Planar germanium heterostructures hosting hole-spin qubits are among the leading platforms for scalable semiconductor-based quantum computing. Yet, device performance is hindered by significant quantum dot variability, which leads to uncertainty in qubit energy levels and random orientations of the spin quantization axis. Tailored control of the g-tensor offers a strategy to overcome these limitations and achieve more reliable qubit operations. Here, we introduce a flexible optimization framework for engineering g-tensor properties. As a benchmark, we numerically obtain the optimal reshaping of the out-of-plane potential in a SiGe-Ge-SiGe quantum well to suppress the in-plane g-tensor components and realize the recently proposed gapless single-spin qubit encoding. This reshaping is achieved through heterostructure engineering, specifically by adjusting the silicon concentration within the quantum well, though the framework remains readily adaptable to alternative design objectives. Our results provide practical design principles for improving the tunability of the spin response, paving the way towards large-scale germanium-based quantum computers.

This paper studies a key research question: how to achieve perfect privacy in over-the-air computation (AirComp)? The problem is particularly intriguing due to a dilemma. Real-field operations can ensure invertibility but generally introduce statistical dependence, resulting in inevitable privacy leakage. In contrast, modulo operations can decorrelate the output from the original message, but suffer from the ill-posed invertibility when applied over non-prime groups (e.g., the real field). This raises a subtle yet fundamental question: Does perfect privacy intrinsically conflict with AirComp? We show that the answer is no. By carefully leveraging the interplay between real-field and modulo operations, perfect privacy and accurate computation can, in fact, be achieved simultaneously, enabling perfectly private aggregation.

Relational database management systems (RDBMSes) can process general-purpose queries, but often have lower performance compared to custom-built solutions for specific queries. For example, consider a group-by query over a few known groups (e.g., grouping by country). While an RDBMS would likely use a hash map to do the grouping, a faster method could hard-code the expected groups into the query executor. But such workload-specific techniques, which we call query accelerators, are not widely used in practice because the engineering effort (optimizer and engine changes, potential bugs) does not justify the isolated performance gains (speedup on a single specific query). We propose Tailwind: an external query planner that brings accelerators into any RDBMS that supports data import/export. Users define their accelerators using abstract logical plans (ALPs): a new mostly-declarative abstraction over relational operators built on regular tree expressions. ALPs allow Tailwind to automatically build customized neural network models to estimate when using a particular accelerator is beneficial. At runtime, Tailwind sits atop an RDBMS and transparently rewrites queries to run across one or more accelerators when predicted to be beneficial, falling back to the underlying RDBMS when not. On Redshift and DuckDB with a library of four diverse accelerators, Tailwind accelerates TPC-H queries by 1.38x on average (up to 29x).

Superconductivity in strontium titanate has remained enigmatic for more than 50 years. The LaAlO$_3$/SrTiO$_3$ (LAO/STO) heterointerface enables systematic dimensional confinement, from a two-dimensional electron gas to quasi-one-dimensional nanostructures, providing access to this quantum state. Transport measurements in patterned devices reveal puzzling phenomena, including width-independent critical currents and anomalous pairing suggestive of one-dimensional behavior, but direct local probes of the patterned interface and its superconducting response have been lacking. Here we use ultralow-temperature non-contact atomic force microscopy, dissipation spectroscopy, and Kelvin probe force microscopy to locally probe signatures of superconductivity in patterned LAO/STO devices. Spatially resolved energy-dissipation measurements reveal superconducting signatures, with features confined in some devices to edge channels approximately 200 nm wide. Dissipation spectra exhibit a characteristic nonlinear bias dependence that provides a local diagnostic of superconductivity, consistent with the intermediate carrier-density regime near the superconducting dome, and persisting up to the critical field. These results establish atomic force microscopy as a local probe of superconductivity in patterned LAO/STO structures and provide a route to addressing longstanding questions about quantum confinement and transport anomalies in correlated oxide nanostructures.

