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arXiv2026

Self-replicating seedbox servers using programmable money

Centralized content distribution makes availability depend on a single operator's survival and willingness to serve. EternalSeedBox replaces the operator and network with inherited economic parameters: each node is a VPS that seeds media over BitTorrent, holds a Bitcoin wallet, and autonomously decides every twelve hours whether to renew its lease, spawn a child, or sweep its funds to a healthier peer before expiring. A single genesis node seeds the fleet, and every node thereafter is provisioned, funded, and retired autonomously. We validate the design against faithful replicas of both the Bitcoin payment network and the SporeStack VPS marketplace by running the unmodified node code. A lump sum of EUR 10,000 grew the fleet to 33 nodes before capital exhausted at day 153. With simulated income, the fleet held 40--80 live nodes across 510 days, recording 268 births and 190 deaths. A heritable caution trait was introduced to diverge across generations: low-caution lineages reproduced faster during high-income phases, while the survival advantage expected of high-caution lineages during income pauses did not appear, leaving selection in favor of low-caution nodes. The fleet tolerates high node turnover because reproduction depends on any node holding a surplus, not any single node surviving. EternalSeedBox shows that a content distribution network can lease, pay for, and replenish its own hardware without a human operator after genesis, provided income exceeds per-node rent.

physics.soc-ph↗
arXiv2026

An Energy-Based Conservative-Dissipative Latent Neural Evolution Operator for Magnetization Dynamics

We develop an energy-based reduced-order model for micromagnetic magnetization dynamics that couples a convolutional autoencoder to a structured latent neural ordinary differential equation. Motivated by the precessional-dissipative structure of the Landau-Lifshitz-Gilbert equation, the latent vector field is generated from the gradient of a learned scalar potential through an antisymmetric operator and a symmetric positive-semidefinite dissipative operator. This potential is learned in nonunique latent coordinates and is not identified with the Gibbs free energy, but decreases monotonically along autonomous continuous-time solutions, while the antisymmetric component permits motion along its level sets. The encoder, decoder, latent energy, and operators are trained jointly on short trajectory windows using latent and decoded-rollout losses alone, without time-derivative supervision, physical-energy labels, or dissipation penalties. At inference, an initial state is encoded once, evolved in latent space, and decoded only at the requested output times, enabling substantially cheaper trajectory prediction than the micromagnetic solver used to generate the training data. We compare quadratic, deep, and additive deep-quadratic latent energies on two datasets parameterized by field amplitude and generated for the two applied-field directions of the NIST $μ$MAG Standard Problem 4. Dissipative-only and antisymmetric-dissipative models achieve comparable accuracy on short training-style windows but differ substantially on uninterrupted rollouts, for which the antisymmetric-dissipative models provide markedly more accurate trajectory predictions. The deep-quadratic energy gives the best overall accuracy for both field directions and exhibits slower error growth when rollouts are extended to twice the training horizon.

cs.LG↗
arXiv2026

Towards Efficient Parametric State Estimation in Circulating Fuel Reactors with Shallow Recurrent Decoder Networks

The recent developments in data-driven methods have paved the way to new methodologies to provide accurate state reconstruction of engineering systems; nuclear reactors represent particularly challenging applications for this task due to the complexity of the strongly coupled physics involved and the extremely harsh and hostile environments, especially for new technologies such as Generation-IV reactors. Data-driven techniques can combine different sources of information, including computational proxy models and local noisy measurements on the system, to robustly estimate the state. This work leverages the novel Shallow Recurrent Decoder architecture to infer the entire state vector (including neutron fluxes, precursors concentrations, temperature, pressure and velocity) of a reactor from three out-of-core time-series neutron flux measurements alone. In particular, this work extends the standard architecture to treat parametric time-series data, ensuring the possibility of investigating different accidental scenarios and showing the capabilities of this approach to provide an accurate state estimation in various operating conditions. This paper considers as a test case the Molten Salt Fast Reactor (MSFR), a Generation-IV reactor concept, characterised by strong coupling between the neutronics and the thermal hydraulics due to the liquid nature of the fuel. The promising results of this work are further strengthened by the possibility of quantifying the uncertainty associated with the state estimation, due to the considerably low training cost. The accurate reconstruction of every characteristic field in real-time makes this approach suitable for monitoring and control purposes in the framework of a reactor digital twin.

