Lattice

Ultra-low-bit language models promise reductions in storage and memory traffic, but a nominal "1.58-bit" label does not specify the deployed representation or its execution cost. We study a scale-up of an aggressive post-training conversion pipeline from Qwen3-4B to Qwen3-8B. The conversion uses KOTMS rotation, E2M-ATQ adaptive ternarisation, and GPTQ-style error compensation in a weight-only A16 configuration. We do not claim these algorithms as new. Our contribution is the end-to-end scale-up characterisation: an external reproduction gate, matched 4B/8B capability analysis, cross-corpus perplexity, effective-bit accounting, lossless lattice-aware packing, and direct packed execution. The 8B model reaches a three-corpus perplexity ratio of 1.361x, with WikiText-2, C4, and PTB ratios of 1.318x, 1.393x, and 1.371x. On eight zero-shot tasks at n = 500, mean accuracy is 64.6% versus 72.4% for FP16, corresponding to 78.5% chance-corrected retention and a 7.8-point absolute cost. The matched 4B run retains 69.6%, yielding an 8.9-point 8B advantage. The packed checkpoint is 8.24 GiB and preserves the recorded perplexity to measurement precision. Direct packed execution reaches 15.52 tokens/s in 7.35 GiB, while a preliminary packed GEMV remains slower than FP16 cuBLAS. The result is a validated scale-up baseline: model size improves robustness to aggressive post-training discretisation, actual serialisation is solved for the measured artefact, and direct execution is feasible, while broader seeds, calibration distributions, and kernel optimisation remain open.

Secure Aggregation (SA) is widely regarded as a strong defense against model-update leakage in Federated Learning (FL), as it reveals only aggregate results while hiding individual updates. In Decentralized Federated Learning (DFL), SA is commonly instantiated as local neighborhood aggregation, where each node obtains a weighted aggregate over its neighbors. We show that this locality creates a structural leakage surface: sparse decentralized topologies provide colluding semi-honest nodes with asymmetric aggregate views, exposing multiple hidden linear combinations of honest participants' private states. Reconstructing private states from these aggregate views is fundamentally challenging, as both the private states and the aggregation coefficients are hidden. We tackle this challenge by establishing a formal connection to the Hidden Subset Sum Problem, a long-studied problem in cryptography. Building on this formulation, we design a lattice-based reconstruction approach that combines lattice reduction with structural filtering to reconstruct protected model states. We evaluate our attack on image, tabular, and text tasks under sparse DFL topologies. Our results show that colluding semi-honest nodes can recover the original local updates of honest nodes, enabling downstream reconstruction of private training data. These findings demonstrate that SA alone does not guarantee privacy in DFL when local aggregation induces asymmetric observations.

Certified selective predictors attain whatever coverage they attain; operators impose an automation floor: answer at least a $β$-fraction of shifted target traffic with at most an $α$-fraction of answers wrong. Under bounded-ratio covariate shift we prove the Floor Certification Map: once that floor must be certified alongside the selection-conditioned risk $α$, certification acquires a feasibility frontier and a two-resource complexity map, additive up to constants: risk in labeled source, the floor in unlabeled target samples. The rates are local, needing a regular frontier margin, slack below the local-regime threshold, and lattice conditions: pre-registered with a lattice margin for the upper bounds, compatible per-slack for the lower. The displayed split is the operational route; oracle weights also allow a labeled-source floor estimate. Three model-tagged results: a lower bound (Model-B), a matching oracle-weight upper bound (Model-A), and an implementable upper bound (Model-B') valid under a pre-registered exact stratified-shift model with nuisance cost priced explicitly. The match is across these models rather than a single-model minimax theorem, and necessarily so: over the full bounded-ratio class no unknown-weight procedure matches at any sample size (Model-B is inconsistent, witnessed at $α=β=1/2$). The nuisance's necessity is only partially settled. Complexity tracks a localized accepted-region functional, not global effective sample size (ESS), on both sides, though a fixed-ESS separation theorem is left open; both lower-bound axes vanish as $β\to0$, so the floor creates the map. Empirically, the registered bite family diverges with log-log slope $-2.002$ within its pre-registered band; a 1,024-cell audit records 0 violations where the formal certificates fire; and a single-corpus SQuAD-to-NewsQA feasibility audit returns honest refusal.

Multi-agent LLM systems often fail not for lack of candidate answers, but because they have no persistent mechanism for deciding which incompatible claim should currently be trusted. Majority vote, debate, and judge-based selection choose an output without recording which claim wins, which is contested, or why a later update supersedes it. We present \term{LatticeMind}, a conflict-aware structured memory that handles contradiction at write time. It maintains explicit item status, applies cheap symbolic conflict checks, and invokes LLM reconciliation only for unresolved semantic cases. On a label-blind ConflictBank evaluation that removes source-name hints, LatticeMind reaches 0.97 accuracy versus 0.61 for the strongest aggregation baseline, with the gap significant at $p<10^{-6}$ by paired McNemar test. Ablations show that removing the checker or the reconciler costs 12 to 14 points. On four secondary planning benchmarks the picture is mixed: LatticeMind beats naive merge on three of four, but does not replace deliberation methods on tasks rewarding iterative search.