The WISE source, J141046.00+740511.2, has been recently observed from radio to $γ$ rays. Although the optical spectrum is consistent with a BL Lacertae (BL Lac) object, the source displays unusually weak radio emission, which challenges standard interpretations. Our aim is to understand the origin of the broadband emission from J141046.00+740511.2, using a leptonic model of an extended jet. To obtain the distribution of electrons along the conical jet, we solved a steady-state convective transport equation. Emissivities were computed along the jet and integrated over the cone volume to obtain the observed flux. Our model successfully reproduces the observed multiwavelength spectral energy distribution from radio to $γ$ rays and naturally accounts for the source's low radio flux without invoking extra emission zones. We also reproduce the mid-IR emission within the same framework. These results demonstrate that extended jet leptonic models can robustly describe the broadband physics of radio-weak BL Lacs.

Computer architecture simulation is essential for evaluating new designs without the need for costly tapeout. The community has developed dozens of valuable simulators that have enabled significant architectural advances. However, using and developing simulators remains a major barrier due to ad-hoc component interfaces, strict deployment requirements, the burden of managing performance optimizations like parallelization at the component level, and limited monitoring and visualization capabilities. The root cause of these limitations is the systematic neglect of user and developer experience in favor of technical functionality. We believe that only by separating technical concerns from user and developer experience concerns -- through a dedicated simulation engine decoupled from hardware models -- can the community overcome these fundamental obstacles and enable more productive architectural research. Akita embodies this philosophy as a dedicated simulation engine that cleanly separates infrastructure from architectural models. Smart Ticking and Availability Backpropagation let developers write simple cycle-based code while achieving event-driven performance. Parallel simulation happens transparently -- developers write single-threaded code while Akita handles multi-core execution. Akita's simple, uniform, yet powerful simulation tracing support enables real-time monitoring and post-simulation visualization. We demonstrate the flexibility of Akita through case studies, including the development of a trace-based DNN simulation and a RISC-V CPU simulation, showing how prioritizing developer experience accelerates architectural research.

Building on the genus-3 reduction $C_A : w^2 = λ^8 + A λ^4 + 1$ established in our companion paper (arXiv:2604.09328), we give an unconditional proof of the perfect-cuboid conjecture ("Conjecture B") on $1{,}072$ explicit master-tuple fibers, excluding all rational $(a,b)$-specialisations on each such fiber. Our three main contributions are: (i) a structural classification theorem showing that every primitive Euler-brick arises from the standard $(a,b,m,n)$-parametrisation up to scaling; (ii) a torsion-intersection argument applied to the elliptic quotients $E_A'$ and $E_A''$: whenever the rank-zero hypothesis and the appropriate torsion condition hold for one of them, $|H_{m,n}(\mathbb{Q})| = 8$ is forced, with the eight points all corresponding to degenerate bricks; (iii) two complementary techniques to verify the rank-zero hypothesis algorithmically -- PARI's ellrank (2-descent) and, where this is ambiguous, Sage's exact rational evaluation of $L(E,1)/Ω_E$ via modular symbols, which combined with the modularity theorem, Kolyvagin's theorem, and Edixhoven's bound on the Manin constant for semistable curves yields an unconditional rank-zero certificate -- together with an explicit lift count refining the naive torsion-intersection bound when the torsion is larger than the leading case. We exhibit $1{,}072$ such fibers with $\max(m,n) \le 100$ on which Conjecture B is thereby established unconditionally.

To couple many independent modes from free space to on chip, the key challenge is not enhancing the many necessary coupling rates (scattering-matrix elements) between targeted mode pairs. Instead, the key is to avoid additional cross-couplings to undesired modes, due to the presence of multiple simultaneously satisfied phase-matching conditions. With this principle, we identify scaling laws for the maximum number of high-efficiency multi-mode couplings that may be achievable for a given refractive index and design region, which are strongly supported by extensive numerical inverse-design experiments in 2D (one-dimensional coupler patterns, scattering in 2D). For such couplers, typical mode counts of 5--10 appear achievable. Three-dimensional couplers (patterned across two dimensions) can be markedly better, with tens of Fourier components in a single-layer device offering the possibility of high-efficiency coupling of hundreds to thousands of modes in relatively compact form factors. Numerical simulations of such a device, without any parameter optimization, predict efficiencies on the order of 5\% for 100 modes -- a collective order-of-magnitude improvement over previous designs.