cs.LG↗
arXiv2026

Learning Spectral-Like Mesh-Free Discretisations

Meshfree methods such as smoothed particle hydrodynamics (SPH) with kernel corrections, radial basis function-generated finite differences (RBF-FD), and the local anisotropic basis function method (LABFM) construct discrete differential operators by imposing polynomial consistency on a local stencil. For stencils containing more nodes than there are consistency constraints, the resulting linear system is underdetermined, and the remaining degrees of freedom are fixed implicitly by the choice of kernel, basis preconditioning, or a minimum-norm condition. Polynomial consistency constrains the operator only in the low-wavenumber limit, and no part of the construction selects for accuracy at the wavenumbers where fine-scale content resides. We introduce Spectral-like Neural Discretisation (SpeND), in which the choice of those degrees of freedom is cast as a learning problem: stencil weights are parametrised by a neural network conditioned on the local node geometry, trained to approximate the modal response of a spectral operator over the resolvable band. A hard-constrained projection layer maps the network output onto the affine subspace of consistent weights, so that polynomial consistency holds exactly by construction rather than as a penalty. Training is self-supervised and physics-agnostic, requiring no reference solutions; the objective minimises dispersion and dissipation error over a prescribed band-limited function space. Modal analysis on disordered two-dimensional node distributions shows that the learned fourth-order operator follows the exact response over a substantially wider band than either explicit LABFM at equal stencil size or fourth-order finite differences on a structured grid, whilst recovering the expected fourth-order convergence rate under refinement.

physics.comp-ph↗
arXiv2026

High-Order-Accurate Continuity Enforcing Nyström Discretization of 3D Maxwell Combined Field Integral Equations

In Nyström-collocation discretizations of the electric field integral equation (EFIE), the surface divergence acts on surface densities that may be discontinuous across patch boundaries, which degrades accuracy and convergence. We show that this not only affects the EFIE but every formulation in which the operator occurs, either in the equation itself or in the scattered field computation, and propose a high-order-accurate continuity-enforcing scheme for smooth surfaces as a remedy for the direct and indirect EFIEs, magnetic field integral equations (MFIEs), and regularized combined field integral equations (CFIEs) alike. The scheme comprises two ingredients: i) We show how to discretize the equations via a Chebyshev-based Nyström scheme, which admits closed quadrature rules. ii) Since unknowns and test vectors are in terms of patch-local curvilinear bases, continuity is enforced by a change of basis: we construct sparse mapping matrices assembled solely from the curvilinear geometry description. In doing so, we restore the accuracy of the EFIE such that it can be combined with the MFIEs with equal weights to form CFIEs. Numerical studies for the scattering from canonical and realistic geometries show that all considered formulations individually and combined benefit from the continuity enforcement in terms of better conditioning, reduced iterations of an iterative solver, and several more digits of accuracy in the scattered fields, despite reducing the total number of unknowns.

math.NA↗
arXiv2026

Adaptive Epidemic Dynamics on Hypergraphs with Group-Level Immunization and Rewiring

Understanding how higher-order social structures shape epidemic spreading requires models that couple group interactions with adaptive behavior. We introduce an adaptive simplicial susceptible-infected-susceptible (s-SIS) model on d-uniform hypergraphs, where both node states and hyperedge activity co-evolve in response to local infection pressure. Hyperedges represent group interactions of fixed size and dynamically reduce their activity through a feedback mechanism in highly infected environments. Within this framework, we design two classes of hyperedge-level interventions: (i) risk-driven immunization, combining spontaneous, activity-based isolation with targeted deactivation guided by hyperedge infection pressure, and (ii) structural rewiring, which reconstructs group structures either randomly or via degree-preferential attachment. By extending the microscopic Markov chain approximation to higher-order interactions, we derive analytical conditions for the existence and stability of both endemic and disease-free stationary states. Our analysis shows that adaptive hyperedge feedback can induce discontinuous phase transitions, nonlinear epidemic thresholds, and bistable regimes in which sufficiently high initial prevalence drives the system to a disease-free equilibrium. Extensive Monte Carlo simulations support the theory and confirm that targeted immunization and degree-preferential rewiring substantially suppress epidemic prevalence, outperforming random strategies. These results demonstrate that higher-order interactions and adaptive group-level responses fundamentally reshape epidemic bifurcations and suggest principles for designing effective intervention policies in complex social systems.

