq c_{G,S}\,\frac{|U||V|}{n+k+1}, \qquad U\subseteq B_n,\; V\subseteq B_k. \] The linear loss is optimal in order whenever \(G\) contains an element of infinite order. In the free group with its standard generating set one may take \(c_{G,S}=1/4\). We also prove that, in a free group, if \(U\subseteq S_n\) and \(V\subseteq S_k\), then \[ |UV|\geq \left(\frac{2}{3}+\frac{1}{3\cdot 4^{\min\{n,k\}}}\right)|U||V|, \] and this constant is sharp for all \(n,k\).

We extend the application of the Guillemin--Abreu formalism to holomorphic sectional and bisectional curvature on $\mathbb C^n$, $\mathcal O(-\ell)$, and Hirzebruch manifolds $M_{n,\ell}$, and further apply it to the total spaces of certain higher-rank vector bundles. The resulting formulas recover known positivity criteria and we show that, when the slope is sufficiently close to $1$, the extremal metrics on $M_{n,\ell}$ have positive holomorphic sectional curvature. We construct complete scalar-flat Kähler metrics on $ \operatorname{Tot}\bigl(\mathcal O(-k)\oplus\mathcal O(-k)\to\mathbb{CP}^n\bigr), $ and identify the Ricci-flat case, which occurs precisely when $2k=n+1$.

We develop a framework for conformal prediction in dyadic regression problems under complex missingness mechanisms. At the theoretical level, we develop general technical tools for establishing finite-sample validity of conformal prediction under distributional invariance conditions weaker than exchangeability. A key result handles the case where the sample itself is a random subset of the index set, a setting not covered by existing theory, via a novel bijection argument that constructs an explicit measure-preserving correspondence between events. In addition, we propose conformal prediction procedures for jointly exchangeable arrays, including full conformal, split conformal, a row-column approach exploiting similarities within rows and columns, and a selective conformal procedure achieving mask-conditional validity. For missing elements, we establish asymptotic validity of a weighted conformal procedure under a nonparametric graphon model for the missingness mechanism. We further establish conditional validity results for both continuous and discrete responses; to the best of our knowledge, this is the first formal proof of asymptotic conditional validity for weighted conformal prediction under a missing-not-at-random assumption. The proposed methods are illustrated on synthetic and real network data.

The deep neural network is a widely used framework in machine learning that has been widely applied in various fields. However, deep neural networks often involve a large number of parameters and inputs, many of which may be irrelevant to the goal or true output. These parameters and input variables not only increase computational complexity, but also contribute to additional computational cost. One solution to this problem is knockoff methods, which have proven successful in controlling false discovery rates in high-dimensional regression. Building on the knockoff methods and using the regularised neural network, this paper proposes three variable screening methods under the condition of controlling false discovery rates: one layer filter, multiple layers filter, and variable weight aggregation filter. In comparison with existing algorithms, we find that our algorithms show satisfactory performance.

Diffusion-based neural solvers have shown strong promise for combinatorial optimization (CO), but existing methods typically rely on supervised training with large collections of near-optimal solutions. In this work, we extend adjoint-based trajectory optimization methods to discrete combinatorial domains. We formulate diffusion-based CO as a stochastic control problem over Continuous-Time Markov Chains and introduce discrete adjoint dynamics for propagating optimization signals through discrete generative trajectories. Building on this formulation, we propose Combinatorial Adjoint Matching (CAM), an unsupervised training framework for discrete diffusion solvers with structured and low-variance trajectory-level optimization signals. Empirically, CAM consistently outperforms existing unsupervised diffusion baselines and achieves performance competitive with strong supervised diffusion solvers and even traditional solvers across diverse combinatorial optimization problems. Our code is available at https://github.com/Shengyu-Feng/CAM.