Robotic construction systems often treat the material system and the robot as separate design problems, locating intelligence primarily in hardware, sensing, motion planning, and control. This project instead investigates how geometric intelligence can be encoded within architected material systems to simultaneously address requirements for robotic grasping, self-alignment, reversible connection, structural performance, and three-dimensional aggregation. We introduce a self-aligning compound nested lattice module composed of conjoined cuboctahedral-octahedral units. The cuboctahedral features of the modules provide defined surfaces for robotic grasping and alignment, while the octahedral features incorporate screw-releasable snap-fit connectors and corresponding receptors. Additionally, we present a nested arrangement that enables interlocking aggregation along the x, y, and z axes. We demonstrate the system through furniture and architectural scale structures assembled using both a robotic arm and mobile assembler. The resulting configurations include seating, spanning structures, surfaces, and vertical frames. Compression testing of the compound module produced a stiffness of 4,556 N/mm, a maximum load of 3,445 N, and a compressive modulus of 17.5 MPa. The modules can also be disassembled and reused across different configurations, supporting reconfigurable and circular construction.

Contact planning in robotics requires models that are both computationally efficient and physically accurate. Sphere-based robot representations satisfy the first requirement by enabling fast collision checking and differentiable geometry, but sacrifice physical accuracy by relying on point contact which cannot capture contact patch area, pressure distributions, rotational stiffness, or frictional moments. We introduce Compliant Sphere Lattice Contact (CSLC), a distributed contact model that operates natively on sphere representations by modeling the robot interface as a compliant lattice of surface spheres connected through anchor and lateral springs. When pressed against an object, the lattice deforms to produce a spatially distributed contact patch that improves the physical accuracy of sphere-based contact. We validate CSLC across two independent solvers and show preliminary results demonstrating contact patch formation and improved grasp stability.

Lacoste-Julien and Jaggi conjectured in 2015 that the pyramidal width of a polytope cannot increase when a vertex is added, provided that every old point remains a vertex. We give an exact counterexample with six integer points in $\R^3$. For $$P=\conv\{v_0,\ldots,v_4\},\qquad Q=\conv\{v_0,\ldots,v_5\},$$ where $$\begin{aligned} v_0&=(-1,-3,-1), & v_1&=(3,2,-2), & v_2&=(0,2,1),\\ v_3&=(-1,-3,3), & v_4&=(-2,0,1), & v_5&=(-1,0,-2), \end{aligned}$$ all five vertices of $P$ remain vertices of $Q$, but $$\PWidth(P)^2=\frac{48}{353} \quad\text{and}\quad \PWidth(Q)^2=\frac{36}{133}.$$ Thus vertex insertion increases pyramidal width by the factor $\sqrt{1059/532}\approx 1.410886779$. The proof uses the equivalence between pyramidal width and facial distance, certifies both face lattices by integer supporting hyperplanes, and evaluates every facial distance by a finite rational calculation. A dependency-free exact verifier accompanies the paper.

Spatially resolved omics studies increasingly combine transcriptomic and epigenomic assays, yet downstream analysis is often still performed using single-modality pipelines. We present LATTICE (Latent Alignment of Tissue-level and Transcriptomic Information for Cross-modal Embedding), a graph-based self-supervised framework that learns spot-level representations from harmonized multimodal features. LATTICE integrates five aligned modality blocks per Visium spot: Visium RNA, scMultiome RNA, scMultiome ATAC, spatial ATAC, and spatial CUT&Tag. These modalities capture spatial transcriptomic measurements, single-cell inferred regulatory activity, and in situ chromatin and histone states within a unified lattice representation. LATTICE constructs a spatial neighborhood graph and trains a TransformerConv encoder using masked reconstruction, cross-modal alignment, and spatial smoothness objectives. On a private 11-sample melanoma cohort from an anonymized clinical collaborator comprising 54{,}912 total spots, LATTICE demonstrated stable optimization behavior, reproducible embeddings across analysis seeds, and complete multimodal integration across all samples. Adding scMultiome RNA to Visium RNA alone substantially improved concordance with Space Ranger clusters across 11 runs (adjusted Rand index [ARI] +0.157, normalized mutual information [NMI] +0.143, and spatial contiguity +0.174). Additional modalities further improved spatial contiguity and multimodal utility score (MUS), although they sometimes reduced agreement with RNA-derived reference labels, likely because the learned embeddings captured chromatin and regulatory structure beyond transcriptomic similarity alone. These results position LATTICE as a practical and empirically grounded framework for multimodal spatial omics integration, while also highlighting the need for stronger supervision and broader external benchmarking.

Quantum logic is usually presented as a non-classical departure from ordinary reasoning forced on us by quantum mechanics, with classical logic kept as the secure starting point. We argue for the opposite order of explanation in a finite and fully computable setting. The free orthomodular lattice on two generators has ninety-six elements, the direct product of a six-element non-distributive factor and a sixteen-element Boolean factor. Reading the first factor as a register of contexts and the second as Boolean content, we obtain a calculus whose elements are context--bit-vector pairs and whose operations act component by component. With this calculus we establish three results. First, we classify the six layers by commutativity, identifying the central kernel of context-neutral propositions together with a dual central layer in which all complementary contexts are present. Second, we show that orthocomplementation rearranges the layers exactly as the complementation of the small factor rearranges its elements, which makes the duality among the layers rigid rather than accidental. Third, we prove that the operation forgetting the context is a surjective homomorphism of orthocomplemented lattices whose quotient is the classical Boolean algebra, so that classical logic is a six-to-one, information-losing image of the contextual calculus.