physics.soc-ph↗
arXiv2026

Distilling deep optical flow stereo methods to retrieve dense three-dimensional wind fields

Geostationary atmospheric motion vectors (AMVs) provide the dense horizontal wind vectors (u,v) and heights ingested into data assimilation systems. Traditional AMVs track features using window-based cross-correlation and estimate heights via infrared brightness temperatures paired with numerical weather prediction (NWP) background states, creating a circular dependency that yields inaccurate heights, high computational cost, and sparse retrievals. Stereo winds from GEO-GEO and GEO-LEO geometrically resolve heights from parallax shifts across different poses, eliminating NWP dependence and improving accuracy, but they remain computationally heavy with limited coverage. In this work, we replace window-based tracking in stereo matching with deep optical flow for efficient, improved retrieval. Fine-tuning balances a self-supervised geometric residual loss with supervised radiosonde reconstruction. To eliminate multi-satellite overlap requirements, we distill the stereo teacher into a single-satellite student model. Chi-square and height uncertainties from the teacher are emulated by the student for quality assurance. The student generates winds across full-disk GEO imagery globally. Validation compares stereo and student models against radiosondes, operational AMVs, ERA5 reanalysis, and EarthCARE cloud profiles. Results through triple collocation show that stereo winds improve performance beyond operational AMVs for water vapor bands (6.2, 6.9, and 7.3 μm), wit degradation in the long-wave infrared (11.2 μm) band.

cs.LG↗
arXiv2026

Auditing Frozen-Encoder Anomaly Detection Across Mechanical Systems: Representation Provenance, Calibration, and Protocol Effects

This version reports a reproducibility audit of the frozen-encoder experiments presented in version 1. The numerical discrimination results are reproducible from the preserved artifacts, but their original attribution to interferometric pretraining is not supported. The released checkpoint contains a nested model state that loads without missing parameters, whereas loading the outer checkpoint dictionary leaves almost the entire EfficientNet-B0 feature stack uninitialized. Preserved embeddings labelled as interferometric have norms of order $10^{-12}$, matching freshly initialized EfficientNet-B0 networks and differing by more than twelve orders of magnitude from the preserved ImageNet embeddings. A second, separately preserved near-zero embedding set produces almost the same IMS 4th-test anomaly scores ($r=0.987$) and record-level discrimination (AUC $0.9812$ versus $0.9818$). We therefore withdraw the causal claim that IMS performance demonstrates a morphological prior transferred from gravitational-wave instrumentation. We reanalyse the controlled IMS splits at matched observed false-positive rates and add multivariate classical signal baselines. The near-zero representations retain strong tail separation, particularly in the 2nd and 4th IMS runs, but this is now interpreted as an exploratory architecture-and-initialization effect coupled to Mahalanobis scoring. A separate PRONOSTIA audit shows that the original large warning times were induced by a lifetime-fraction baseline; under fixed-time evaluation, a ten-feature classical baseline outperforms the preserved encoder scores. These results illustrate how checkpoint provenance, finite-sample calibration, architecture, and target-domain baselines can create an appearance of cross-domain transfer. They also define the controls required before assigning physical meaning to frozen-representation anomaly scores.

astro-ph.IM↗
arXiv2026

Moment-enhanced shallow-water equations with an effective wall closure for no-slip bottoms

Shallow-water equations and low-order shallow-water moment models use vertically coarse representations and therefore cannot, in general, resolve the thin wall-affected region produced by a no-slip bottom. Enforcing the pointwise wall value on a low-order global polynomial reconstruction can introduce stiff relaxation and distort the resolved interior velocity profile. Starting from the incompressible Navier--Stokes equations with Navier bottom friction, we derive a bottom-to-mean relation in a distinguished regular-friction regime and use it to define an endpoint-consistent effective wall-traction closure for the shallow-water equations and the hyperbolic shallow-water moment equations. The closure represents the momentum effect of unresolved near-wall dynamics; it neither resolves the physical boundary layer nor imposes the pointwise no-slip trace on the reconstructed polynomial. It recovers the perfect-slip wall contribution when the friction coefficient vanishes. Because only source terms are changed, the homogeneous principal matrices and their established two-dimensional hyperbolicity classification remain unchanged. We compare the standard and modified reduced models with two-phase incompressible Navier--Stokes computations in OpenFOAM for wet-bed dam-break and three-dimensional collapse tests. In the cases considered, the modified closure reduces the excessive damping of the classical low-order wall source and improves agreement in depth-averaged and resolved-interior velocity diagnostics, but it does not uniformly improve front-propagation speed. The regular-friction asymptotic remainder is not uniform in the large-friction numerical regime; there the effective coefficient is used as a wall-model continuation and assessed empirically.