Range anxiety and long recharging times remain critical barriers to electric vehicle adoption. Dynamic Inductive Charging (DIC) offers a compelling solution by enabling wireless power transfer while driving, potentially reducing battery size requirements and thus vehicle costs. However, DIC infrastructures are expensive and power-constrained, requiring intelligent resource allocation to maximize user satisfaction and economic viability. We propose a Model Predictive Control framework for optimal power allocation in DIC systems, using edge computing and vehicular communications to prioritize vehicles with critical battery states. The framework is implemented and evaluated through SUMO-based simulations on a realistic 10 km urban scenario in Istanbul, Turkey, under varying traffic intensities. Results demonstrate two critical limitations of uncoordinated allocation. First, resource utilization remains suboptimal despite available power when demand saturates system capacity. Second, when demand exceeds capacity, uniform distribution of power leaves a heavy tail of critically unsatisfied vehicles that may require emergency stops. Our MPC-based strategy addresses both regimes -- maximizing power utilization during saturation through dynamic stripe rebalancing, and improving satisfaction fairness under scarcity by aggressively prioritizing depleted batteries at the expense of well-charged vehicles. The framework and simulation tools are released as open-source to support further research in this emerging domain.

We investigate the dynamics of dissipative systems with stochastic forcing and focus in particular on mean-square stability. First we show, under a natural condition on the drift and diffusion, that the stochastic system is mean-square dissipative. Next we examine the linearised system and state conditions ensuring that perturbations of a linear system with affine noise are bounded. We then relate the mean-square dynamics of the nonlinear and linearised systems. The approach gives a straightforward deterministic method to examine the effects of stochastic forcing on the stability of equilibria of deterministic systems and to obtain bifurcation diagrams that can be included into standard numerical continuation packages. The technique is illustrated numerically on some standard and non-standard examples.

The impossibility of eliminating hallucination, understood here as incorrect definite answers, in sufficiently expressive yes-or-no formal domains is an immediate consequence of classical undecidability theorems. This note does not revisit that forced-answer obstruction as its main claim. Instead, it attempts to formally describe the corresponding limitation for abstaining systems. Abstention can trivially avoid hallucination if the system is allowed to abstain on every input; the substantive question is how large the domain of guaranteed correct non-abstaining answers can be. We formulate this question using separation in the arithmetical hierarchy. Given disjoint sets $A$ and $B$, any system that answers Yes on all queries indexed by $A$ and No on all queries indexed by $B$ induces a separator of $A$ from $B$. By combining this observation with the classical existence theorem of $Δ_{n}^{0}$-inseparable pairs of $Σ_{n}^{0}$-sets, we yield a computability-theoretic trade-off between avoiding hallucination by abstention and maintaining a large domain of guaranteed coverage.

In ferroelastic materials, spontaneous symmetry breaking leads to the formation of twin domains. Although the bulk crystal typically remains centrosymmetric, inversion symmetry can be locally broken at the domain walls, potentially changing phonon selection rules and enabling local anharmonic phonon coupling. Here we report direct evidence of such anharmonic coupling in ferroelastic LaAlO$_3$ using two-dimensional Raman-terahertz spectroscopy. We attribute the cross-peaks observed in the two-dimensional spectra to both mechanical and electrical anharmonicity between the $A_{1g}$ Raman-active phonon and the $E_g$ phonon, which acquires finite infrared activity through local inversion symmetry breaking at ferroelastic domain walls. These findings provide new insight into the complex lattice dynamics of ferroelastic materials and highlight the potential of two-dimensional Raman-terahertz spectroscopy to uncover subtle symmetry breaking through the detection of intrinsically weak anharmonic signals.