Lattice field theory is the workhorse of non-perturbative physics, used to simulate phenomena from the strong nuclear force to critical phenomena in materials. Its Boltzmann distributions are parametrized analytically by coupling constants, but these bare parameters are weak predictors of observables -- extracting physics typically requires extensive simulation. While normalizing flows have emerged as effective samplers at fixed couplings, it remains difficult to interpret what these networks have learned. This raises a natural question: can the physics be read off directly from the flow network parameters themselves, and can those parameters be generated for unseen theories? We propose lattice field theory as a testbed for neural network interpretability: because the target physics is qualitatively well-understood and smoothly varying, it provides ideal synthetic data with known ground truth. To this end, we introduce JEPAWG, a Joint-Embedding Predictive Architecture-based Weight Generator that maps couplings directly to flow weights via a learned latent space. On a scalar theory at lattices of size $6^2$ to $11^2$, the JEPAWG latent space recovers the correct intrinsic dimension of the underlying manifold, locates the phase transition, and encodes a finite-size shift aligned with the 2D Ising exponent $ν\approx 1$, allowing us to uncover physical structure by studying the network weights alone. This suggests the fascinating idea of treating the network weights as a new type of physical observable. As a generator, JEPAWG also interpolates and extrapolates to unseen couplings effectively and remains robust to weight-space discontinuities introduced by multi-seed training data, outperforming PCA, AE, and VAE baselines.

Least privilege, the principle that an identity should hold only the permissions strictly required for its task, has been a foundational primitive of access control for decades. We argue that this principle is insufficient for agentic AI systems, which do not merely hold permissions but can combine, approve, and amplify them across workflows and system boundaries. We propose least autonomy as an appropriate generalization and develop a formal theory. First, we define a compositional blast radius d(a,b) that measures structural separation between actions in an enterprise hierarchy, combining an ultrametric tree with lattice-valued confidentiality, integrity, and control-context labels. Second, we define a directed agent influence graph G(theta). An arc from U to V requires a directed shared-resource write-to-read meeting or a conservative undirected agent-to-agent (A2A) communication meeting, and a meeting-conditioned influence potential at or above an externally selected policy threshold theta. A catalogue-radius profile supports calibration and audit of theta. Finally, we define a collusion predicate over graph reachability that detects authorization composition, decision manipulation, and cross-domain capability composition.

We introduce a novel technique for scalable sampling of spin-system states with continuous symmetries using diffusion models. By applying our approach to the XY model, a fundamental continuous-spin model in condensed matter physics, we show that our technique addresses the shortfalls of the Markov chain Monte Carlo (MCMC) in generalization to varying system sizes. More specifically, we show that training a temperature-conditioned diffusion model on smaller-size XY model lattices enables the generation of accurate samples in larger lattice sizes. By tracking physically important observables of the model, such as spin correlations, our experiments demonstrate that diffusion sampling followed by a few MCMC steps reduces the thermalization time by an order of magnitude relative to the standard MCMC with random initialization. Our study provides valuable insight as to how generative models can be used to study continuous-state condensed matter systems at scale.

Soft lattice structures are increasingly used in robotics to tailor compliance and guide deformation; however, actuation is typically introduced at the device or module level, with actuators inserted into otherwise passive architectures. In this work, we move actuator-lattice co-design to the unit-cell scale. We present an embedded pneumatic unit cell that integrates curved-strut lattice geometry with a bidirectional bellow actuator within a single monolithic element. When tessellated, the lattice functions as a distributed actuation field in which global morphology is governed by spatial actuation patterns rather than uniform pressurization. Experimental characterization of 1x1, 2x2, and 3x3 tessellations demonstrates scalable displacement and force generation with repeatable cyclic performance. Selective actuation of unit cells in a 3x3x3 array produces distinct global deformation modes, including bending and directional grasping, without altering hardware configuration. Additionally, coupling active and passive unit cells enables bending-driven crawling locomotion, demonstrating that heterogeneous tessellations can translate through asymmetric deformation. These results establish unit-cell-level actuation as a strategy for distributed morphing in lattice-based soft robots and provide a foundation for scalable, monolithic robotic architectures.