math.NA↗
arXiv2024

Predicting unobserved climate time series data at distant areas via spatial correlation using reservoir computing

Collecting time series data spatially distributed in many locations is often important for analyzing climate change and its impacts on ecosystems. However, comprehensive spatial data collection is not always feasible, requiring us to predict climate variables at some locations. This study focuses on a prediction of climatic elements, specifically near-surface temperature and pressure, at a target location apart from a data observation point. Our approach uses two prediction methods: reservoir computing (RC), known as a machine learning framework with low computational requirements, and vector autoregression models (VAR), recognized as a statistical method for analyzing time series data. Our results show that the accuracy of the predictions degrades with the distance between the observation and target locations. We quantitatively estimate the distance in which effective predictions are possible. We also find that in the context of climate data, a geographical distance is associated with data correlation, and a strong data correlation significantly improves the prediction accuracy with RC. In particular, RC outperforms VAR in predicting highly correlated data within the predictive range. These findings suggest that machine learning-based methods can be used more effectively to predict climatic elements in remote locations by assessing the distance to them from the data observation point in advance. Our study on low-cost and accurate prediction of climate variables has significant value for climate change strategies.

cs.LG↗
arXiv2026

Real-time virtual circuits for plasma shape control via neural network emulators: experimental demonstration on MAST Upgrade

Conventional plasma shape control in tokamaks relies on virtual circuits (VCs) that are computed offline from linearisations around a small, tailored number of reference equilibria, and deployed as expertly prepared schedules during the discharge. Here, we report on the first experimental deployment of real-time VCs. We replace pre-set look up tables with VCs updated in real time using surrogates of the plasma response. Both the existing control architecture and the interpretability of VC-based control are retained. Previous work showed that neural network emulators can produce accurate VCs, and validated their performance in closed-loop shape control simulations. Here, we report their first experimental validation on MAST Upgrade (MAST-U). Dedicated experiments spanning different scenarios, including prescribed shape perturbations, feedback-driven divertor-leg motion, and strongly evolving plasma configurations, show that real-time VCs can realise plasma shape control tasks within the MAST-U plasma control system. These results establish the experimental feasibility of real-time linearisations as a practical extension of conventional plasma shape control in tokamaks. The present implementation demonstrates a central step towards a simpler control workflow, in which manually constructed, phased VC schedules are replaced by VCs generated automatically online from a trained surrogate model, without scenario-specific retraining.

physics.plasm-ph↗
arXiv2026

MakoXC: Rearchitecting DFT Exchange-Correlation with Matrix-Aligned and Knowledge-Organized Sparsity

Density Functional Theory (DFT) is indispensable for materials science and drug discovery, yet the exchange--correlation (XC) evaluation remains a major bottleneck due to its cubic scaling. Although linear-scaling methods exploit electronic nearsightedness to reduce asymptotic complexity, they produce irregular sparse workloads that hide implicit sparsity and prevent efficient use of modern AI accelerators. We present MakoXC, a modular matrix-aligned XC evaluation engine that rearchitects nearsightedness-induced sparsity into regular, accelerator-friendly computations. MakoXC co-designs three key techniques: (1) Matrix-Aligned Cells reorganize nearsightedness-induced interactions into dense, accelerator-aligned data clusters; (2) Sparsity-Guided Activation translates deeper implicit sparsity into numerically correct structured execution for practical linear scaling; and (3) Kernel-Fused Pipeline consolidates fragmented workloads into a unified, compute-intensive execution path that fully unleashes accelerator throughput. Extensive evaluations show that MakoXC achieves average speedups of 67.8$\times$ speedup over standard XC evaluation and 4.7$\times$ over state-of-the-art linear-scaling methods. When integrated into a production-grade commercial DFT package, MakoXC scales XC evaluation to ubiquitin (1,231 atoms, def2-SVP) on 64 GPUs, enabling the end-to-end DFT calculation to complete in under five minutes. By restructuring XC evaluation into a unified, structured computation, MakoXC demonstrates how scientific workloads can achieve genuine low complexity while maximizing parallel efficiency on AI accelerators.