We investigate the photoproduction of the tensor meson $f_2(1270)$ on the proton within a Regge-based framework, focusing on the reaction $γp \to p f_2(1270)$ in the few-GeV energy region. The production mechanism is modeled through the exchange of vector-meson Regge trajectories in the $t$-channel, including both $ρ$ and $ω$ exchanges with phenomenologically motivated couplings. The scattering amplitudes are derived using Reggeized propagators and effective hadronic vertices, allowing for the calculation of differential cross sections in the narrow-width approximation. We further extend the analysis to the $π^+π^-$ invariant mass distribution by incorporating a relativistic Breit--Wigner description of the $f_2(1270)$ resonance.

Core-collapse supernovae provide natural laboratories for the production of new light particles. In particular, axion-like particles (ALPs) can be constrained via SN1987A cooling arguments. However, significant astrophysical and nuclear uncertainties imply that such bounds may vary strongly depending on modeling choices, even when expensive simulations are employed. In this context, semi-analytic methods offer a simple and fast alternative for deriving new-physics constraints. Building on a previous semi-analytic framework, in which proto-neutron star (PNS) observables are expressed in terms of six global PNS parameters, we include a finite ALP mass in the calculation and derive bounds in the axion-nucleon coupling versus mass plane. The obtained bounds are in good agreement with previous results from numerical simulations, demonstrating the robustness of the method. We also illustrate the sensitivity of the bounds to different PNS parameter calibrations, nuclear effects and cooling exclusion criteria.

We study Turán-type extremal problems for distance graphs, motivated by work of Csikvári, Bollobás, Tyomkyn, and Uzzell. We determine the maximum number of vertex pairs at distance three in an $n$-vertex graph with no triangle formed by these pairs, resolving the first case of a conjecture of Tyomkyn and Uzzell. We also determine the maximum number of vertex pairs at distance two in an $n$-vertex graph with no triangle formed by these pairs and give a complete characterization of the extremal graphs, settling another problem of Tyomkyn and Uzzell.

We apply a systematic inverse design approach to discover foundry-compliant, multilayer grating couplers that can efficiently couple a number of independent waves from free space to on-chip propagating modes. For visible- and near-infrared couplers, we find that minimum feature sizes are by far the most important constraint to tailor the design algorithms around. If, additionally, one forces the optimization to be robust to over- and under-etch errors, the resulting designs exhibit stable optimal efficiencies in the presence of other imperfections (critical dimension variations, overlay mismatch, and sidewall angle variation). The foundry-compliant designs exhibit moderate efficiency penalties as feature sizes increase, but no change to simple underlying scaling laws with respect to requisite numbers of layers and layer thicknesses. These results establish a practical, generalizable framework for high-efficiency multimode coupling within the constraints of modern semiconductor foundries.

Emergent magnetic states at oxide interfaces arise from the interplay of charge transfer, orbital reconstruction, and dimensional confinement, offering a route to engineered correlated-electron behavior in nanoscale spintronic materials. Here, we combine in situ synthesis, polarization-dependent angle-resolved photoelectron spectroscopy, X-ray magnetic circular dichroism, and first-principles electronic-structure calculations to investigate LaNiO3/CaMnO3 superlattices. We show that reducing the LaNiO3 thickness drives a metal-insulator transition accompanied by loss of electronic coherence and an orbital-polarization crossover in the ultrathin limit. These changes weaken charge transfer across the interface and suppress the interfacial Mn magnetic moment in CaMnO3, revealing that the emergent ferromagnetic state is directly governed by electronic confinement in LaNiO3. The insulating state and orbital reconstruction are reproduced by density functional theory combined with dynamical mean-field theory. Together, these results establish a direct and tunable coupling among electronic, orbital, and magnetic degrees of freedom in oxide heterostructures.