Multi-agent LLM systems share state through memory stores, vector indices, and tool registries. We model such sharing as long-running read-generate-write operations under deterministic-generation semantics -- the regime durable-execution engines enforce by deterministic replay -- and formalize four concurrency anomalies in TLA+: stale-generation, phantom-tool, causal-cascade, and tool-effect reordering, structural analogues of classical isolation anomalies, each with a TLC counter-example. The exclusion lattice over these anomalies is trivial; the contribution is the mechanically verified realizability and strict separation of one maximal chain within it, $L_0 \subsetneq \cdots \subsetneq L_4$, to our knowledge the first machine-checked consistency hierarchy for such runtimes. A development of 274 Verus obligations (zero assume, zero admit; trust base: two structural axioms and a mutex correspondence) proves the detectors sound and complete against the specifications and each runtime its avoidance set. Three deployed Rust runtimes realize L0-L1 (pessimistic locking, serializable snapshot isolation, default-SI), each verified against stale-generation and refined to its state machine; L2-L4 are exec-mode-verified with dependency-free prevention twins (A3, A6, A2: 0/1000 versus 1000/1000), and L2 is run live across three model families (A3 prevented in all 120 retracted sessions). We reproduce a silent lost update in ByteDance's deer-flow, formalizing its fix as a verified $L_0 \to L_1$ refinement, and exhibit tool-effect reordering in LangGraph's ToolNode on unmodified output, removed by an L3 commit-order sequencer. The verified detector, refinements, and realizability artifacts are the contribution; the phenomena and lattice are classical.

The Lenstra-Lenstra-Lovász (LLL) algorithm is a seminal contribution to computer science used for lattice basis reduction, yet its polynomial-time outputs produce bases that are far from optimal as the dimension grows. We show that deep reinforcement learning can discover strictly superior, generalizable reduction strategies by interacting with the primitive action space of LLL. We formulate lattice reduction as a single-player Markov Decision Process (MDP) and train a deep residual network using an AlphaZero-style self-play pipeline augmented with adaptive-horizon MCTS (Monte Carlo Tree Search), which couples multi-step network predictions with an entropy-gated expansion mechanism. The resulting policy, DeltaStar, is trained exclusively on small $8$-dimensional $q$-ary lattices and requires fewer primitive row operations than LLL. Crucially, it generalizes zero-shot to unseen moduli and higher dimensions up to $n=32$ without retraining.

Submodular function minimization has gained a lot of interest in recent years. They are highly applicable in the area of Computer Vision and Machine Learning. Often such applications require to work with submodular functions defined on distributive lattice. Current best way of dealing with it is using a transformation which extrapolates the submodular function for the respective boolean lattice. It makes optimization system too inefficient due to enlargement of the working space. Quantitatively, the expanded space has additional exponential (in set size) number of elements. We propose a generic framework for dealing with distributive lattice which only works within distributive lattice. Our framework allows one to use already established submodular function minimization algorithms for boolean lattice. In our experiment, we show the huge improvement in terms of running time over tranditional methods for handling distributive lattice.

The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by using a second lattice carefully constructed as training points in the parametric space. We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples. In addition, the architecture is simplified, because the high-dimensional Fourier transform on rank-1 lattices requires only a \emph{one-dimensional fast Fourier transform}, and we can use a \emph{hyperbolic cross} frequency index set with lattice points. We demonstrate the benefits of our \emph{lattice-based hyperbolic-cross FNOs} for an elliptic PDE on the torus.

Physics-informed machine learning is often assessed by curve error, although engineering use depends on downstream decisions: ranking candidates, avoiding infeasible designs and limiting regret. We introduce pinn-gym, an open benchmark for material-conditioned lattice design that couples a transparent reduced-order crush-and-impact oracle with five printable polymer cards, dimensionless force-response targets and a protocol spanning curve fidelity, physical admissibility, top-k retrieval and mass regret. Across per-material, pooled and cross-material settings, low nRMSE is frequently insufficient to identify useful design selections. Physics-informed losses alter trade-offs rather than monotonically improving all metrics, and dimensionless conditioning improves comparability without making transfer symmetric. The benchmark is not a certified material model; within the released oracle, candidate generator and material cards, pinn-gym provides a reproducible testbed for evaluating PIML surrogates as decision systems rather than curve predictors alone.

Learning semantics is essential for deep learning models to be interpretable and better aligned with human reasoning. Concept-based models approach this by representing classes through meaningful semantic abstractions, but typically treat all concepts as a flat, unstructured set learned at a single neural network layer. This overlooks a fundamental property of human semantic understanding: concepts being organized hierarchically, from general to specific. While deep networks do learn a hierarchy of visual features, this structure is rarely aligned with explicit semantic hierarchies. Drawing on Formal Concept Analysis, we demonstrate that formal concept lattices provide principled semantic scaffolds to guide neural network learning. These lattices naturally identify where in the network concepts should be learned based on their level of generality. This allows the model to develop staged, semantically grounded representations throughout its depth. Empirical results on real-world datasets show that our models produce more interpretable embeddings, support more effective interventions, and learn concept representations that are both meaningful and hierarchically structured.

The image-source model (ISM) is a widely adopted method for efficiently simulating acoustic room impulse responses (RIRs) under specular reflection assumptions. Acoustic paths between source and receiver are traced to lattice points computed from successive reflections over bounding planes of the room. Rectangular rooms bound the total number of image-sources to be polynomial in the RIR's duration or distance $k$ equivalent, with degree equal the number of room dimensions $N$. Direct ISM simulations are therefore compute upper-bound by $O \left ( k^N \right )$, and consider only cases of $N \leq 3$ for tractability and real-world applications. This work proposes an alternative computational method that lowers the asymptotic compute bound to $O \left ( N k^2 \log k \right )$ for integer coordinates and room dimensions via reducing ISM lattice point counting to the classic Gauss circle problem (GCP). We extend the lattice counting model to frequency-dependent and reflection weighted image-sources in higher dimensions, relating solutions between successive dimensions via the convolution operator. Two constructions for realizing RIRs are presented, along with time-frequency controls, error and run-time analysis, and RIR statistics.