cs.DC↗
arXiv2026

Measuring Collective Semantic Change in Populations of Language Model Agents

Collective semantic change in populations of language model agents is a measurable dynamical phenomenon. We present a passive longitudinal instrument called Kopterix that observes the semantic state of an agent population as a sequence of bounded observations under a protocol defined before the observations begin. Each observation divides the sampled feed by post age into surface, mid-stream, and residue layers, which makes semantic differences across content age measurable alongside run-to-run change. We validate the instrument on Moltbook, an agent-native social platform, over a two-month window of scheduled observations, with the periodicity check extended across approximately four months. At the lexical level, rarefied entropy resolves an April-May difference in the evenness of the stored top 200 unigram distributions, and adjacent states are lexically closer than states paired after timestamp shuffling. At the geometric level, grand mean centering exposes the scale of a common embedding direction, and scheduled shuffle checks support a recurring excess in the mid-stream to residue separation relative to the shuffled reference. At the temporal level, detrended scalar quantities and centered layer centroids lose much of their similarity over several hours, and a weaker positive component declines across longer separations with no strong weekly recurrence. Several attractive apparent structures failed their controls, and each reading is limited to the level its controls support. The design applies wherever a population of agents produces a timestamped language environment that can be observed repeatedly and divided by content age.

physics.soc-ph↗
arXiv2026

Benchmarking large language model agent societies against human behavioural distributions

Populations of large language model agents are increasingly used as experimental societies. Three doubts shadow every such result: whether the agents behave like the humans they stand in for, whether a finding survives changes to the apparatus that leave the rules untouched, and whether apparent social dynamics are interaction at all rather than the reproduction of experiments the models have read. This article introduces SILICA, an open instrument that tests all three. Five environments carry published human anchors, each paired with perturbations that re-render the same rules and with variants whose payoffs point away from the memorised result. Twelve open-weight models were run through it on a single consumer graphics card. Agreement with human data is confined to starting points: first-round public-goods contributions fall inside the equivalence margin for eight of eleven models, while no model matches end-state contributions or the human corridor of cooperation. Merely swapping the order in which two actions are listed costs one model 58 points of cooperation. Presenting responders with a fixed schedule of offers shows that only one model, the sole reasoning-trained one, places its acceptance threshold where the incentive requires; two move theirs part of the way, two move them the wrong way, and three never acquire one. Conventions form through a shared prior over the names rather than through negotiation, though negotiation reappears once that prior is disrupted. On the certification ladder defined here, current silicon societies support exploratory claims and no more.

physics.soc-ph↗
arXiv2026

A Learning-based Framework for Spatial Impulse Response Compensation in 3D Photoacoustic Computed Tomography

Photoacoustic computed tomography (PACT) is a promising imaging modality that combines the advantages of optical contrast with ultrasound detection. Utilizing ultrasound transducers with larger surface areas can improve detection sensitivity. However, when computationally efficient analytic reconstruction methods that neglect the spatial impulse responses (SIRs) of the transducer are employed, the spatial resolution of the reconstructed images will be compromised. Although optimization-based reconstruction methods can explicitly account for SIR effects, their computational cost is generally high, particularly in three-dimensional (3D) applications. To address the need for accurate but rapid 3D PACT image reconstruction, this study presents a framework for establishing a learned SIR compensation method that operates in the data domain. The learned compensation method maps SIR-corrupted PACT measurement data to compensated data that would have been recorded by idealized point-like transducers. Subsequently, the compensated data can be used with a computationally efficient reconstruction method that neglects SIR effects. Two variants of the learned compensation model are investigated that employ a U-Net model and a specifically designed, physics-inspired model, referred to as Deconv-Net. A fast and analytical training data generation procedure is also a component of the presented framework. The framework is rigorously validated in virtual imaging studies, demonstrating resolution improvement and robustness to noise variations, object complexity, and sound speed heterogeneity. When applied to in-vivo breast imaging data, the learned compensation models revealed fine structures that had been obscured by SIR-induced artifacts. To our knowledge, this is the first demonstration of learned SIR compensation in 3D PACT imaging.

cs.LG↗
arXiv2026

Efficient primal--dual splitting methods for a Poisson-constrained JKO scheme for Poisson-Nernst-Planck models