Despite many decades of research, economically grounded models that analyse energy consumption and energy-efficiency adoption within a unified framework remain underdeveloped. This article addresses this gap by proposing a model of consumption, investment, and energy-efficiency adoption under uncertainty. It develops new definitions of the rebound and backfire effects, and integrates their welfare implications into a model of optimal subsidy design. Macro-level technology diffusion and energy consumption across heterogeneous agents are also formalised. Explicit results for core objects are derived, including the adoption threshold and post-adoption strategies, and these are shown to depend on agent wealth, introducing a novel channel through which financial conditions influence technology-adoption decisions. An approximation scheme is proposed to estimate welfare implications explicitly. Adoption of energy efficiency is shown to be welfare improving in the main. A detailed case study of a representative German single-family home illustrates the theoretical results. Numerical analysis indicates that the subsidy policy effectively steers aggregate energy consumption.

We address the problem of numerically approximating the velocity and pressure governed by the Stokes system when the boundary conditions are only partially known and thus do not uniquely determine the velocity-pressure couple. We propose an algorithm that takes advantage of available linear measurements of the velocity and pressure to construct a numerical approximation. This approximation is guaranteed to be near-optimal in the sense that it approximates the velocity-pressure couple that minimizes, in the energy norm, the distance to all other solutions satisfying the measurements and the Stokes system.

The black hole no-hair theorem is traditionally derived from the uniqueness theorems of general relativity. We show that a quantitative form follows from unitarity together with the standard semiclassical assumptions of horizon causality and interior accessibility. For a semiclassical black hole, we prove that the trace distance between exterior states corresponding to two same-charge infalling states is bounded by $2\sqrt{2\varepsilon}$, where $\varepsilon$ quantifies the diamond norm departure of the interior channel from a perfect isometry which is a quantitative measure of horizon-smoothness violation that upper-bounds $1 - F_I$, where $F_I$ is the interior fidelity capturing how faithfully the infalling state is retained. Inverting this relation yields a trade-off inequality, $\varepsilon \geq D_{\max}^2/8$, between the maximum exterior distinguishability $D_{\max}$ and the degree of horizon smoothness. This establishes that observable exterior quantum hair is quantitatively incompatible with exact horizon smoothness under unitary evolution: any model predicting nonzero exterior hair must violate the equivalence principle at the horizon by a quantifiable amount. Pre-existing entanglement with the infalling system is the only channel for quantum hair compatible with both unitarity and horizon smoothness.

Time-to-event outcomes are commonly used as primary endpoints in randomized clinical trials. Despite this, relatively little work incorporates baseline covariate information while also accounting for stratified randomization, a common form of randomization. Moreover, leveraging efficiency gains using these approaches typically requires pre-specifying a subset of covariates that are most predictive of the outcome -- a challenging task in practice, as most trials collect dozens of potentially prognostic baseline variables. In this work, we build on existing literature to propose a data-adaptive and model-robust covariate adjustment method for time-to-event outcomes. Our approach, based on targeted minimum loss-based estimation, allows for data-adaptive covariate selection and model-robust efficient inference on functionals of the survival curve while accounting for stratification. Through extensive simulations and analysis, we showcase the simplicity and improved precision of our method when the covariate set is not known a priori.

This paper presents a real-world experimental analysis of a modular reconfigurable intelligent surface (RIS) prototype designed to operate in the 5G N78 band. Unlike most RIS studies in the literature that focus on simulations or controlled setups, the proposed system is validated through three phases consisting of indoor measurements, outdoor long-range tests, and deployment in a live commercial 5G standalone network. The RIS is exploited to enhance coverage in a non-line-of-sight (NLoS) zone, identified through baseline drive tests. Results show promising gains in RSRP and SINR, while also restoring 5G service at user locations where access was previously not available. The results highlight the practical potential of RIS for coverage enhancement in operational 5G networks.