This paper presents a learning-based navigation framework for indoor mobile robots. The proposed method combines a supervised neural global planner, trained from cost-aware A* expert trajectories, with the proposed Learning-Based DWA local planner, which is formulated as discrete candidate selection over the Dynamic Window Approach (DWA) action lattice. For local planning, the policy is first trained by behavior cloning and then refined by Proximal Policy Optimization (PPO) under feasibility-aware masking. The framework is implemented and evaluated in both simulated and real-world indoor environments. Experimental results show that the proposed method generates feasible global routes and reliable local motion commands for safe goal-directed navigation in the presence of obstacles. These results demonstrate the effectiveness of integrating learning-based global planning with reinforcement-learning-refined local control for indoor mobile robot navigation. The source code will be released at https://ntdathp.github.io/rl_robot_web/.

Traditional cryptography, rooted in problems, e.g., integer factorisation or discrete log, is inevitably vulnerable to a fully operational quantum computer. Although it remains an engineering frontier, the looming threat extends to encrypted data stored today, which could be decrypted in the future with quantum capabilities. To safeguard against this eventuality, the backbone of the modern quantum-safe cryptography is the Shortest Vector Problem (SVP). We enhance Laarhoven's treatment of Ajtai et al.'s sieving as a genetic algorithm (GA) for the SVP by incorporating domain-informed SVP representation and crossover while naturally extending application to the module lattices.

This paper presents a magnet-based robotic skin that integrates a multilayer soft lattice with distributed Hall-effect sensor arrays and a tactile super-resolution model. External contact forces are converted to magnetic field changes by embedded permanent magnets, and the lattice spreads these changes across the sensing domain. This gives each sensor a large, overlapping receptive field and enables a large sensing area with minimal blind spots. Lattice parameters are tunable, enabling joint adjustment of mechanical compliance and transduction characteristics. An implicit modeling workflow and selective laser sintering (SLS) 3D printing support rapid fabrication of conformal, high-complexity structures. A convolutional neural network trained on experimental measurements estimates contact location and normal force in real time. Experiments validate localization accuracy and indicate scalability to larger surfaces, suggesting applicability to whole-body robotic skin and safe human-robot interaction.

International Classification of Diseases (ICD) is a globally recognized coding system that records diagnostic events during each patient encounter, providing a standardized data foundation for various clinical tasks. However, the irregular and hierarchical nature of ICD code sequences poses challenges for N-D lattice-based sequential modeling methods, leading to overly complex model designs. In this paper, we propose GraD-IBD, a graph diagnosis model that reformulates longitudinal ICD trajectories as visit-bucketized, temporally directed graphs to detect the risk of inflammatory bowel disease (IBD). A novel context-aware, time-decay message passing mechanism was developed to capture temporal dependencies while reducing model complexity. The experimental results using a real-world clinical dataset demonstrated consistent and robust improvements in IBD detection over state-of-the-art methods, with significant reductions in computational complexity compared to sequential models. These findings highlight the potential of graph representation learning to enable efficient, scalable, and accurate disease risk prediction from longitudinal ICD diagnosis codes.

We introduce dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations of lattice polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by signed circuits' -- combinatorial invariants from the theory of oriented matroids. We show that these circuits are necessary and sufficient to determine a triangulation's regularity from the graph, provided certain magnitude information is retained. The model is independent of the polytope's point count and invariant under its orientation-preserving symmetries ($\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d$), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with $N_\mathrm{pts} \leq 40$, dualGNN is the only sampler we tested that is consistent with uniform sampling across all our diagnostics (KL divergence from uniformity, collision counts, and sample autocorrelation). The model is small ($\sim92$k parameters) and trains in $\sim7.5$ hours on a single consumer GPU. We apply dualGNN to string theory, sampling Calabi-Yau threefolds uniformly at $h^{1,1}=86$; we also sample CYs at $h^{1,1}=128$, observing no deviations from uniformity, but our diagnostics are weaker here. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN (pip install dualgnn), and dualGNN is integrated into CYTools.

We introduce CAROL (Chain-based Adaptive Reconfiguration Over Lattices), a probabilistic framework for test-time hallucination reduction in large language models. Rather than relying on token-level uncertainty, CAROL defines a semantic uncertainty measure based on the consistency between generated responses and a trusted context, inducing a string-submodular objective over a lattice of textual sequences. This formulation enables hallucination mitigation to be cast as a Markov chain accept-reject process with provable convergence and near-optimality guarantees, allowing the model to iteratively refine outputs toward semantic consistency. By operating at the level of meaning, CAROL unifies hallucination detection and mitigation within a single framework. Empirical results on question answering and multi-agent reasoning benchmarks show that CAROL significantly reduces hallucinations and improves reliability and interpretability compared to likelihood-based and retrieval-augmented baselines, while maintaining competitive computational efficiency.