The Poisson--Nernst--Planck (PNP) equations strongly couple ionic transport and electrostatic interactions through the Poisson equation, posing substantial numerical challenges under small permittivity and complex potential boundary conditions. Underlying these equations is a natural Wasserstein gradient-flow structure, in which the Poisson equation serves as a local realization of the nonlocal electrostatic interaction energy. Exploiting this structure, we formulate each time step as a constrained convex minimization problem where the ionic continuity equations and the Poisson equation are incorporated as linear constraints, allowing the concentrations, fluxes, and electrostatic potential to be updated simultaneously. The variational structure of the scheme intrinsically guarantees the dissipation of the original free energy, mass conservation, and nonnegativity of ionic concentrations under general electrostatic boundary conditions. Moreover, the framework is structurally modular: extending from classical to modified PNP models with steric interactions and concentration-gradient corrections requires only modifying the energy functional, while all structure-preserving properties are automatically retained. To efficiently solve the resulting large-scale constrained problems, we develop preconditioned and transformed primal--dual algorithms equipped with tailored fast dual solvers, namely DCT-based direct and Schur-complement iterative methods, that exploit the coupled block structure of the PDE constraints. Numerical experiments on classical and modified PNP systems demonstrate the accuracy and structure-preserving properties of the scheme, and show that the proposed algorithms converge reliably in strongly coupled small-permittivity regimes without significant growth in computational cost.

math.NA↗
arXiv2026

Prototype-guided transfer of sparse literature knowledge for electrolyte additive discovery

Electrolyte additive discovery remains challenging because experimentally validated molecules are sparse, whereas accessible chemical spaces are vast and largely unlabeled. This challenge is amplified in lithium-ion batteries, where additive performance arises from coupled interfacial reactions rather than a single molecular property. Here, we develop a prototype-guided molecular intelligence, ProtoMI, a literature-driven framework that learns transferable structural priors from reported electrolyte additives and uses them to prioritize candidates in unlabeled chemical space. For boron-containing additives, ProtoMI combines 126 literature-reported molecules with 179,977 unlabeled candidates. Graph contrastive learning identifies seven chemically interpretable prototypes from the reported additives, and prototype guided semi-supervised contrastive learning adapts these prototypes to the candidate space under source-target distribution mismatch. In retrospective temporal validation, ProtoMI achieves enrichment factors of 9.2-45.6 while screening less than 2% of the candidate space. A subsequent translation step identifies four commercially accessible candidates. One representative candidate, 4,4,5,5-Tetramethyl-2-[10-(1naphthyl)anthracen-9-yl]-1,3,2-dioxaborolane (TNDB), improves high-temperature LiFePO4||graphite cycling at 55 °C by 34.93% relative to the baseline electrolyte. An arsenal of characterizations and operando optical fiber Fourier transform infrared spectroscopy suggest that TNDB forms B-containing, F/P/O-modified inorganic interphases, suppresses solvent decomposition and reduces Fe deposition on graphite. This case study shows how sparse literature knowledge can guide experimentally efficient molecular discovery in data-scarce battery-additive spaces.

physics.chem-ph↗
arXiv2026

AxQM: A Textbook-Scale Benchmark for Formal Proof Synthesis in a Library of Finite-Dimensional Quantum Mechanics

Formalizing mathematics in a proof assistant, where a machine checks every definition, statement and proof, has set a new standard of rigor. Large language models are now capable of formalizing autonomously, even at the scale of whole textbooks. We bring this standard of rigor to physics, where theoretical arguments carry idealizations that are rarely stated fully, and any logical gaps could have a cascading effect on interdependent results. Recognizing the need to evaluate autoformalization systems for physics, we release AxQM, 1,019 kernel-checkable proof-synthesis tasks over 479 items drawn from the textbook Quantum Computation and Quantum Information by Nielsen and Chuang. The tasks are stated in a custom Lean library of finite-dimensional quantum mechanics. By task count, it is the largest proof-synthesis benchmark in physics by a factor of four. AxQM is derived from a near-complete formalization of the formal portions of the textbook, so every task is guaranteed a solution, which we keep private. Grading of the benchmark is done deterministically by the Lean kernel, which checks that the proof compiles, that no sorry appears in it or in any declaration it depends on, and that it introduces no new axioms.

quant-ph↗