Deep strong light-matter coupling represents an extreme non-perturbative regime of quantum electrodynamics, in which the interaction strength exceeds the bare frequencies of the uncoupled systems. The ground state features strong quantum correlations between photons and matter excitations, and new cavity-driven phase transitions are expected to occur. Whether a superradiant quantum phase transition, marked by spontaneous dipole ordering and photon condensation, is possible has remained a long-standing and controversial question. Such phenomena have been proposed to arise in exotic electronic systems hosting Dirac and Kane fermions, owing to the formal absence of an $A^2$ term in their low-energy Hamiltonian. Here we exploit the ultralow effective mass of Kane fermions to realise Landau polaritons in a bulk mercury cadmium telluride layer coupled to a Fabry-Perot resonator. Using thermally tunable carrier density, we continuously tune the coupling from the weak to the deep-strong regime, achieving a record normalised coupling ratio exceeding 1.6 above room temperature. The measured polariton spectra are in excellent agreement with a rigorous, gauge-invariant microscopic theory. Despite the nonlinear Landau level structure of relativistic Kane fermions, we show that a diamagnetic $A^2$ term naturally emerges and precludes a superradiant phase transition. These results resolve the long-standing controversy surrounding cavity quantum electrodynamics of relativistic-like matter systems, extend deep-strong-coupling physics to Kane fermions, and open new opportunities for polaritonic semiconductor devices operating in extreme light-matter coupling regimes.

We discuss a Quantum-Enhanced Computing Continuum, a heterogeneous, hybrid architecture that integrates quantum processing units (QPUs) within an Edge-Cloud-HPC fabric. Promote sustainability by shifting from performance to "energy-aware integration.' The architecture has three layers: a Physical Layer with shared fiber-optic infrastructure, a Control and Orchestration Layer managed by the user, and an Application Layer with an Adaptive Quantum Classical Fusion (AQCF) framework. Tighter system integration, like moving from cloud coupling to cryogenic logic, reduces energy waste and "thermal footprints.' The aim is a Green Performance Advantage: energy per problem solved in the era of Advanced Computing.

Vehicular sensing-based intelligence has made substantial progress in transportation systems, leading to higher levels of safety and sustainability for smart cities and autonomous systems. This paper proposes a new approach to learn an interactive generalized dynamic Bayesian network (I-GDBN) model aiming to predict future LiDAR sensor blockages from time-sequence-based 3D point cloud perception. During learning, separate GDBN models are trained for various vehicles in normal and blockage situations. To perform the interaction between multiple vehicles, a high-level vocabulary is formed. Initially, during testing, the best generative model for either normal or blockage situations is selected. An interactive Markov jump particle filter (I-MJPF) is then proposed to leverage the probabilistic information provided by the I-GDBN to infer the blockages and detect the abnormalities at the high abstraction level. The proposed interactive model allows better self-aware and explainable capabilities that can adapt to blockage scenarios, which is also helpful when sensors fail to provide observations.

For a graph \(G\), let $avm(G)$ denote the average size of its maximal matchings. This parameter was introduced by Engbers and Erey in the study of extremal problems for maximal matchings, and they asked for extensions from trees and unicyclic graphs to \(k\)-cyclic graphs. In this paper, we solve the first non-unicyclic case by determining the minimum value of $avm(G)$ over all connected bicyclic graphs with \(n\) vertices and \(n+1\) edges. We prove that, for every connected bicyclic graph \(G\) of order \(n\ge 5\), \[ \operatorname{avm}(G)\ge \frac{4n-11}{2n-5}. \] Moreover, equality holds uniquely for the graph obtained from two triangles sharing a common edge by attaching all remaining \(n-4\) pendant edges to one of the two vertices of degree \(3\). The key point is to translate the minimization of \(\operatorname{avm}(G)\) into structural restrictions on small maximal matchings, which are then analyzed through the three possible bicyclic core types.