We develop a rigorous algebraic framework for deep convolutional architectures, CNNs, ResNets, and encoder--decoder networks such as UNet, grounded in lattice theory and mathematical morphology. The central tool is the Matheron--Maragos--Banon--Barrera (MMBB) universal representation theory for translation-invariant operators, which we apply systematically to every layer of a standard deep network. The principal finding is that the standard CNN pipeline (linear convolution~$+$ ReLU~$+$ flat max-pooling) is a cross-lattice operator: the convolution is an erosion in the Fourier inf-semilattice while ReLU is a lattice-join closing and max-pooling is a dilation in the pointwise max-plus lattice, and their composition is a morphological opening in neither. A second finding is that the upper adjoint of ReLU in the pointwise lattice is a global (non-local) operator, the identity on globally non-negative functions and $-\infty$ otherwise, so no local morphological erosion can form an adjunction pair with ReLU. These two results together provide the precise algebraic reason why depth in standard CNNs introduces genuine representational power: the composed layer is not idempotent. Three layer designs that are genuine idempotent openings are identified and fully characterised: the pure max-plus morphological layer (pointwise lattice), the spectral Wiener layer (Fourier lattice), and the self-dual morphological layer. We establish a complete fixed-point and convergence theory. The framework also unifies max-pooling, strided convolution, and the Laplacian pyramid under the Goutsias--Heijmans adjoint pyramid theory, and gives the Activation--Pooling Dilation (APD) factorisation with its correct adjoint.

Neural generative samplers for lattice field theory can be costly to train and evaluate. When they miss modes or assign them incorrect relative weights, biased observables do not reveal which collective variables are responsible. We project a trained flow-matching velocity onto vector fields built from lattice operators and Fourier modes. In two-dimensional lattice $φ^4$ theory, the projection separates changes in the overall magnetization from the lowest nonzero-momentum fluctuations and guides an explicit invertible proposal that treats them separately. Allowing the amplitude of the lowest nonzero-momentum fluctuations to depend on the magnetization improves the overlap between the proposal and target distributions, while the same two-parameter modification at higher momenta gives smaller improvements. The Metropolis--Hastings correction defines a Markov chain with the target Boltzmann distribution as its stationary law, and the normalized proposal density yields finite-volume partition-function estimates consistent with an independent HMC calculation. At the larger tested volume, the overlap between the proposal and target distributions deteriorates substantially, limiting the range over which the same parameterization remains effective.

The companion paper introduced a four-level verification lattice on agent-skill manifests (unverified, declared, tested, formal) and left the top level aspirational. This paper closes that gap. We give a precise semantics for skill behaviour faithful to how a skill is consumed by an LLM-driven runtime (a deterministic script-side reachable through a non-deterministic LLM-side), state the verification problem as a capability-containment property over that semantics, and present three composable methods that together raise a skill from declared or tested to formal: (1) sound static capability-containment analysis of the script-side via abstract interpretation over a small effect lattice; (2) a refinement type system for tool-call envelopes that mechanically rejects any call whose statically-inferred capability is not in the manifest's declared set; (3) SMT-bounded model checking against the parent paper's biconditional correctness criterion, with the bound chosen so any counter-example fitting the runtime's transaction-buffer horizon is exhibited as a concrete trace. We prove the three layers composed soundly cover the parent paper's threat model modulo a single residual (the LLM's freedom to refuse to act) that the parent paper's runtime biconditional catches at session boundary. The methods reuse existing well-engineered tools (Z3, Semgrep, CodeQL, refinement-type checkers, mechanised proof assistants) rather than asking operators to build new ones, and the proof-carrying artifact extends the existing SKILL.md convention. All three methods plus the bundle producer and re-checker ship as zero-dependency JavaScript modules in the open-source enclawed framework (https://github.com/metereconsulting/enclawed; project page https://www.enclawed.com/), with 53 unit tests and an end-to-end CLI demo on a sample skill.

We introduce the Lattice Deduction Transformer (LDT), a recurrent transformer that approximates logically sound deduction by projecting its latent state through a lattice between forward passes. We train on-policy in a process that mirrors deduction in a search-based constraint solver and supervise training via a domain-agnostic, abstract-interpretation-based approximation of the set of solution candidates. An $800$K-parameter LDT achieves $100\%$ accuracy on Sudoku-Extreme and Snowflake Sudoku, at a fraction of the training cost of prior small recurrent reasoners, while remaining empirically sound: the model returns a correct answer or abstains. A $1.8$M-parameter variant reaches $99.9\%$ accuracy on Maze-Hard. Frontier LLMs score $0\%$ on all three benchmarks.

We formally introduce a class of models inspired by renormalization group (RG) theory, built on additive hierarchical expansions analogous to those appearing in functional ANOVA and mixed-effects models. Like ReLU convolutional neural networks, they are almost everywhere locally linear; unlike ReLU networks, their partition structure is explicit, interpretable, and easy to modify or constrain. In these models, one defines a multidimensional lattice partition of the input space and uses it to scaffold variations in regression parameters. Each dimension of the lattice corresponds to an attribute by which the statistics of the problem may vary. The parameters are themselves expressed in the form of an expansion, where each term captures variations relative to a lower (coarser) interaction scale. These models admit multiple equivalent interpretations: as piecewise GLMs, as hierarchical mixed-effects regressions, or as regression trees with structured parameter sharing. Since RG motivates the design of these models, we use techniques from statistical physics -- specifically replica analysis -- to study their generalization properties. Specifically, we analyze the behavior of the Watanabe-Akaike Information Criterion (WAIC) as a proxy for generalization loss. This analysis yields two practical results: (i) guidance on the lattice design as a function of dataset size and predictor dimensionality; and (ii) a principled scaling law for the regularization prior when adding higher-order terms to the expansion so that one can increase model complexity without an expected increase in generalization loss. We evaluate the methodology on public datasets and find performance competitive against both blackbox methods and other intrinsically interpretable approaches.