Cryptocurrencies are increasingly adopted as investment assets, making their interactions with traditional financial markets central to cross-asset diversification and systemic risk. This paper studies the integration of cryptocurrencies, fiat currencies, and S&P500 equities using a balanced panel of 381 assets from October 2017 to February 2024. We combine rolling correlation networks, community structure, market-specific and system-wide Turbulence Indices, and VAR-based connectedness analysis to examine how market stress, network structure, and shock transmission vary across financial regimes. The results show that cross-asset integration is episodic. In calm periods, the three asset classes remain relatively segmented, whereas under stress, local clustering increases, modular separation weakens, and communities become more compositionally mixed across asset classes. Connectedness analysis further shows that regime shifts alter the structure of transmission rather than simply increasing spillover magnitudes. In high-turbulence states, fiat-market turbulence becomes the dominant propagation channel, while network clustering and modularity play a greater role in transmitting forecast uncertainty. These findings support the interpretation of network structure as an emergent, state-dependent transmission layer rather than a persistent exogenous driver of turbulence. The results highlight the need for regime-aware risk monitoring, since full-sample connectedness estimates can understate the cross-asset coupling that emerges precisely when diversification benefits are most fragile.

The 2024-2025 Nobel and Turing awards recognised AI and quantum science simultaneously. Yet no deployed system has brought these streams together for the public. This paper presents QSignAI, a production-deployed platform demonstrating a bidirectional AI-quantum relationship in a real-time event participation system. We address three questions: can quantum-randomness generation via a two-source extractor be embedded in an AI-driven social platform with acceptable latency; can an AI bot make quantum phenomena perceptually legible to general audiences; and does the combined system work in practice? A conversational bot routes each participant's first message through a quantum pipeline comprising a Toeplitz two-source extractor over independent single-qubit Hadamard measurements on SV1 and DM1 simulators, plus a 2-qubit Bell state, producing a unique quantum-randomness-seeded identity signature per participant. The first two questions are answered through system architecture and qualitative deployment evidence from live events; the third through successful production deployment. The current deployment uses cloud quantum simulators; physical QPU randomness is the near-term extension. Measurable benchmarks are identified as priority future work.

Schrödinger bridges for time series (SBTS) generate synthetic paths by projecting, in relative entropy, a Brownian reference onto the path laws that match the joint distribution of the data on the observation grid. The Brownian reference, however, fixes the quadratic variation of the generated paths, which is restrictive when stochastic volatility, correlated noise, or rank-deficient covariance structures must be reproduced. We introduce "Triangular-Reference Schrödinger Bridges for Time Series" (TR-SBTS), which keeps the entropy-projection backbone of SBTS but replaces the Brownian reference by a triangular, volatility-informed, intervalwise frozen reference on a state augmented with latent covariance descriptors. The construction remains a single entropy projection on the augmented state: the minimiser is the \(h\)-transform of the reference, and on each frozen interval the optimal drift has the logarithmic-gradient form \(b^\star(t,x)=A\,\nabla\log H(t,x)\), intrinsic to the active covariance directions when the frozen covariance \(A\) is degenerate. We prove stability of the frozen approximation and consistency of the associated regularised kernel estimators, describe a reference-aware Nadaraya--Watson implementation of the conditional next-increment law, and evaluate the construction on numerical experiments.

We introduce the \emph{Morse ensemble polynomial} $\ME_K(z_0,\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\ME_K = \sum_M \prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete distribution of Morse vectors across all discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes. Our main results are the following. \textbf{(I) The Laplacian Formula}: for any connected graph $G$, $\ME_G = z_1^{m-n}\det(z_0z_1\,I_n + L_G)$, identifying $\ME_G$ as a complete Laplacian spectral invariant and showing $\ME_G$ to be incomparable with the Tutte polynomial. \textbf{(II) The Top-Face Recursion}: adding a $d$-simplex $σ$ (with $\partialσ\subset K$) to a complex $K$ gives a recursion $\ME_{K\cup\{σ\}} = z_d\cdot\ME_K + \sum_{τ\precσ}(\ME_{P(K')\setminus\{σ,τ\}}-F(K,σ,τ))$. The correction term is controlled by the top incidence graph: an incidence-separation criterion detects exactly when $F=0$, and the incidence distance gives the leading obstruction term. As a topological application, this recursion gives exact coefficient recursions for perfect and optimal discrete Morse vectors. \textbf{(III) The independence ME polynomial} $Φ(G) := \ME_{\mathrm{Ind}(G)}$ is a fine graph invariant which strictly refines the graph-level Morse ensemble $\ME_G$, separates examples not distinguished by $T_G$ and $I(G;t)$, and records collapse-level information of $\mathrm{Ind}(G)$ through coefficients such as $[z_0]Φ(G)$.