We report on a new type of rectifier which is in full contact equilibrium and thus, if down-sized to the nanoscale, shows no drift even if exposed to elevated temperatures and/or extreme waiting times. This is in contrast to existing diodes which rely on frozen doping profiles and are hence non-equilibrium devices. Our rectifiers are related to Schottky diodes but employ "dopants" whose mobilities are high enough to follow the electrical field quickly but low enough to not compete with the electrons in terms of conductivities. In order to realize such a device based on mixed conductors, we use nanosized TiO2 films on Ru as a substrate which can store Li at the interface according to a job-sharing mechanism (Li-ions on the TiO2 side, electrons on the Ru side). The excellent functionality of this nanoionic device is demonstrated (e.g., current on-off ratio can exceed 6-7 orders of magnitude) and the additional advantages stressed (such as ease of preparation and tuning the characteristics electrochemically).

In a recent preprint (Mosegaard and Curtis, 2024, arXiv:2411.13570v2) we analyzed the consequences of ignoring the well-known inconsistency of classical conditional probability densities. We explained how this inconsistency, together with acausality in hierarchical methods, invalidate a variety of commonly applied Bayesian methods when applied to problems in the physical world. Yan et al., 2026, (arXiv:2603.27038v1) published a note, in which they claim, contrary to our preprint, that there are no inconsistencies if one uses the method of conditional expectations to derive probabilities. Furthermore, they believe that there are mathematical errors in our exposition and in our use of the Bayesian framework. This note is a response to the claims made by Yan et al. Yan et al. do not discriminate between physical and statistical consistency. Their note addresses statistical consistency of a solution under a change of variables; this is already known to be resolved by using the theory of conditional expectations. By contrast, our preprint concerns the physical consistency of any solution under a change of mathematics used to derive that solution. It demonstrates that widely used methods to compute Bayesian posterior solutions are physically inconsistent under a change of variables. Their note does not, therefore, address the tenet of our preprint. We show herein that the theory of conditional expectations does not resolve physical inconsistency, and that Yan et al. make mathematical errors. We conclude that their claims are unfounded, and in some cases we show that their critique is meaningless. The conclusions of our preprint therefore stand.

Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over all possible multiplication orders in the Leibniz formula for the determinant. He used the symmetrized determinant to design algorithms estimating the permanent of a matrix. To this end, he showed that there is a $O(n^{r+3})$ algorithm computing $\sdet$, where $r$ is the dimension of the algebra, and is therefore polynomial-time computable for fixed $r$. In this work, we study the algebraic properties and complexity of $\sdet$. While most of the properties of the ordinary determinant don't generalize to $\sdet$ defined on non-commutative algebras, we show that the principal minor expansion of the $\sdet$ is analogous to the ordinary determinant. Second, we prove that there exists a polynomial-sized algebra such that computing the symmetrized determinant is $\sharpP$-hard. Third, we show that the associated polynomial family is $\VNP$-complete over a suitable polynomial-dimensional algebra in the non-commutative setting. Further, when seen as a family of polynomials over the matrix algebra, it is also $\VNP$-complete in the commutative setting. This places the symmetrized determinant among the natural complete families arising from algebraic computation.

Constant Product Market Makers use fees that are typically fixed proportions of trade size. When these fees are automatically reinvested into the pool, as in Uniswap~V2 and some designs of Uniswap V4, the final state after a trade can depend on how the trade is split into smaller transactions. This path dependence complicates the risk assessment for liquidity providers and affects composability guarantees. We characterize the functional class of fee structures that ensure path independence: the combined fee factor must depend only on the current pool invariant k=xy. For this class, we derive a system of ordinary differential equations governing pool dynamics and obtain a closed-form integral exchange formula. Within this class, we construct a parametric family of fee functions that achieve zero Impermanent Loss for a given initial pool state, and prove that no universal fee function can eliminate Impermanent Loss for all initial states simultaneously. We analyze implications for arbitrage windows and slippage, and validate our theory through controlled simulations. Our framework provides protocol designers with a principled approach to fee optimization that aligns liquidity provider and trader incentives while preserving composability.