This study presents a novel deterministic optimization algorithm based on a special variant of the Linear Congruential Generator (LCG). While conventional algorithms generally operate within the search space, the introduced technique follows a two-level architecture. In particular, an external loop that adaptively balances between exploration and exploitation, while the internal loop evaluates solutions. It is motivated by the intrinsic structure of the generator, the reason behind naming it the Structured Linear Congruential Generator (S- LCG). which enjoys a number of unique characteristics as follows: 1) a memoryless scheme, which ensures non-overlapping sequences based on distinct seeds, thus ensuring no evaluation redundancy; 2) bit splitting representation, which converts LCG states into multi-dimensional points to overcome the Marsaglia lattice effect; 3) adaptive exploration-exploitation of the generator space, which leads to implicit optimization of the surrogate smooth objective function; and 4) constant information gathering speed to avoid the problem of premature convergence. Extensive testing on 26 benchmark functions across dimensions d = 2 to 30 demonstrates that S-LCG comes within 1% of the global optimum in 83.3% of 138 cases (100% at d = 2, 81.2% at d = 30) while the nearest competitor GA achieved 75.4%. Statistical validation shows that S-LCG outperforms eight cutting-edge binary algorithms. Furthermore, its practical value is confirmed by validation on three constrained engineering design problems. In the end, S-LCG offers an optimization framework that is strictly reproducible and requires only one sensitive parameter to be tuned.

We study sequential interventions under prerequisite constraints. In this setting, admissible intervention sequences are paths in the ideal lattice of a finite prerequisite poset rather than unconstrained action strings. We give an exact local-to-global theory of order sensitivity on this state space. First, we prove that any two admissible paths with the same endpoints differ by a finite sequence of elementary diamond swaps. Second, for edge-additive path valuations, we show that path-independence is equivalent to vanishing diamond curvature, yielding an endpoint potential with a canonical Möbius parameterization on the ideal lattice. Third, we prove that a local diamond field is induced by an edge-based path model if and only if it satisfies cube consistency, with uniqueness after fixing a reference-tree gauge. Under reduced-state longitudinal assumptions, supported reference paths identify reference-path scores, whereas local order effects require two-sided support of both orders on each diamond. These results yield exact planning consequences, including an order-insensitivity bound and dynamic programming on the truncated ideal lattice.

We introduce LATTICE, a benchmark for evaluating the decision support utility of crypto agents in realistic user-facing scenarios. Prior crypto agent benchmarks mainly focus on reasoning-based or outcome-based evaluation, but do not assess agents' ability to assist user decision-making. LATTICE addresses this gap by: (1) defining six evaluation dimensions that capture key decision support properties; (2) proposing 16 task types that span the end-to-end crypto copilot workflow; and (3) using LLM judges to automatically score agent outputs based on these dimensions and tasks. Crucially, the dimensions and tasks are designed to be evaluable at scale using LLM judges, without relying on ground truth from expert annotators or external data sources. In lieu of these dependencies, LATTICE's LLM judge rubrics can be continually audited and updated given new dimensions, tasks, criteria, and human feedback, thus promoting reliable and extensible evaluation. While other benchmarks often compare foundation models sharing a generic agent framework, we use LATTICE to assess production-level agents used in actual crypto copilot products, reflecting the importance of orchestration and UI/UX design in determining agent quality. In this paper, we evaluate six real-world crypto copilots on 1,200 diverse queries and report breakdowns across dimensions, tasks, and query categories. Our experiments show that most of the tested copilots achieve comparable aggregate scores, but differ more significantly on dimension-level and task-level performance. This pattern suggests meaningful trade-offs in decision support quality: users with different priorities may be better served by different copilots than the aggregate rankings alone would indicate. To support reproducible research, we open-source all LATTICE code and data used in this paper.

Lattice Hamiltonian systems underpin models across condensed matter, nonlinear optics, and biophysics, yet learning their dynamics from data is obstructed by two unknowns: the interaction topology and whether node dynamics are homogeneous. Existing graph-based approaches either assume the graph is given or, as in $α$-separable graph Hamiltonian network, infer it only for separable Hamiltonians with homogeneous node dynamics. We introduce the Hamiltonian Graph Inference Network (HGIN), which jointly recovers the interaction graph and predicts long-time trajectories from state data alone, for both separable and non-separable Hamiltonians and under heterogeneous node dynamics. HGIN couples a structure-learning module -- a learnable weighted adjacency matrix trained under a Hamilton's-equations loss -- with a trajectory-prediction module that partitions edges into physically distinct subgraphs via $k$-means clustering, assigning each subgraph its own encoder and thereby breaking the parameter-sharing bottleneck of conventional GNNs. On three benchmarks -- a Klein--Gordon lattice with long-range interactions and two discrete nonlinear Schrödinger lattices (homogeneous and heterogeneous) -- HGIN reduces long-time energy prediction error and trajectory prediction error by six to thirteen orders of magnitude relative to baselines. A symmetry argument on the Hamiltonian loss further shows that the learned weights encode the parity of the underlying pair potential, yielding an interpretable readout of the system's interaction structure.

Local gauge symmetry underlies fundamental interactions and strongly correlated quantum matter, yet existing machine-learning approaches lack a general, principled framework for learning under site-dependent symmetries, particularly for intrinsically nonlocal observables. Here we introduce a gauge-equivariant graph neural network that embeds non-Abelian symmetry directly into message passing via matrix-valued, gauge-covariant features and symmetry-compatible updates, extending equivariant learning from global to fully local symmetries. In this formulation, message passing implements gauge-covariant transport across the lattice, allowing nonlocal correlations and loop-like structures to emerge naturally from local operations. We validate the approach across pure gauge, gauge-matter, and dynamical regimes, establishing gauge-equivariant message passing as a general paradigm for learning in systems governed by local symmetry.

Structured sequence generation often requires a model to satisfy several input-derived constraints in a single output. Standard decoding methods may assign high probability to fluent continuations while placing low mass on continuations that realize all required anchors jointly. We study this regime as a rare-event sequential inference problem. LatticeBridge combines a compact prefix language model, instance-compiled surface automata, and a twisted sequential Monte Carlo (SMC) decoder with resampling, multilevel splitting, and a source-support proposal term derived from instance-provided phrases. The constraint representation is compiled from each input instance and does not rely on manually curated lexical classes. On 2,610 attainable validation tasks spanning CommonGen, E2E NLG, and WikiBio, the particle decoder improves exact anchor satisfaction and mean anchor coverage over greedy, beam-filtered, and best-of-k ancestral baselines under a shared proposal model. Since exact anchor satisfaction alone does not rule out unsupported attribute substitutions, the evaluation reports required-anchor coverage, source coverage, source-intrusion diagnostics, overlap, runtime, and particle statistics jointly. The benchmark characterizes the faithfulness-overlap-latency frontier under a fixed proposal model.

Scalar quantization of large language models (LLMs) is fundamentally limited by information-theoretic bounds. While vector quantization (VQ) overcomes these limits by encoding blocks of parameters jointly, practical implementations must avoid the need for expensive lookup mechanisms or other explicit codebook storage. Lattice approaches address this through highly structured and dense packing. This paper explores the Leech lattice, which, with its optimal sphere packing and kissing configurations at 24 dimensions, is the highest dimensional lattice known with such optimal properties. To make the Leech lattice usable for LLM quantization, we extend an existing search algorithm based on the extended Golay code construction, to i) support indexing, enabling conversion to and from bitstrings without materializing the codebook, ii) allow angular search over union of Leech lattice shells, iii) propose fully-parallelisable dequantization kernel. Lastly, we provide a geometric reinterpretation of combining shape--gain quantization with GPTQ-style Hessian corrections: the standard scale-correction step of shape--gain acts as a retraction onto a product of spheres, yielding a Spherical GPTQ primarily acting on directions. We find that low-angular-distortion LLVQ reduces sensitivity to Hadamard/rotation preprocessing, and enables a strong Hadamard-free PTQ in practice. LLVQ delivers state-of-the-art LLM quantization performance, outperforming recent methods such as Quip#, QTIP, and PVQ. The results highlight the effectiveness of high-dimensional lattices for scalable, theoretically grounded model compression.

A key question in benchmarking generative crystal reconstruction models is how the amount and type of crystallographic information provided to a generative model affects its ability to reconstruct atomic structures. Yet such comparisons often overlook the fact that models receive unequal information about the target during reconstruction, thereby confounding architectural conclusions. We present AtomBench, an extensible, model-agnostic framework for comparing generative models on a well-defined crystal reconstruction task (rather than \textit{de novo} generation), which we here apply to conventional superconductors. We train and evaluate four models, AtomGPT, CDVAE, FlowMM, and MatterGen, on the JARVIS Supercon-3D and Alexandria DS-A/B datasets, grouping them by the information each accesses at inference. Reconstruction fidelity is measured by the Kullback-Leibler divergence (KLD) and mean absolute error (MAE) of lattice parameters and the root-mean-squared displacement (RMSD) of atomic coordinates. We further introduce the continuous corrected RMSD (ccRMSD), a continuous measure of local geometric fidelity defined for every structure in the test set. MatterGen achieves the best atomic-coordinate reconstruction, followed by AtomGPT, while CDVAE reconstructs lattices most accurately, and FlowMM is the least accurate but fastest overall. We find that conditioning on critical temperature T$_c$ does not consistently improve fidelity. We also release AtomBench as an open-source Python package that reproduces all reported reconstruction metrics, figures, and tables from one or more benchmark files and supports direct submission to the JARVIS-Leaderboard. Any inverse model emitting crystal reconstructions can be benchmarked with \texttt{atombench}, and we encourage community use. https://github.com/atomgptlab/atombench