Search: “euler-equations”

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Unsolved Problem by Fields Medalist Breached by Two High School Students

Two high school students used Claude Opus 5 and GPT-5.6 Sol to help solve an open Lorentzian polynomials problem, posting a 75-page arXiv proof.

Aayush Bathija and Prince Rohatgi of Oak Park High School, mentored by UCLA postdoc Daniel Soskin, published the 75-page paper 'Bounded Ratios for Lorentzian Polynomials' (arXiv 2609.05341), solving an open problem in Fields Medalist June Huh's Lorentzian polynomial theory. The main structural theorem extends bounded coefficient-ratio characterization from quadratic to arbitrary-degree polynomials via discrete convexity conditions. The students used Claude Opus 5 and GPT-5.6 Sol for exploration and proof ideas but independently verified all arguments; the result follows an open letter from 25 Fields Medalists voicing concerns about AI's impact on mathematical rigor.

Hacker News · security·Sep 14, 12:23 UTC · 2d agoAI researchHN 22↑ · 19 comments

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Embedded Graph Flows for Categorical Graph Generation

Researchers propose Embedded Graph Flows, a generative model with learned categorical embeddings that beats DiGress and GruM on molecular graph benchmarks.

Embedded Graph Flows (EGF) learns continuous embeddings for node and unordered-edge categories and transports Gaussian noise toward these endpoints using a permutation-equivariant graph transformer. On QM9 it achieves the best result on all four reported metrics, with a Fréchet ChemNet Distance of 0.150 versus 0.717 for DiGress and 0.812 for GruM. On ZINC250k it retains the lowest NSPDK MMD, indicating close agreement with local substructures of reference molecules. Code is released on GitHub.

arXiv cs.AI / cs.LG / cs.CL·Sep 4, 16:26 UTC · 12d agoAI research1

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Towards a Deterministic Math Solver for Clinical Language Models

Paper shows handing arithmetic to a deterministic Python solver beats direct model calculation at 32B but not reliably at 7B on MedCalc-Bench.

Researchers test a Program-Solve interface where clinical LLMs write case-specific Python executed by a restricted local solver instead of doing arithmetic directly. On MedCalc-Bench Verified (1,100 cases, 55 calculators), Qwen2.5-32B-AWQ scored 90.53% with solver handoff versus 83.47% with direct arithmetic (+7.05 points), while Qwen2.5-7B gained an unreliable +3.29 points with a confidence interval spanning zero. The authors audited the benchmark against clinical guidelines and flagged 16 of 55 calculators for version, use, or coefficient concerns.

Hugging Face daily papers·Sep 8, 20:00 UTC · 8d agoAI research

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Controversy over OpenAI's Maths Breakthrough

OpenAI claims its internal model proved the Navier-Stokes equations 'blow up' — a Millennium Prize Problem — amid allegations it borrowed mathematicians' methods.

OpenAI announced that an internal model produced a proof, certified in the Lean proof assistant, showing the Navier-Stokes equations can 'blow up,' implying infinite fluid speeds — a claimed solution to one of the seven $1-million Millennium Prize Problems. Mathematician Tristan Buckmaster alleged OpenAI, after learning of progress by him and Anthropic employee Levent Alpöge on 'blowing up' the related Euler equations, adopted a similar 'forcing' method; OpenAI's Sébastien Bubeck denied this, saying the model independently solved Euler by different means and produced the full Navier-Stokes proof over one weekend. Mathematicians including Diego Córdoba, co-developer of the forcing approach, remain cautious, and the community is still evaluating the competing proofs.

Hacker News · security·Sep 8, 16:57 UTC · 8d agoAI researchHN 21↑ · 5 comments1

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Variational Continuation for Double Pendulum Periodic Orbits

Researchers introduce a Hessian-based, integrator-free method using automatic differentiation to continue periodic orbits in the double pendulum, uncovering previously unreported orbit families.

A new arXiv paper presents a variational, Hessian-based framework for numerically continuing periodic orbits in dynamical systems, parametrizing candidate loops as Fourier series and minimizing deviation from the governing differential equations. Automatic differentiation replaces hand-derived Jacobians, and flat directions of the loss landscape guide the continuation search. The method is demonstrated on the double pendulum, mapping bifurcations along orbit families, including periodic orbits where neither pendulum mass is ever simultaneously at rest.

arXiv cs.AI / cs.LG / cs.CL·Sep 4, 16:42 UTC · 12d agoAI research

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A Ranking Approach for Measuring Calibration

Researchers propose rankECE, a ranking-based calibration error measure with theoretical guarantees that outperforms binned ECE approximations.

The paper introduces rankECE, an alternative to Expected Calibration Error (ECE) that measures miscalibration by comparing points with neighboring predicted-probability values. It addresses the impossibility of estimating ECE with guaranteed accuracy in assumption-free settings. Theoretical guarantees and empirical results establish rankECE as a better proxy for ECE than the binned approximations most commonly used in practice.

arXiv cs.AI / cs.LG / cs.CL·Sep 11, 17:34 UTC · 5d agoAI research1

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Proximity Gaps for Gabidulin Codes and Applications

Researchers prove proximity-gap bounds for rank-metric and Gabidulin codes, enabling the first polynomial commitment scheme framework based on rank-metric error-correcting codes.

The paper proves every linear rank-metric code admits a proximity gap for deltas up to (d-1)/(3n) with error at most q^(e+1)/q^m, and improves the gap to (d-1)/(2n) for Gabidulin codes with error at most 10q^(n-1)/q^m, matching bounds for Reed-Solomon codes. A constructed infinite family of constant-rate Gabidulin codes shows the (d-1)/(2n) bound is tight, and a counterexample establishes a lower bound on the error at the d/(3n) gap. Applications include an IOPP for interleaved Gabidulin codes adapted from the Ligero IOPP and a q-linearized polynomial commitment scheme adapted from Ligero-based PCS, reportedly the first PCS framework based on rank-metric codes.

arXiv cs.CR·Sep 9, 07:45 UTC · 7d agoResearch

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GPT-6 Astra gives mathematicians a breather, and OpenAI says that's by design

OpenAI's GPT-6 Astra tops ulam.ai's ErdosBench math benchmark with 106 of 226 problems solved, while the company prioritizes recursive self-improvement over math optimization.

OpenAI's GPT-6 Astra leads ulam.ai's ErdosBench with a score of 3.23, solving 106 of 226 open math problems and fully solving 43, ahead of GPT-5.6 Sol's 78 solved problems. Chief scientist Jakub Pachocki said OpenAI deliberately avoided targeted math optimization to prioritize recursive self-improvement and automated alignment research. Benchmark developer Przemek Chojecki estimated the gain at 5-10% across tested math-research skills. Mathematician Terence Tao warned at the 2026 International Congress of Mathematicians that AI-generated proofs could shift mathematics from proof scarcity to proof overload.

The Decoder·updated Sep 12, 20:00 UTC · 4d ago·first Sep 10, 13:45 UTC · 6d agoAI industry 10 sources1

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The Pelican comparison grid for Astra is pretty interesting

Simon Willison's pelican SVG comparison shows GPT-6 Astra producing markedly better images than GPT-5.6 Sol, Terra, and Luna across reasoning levels.

Willison generated pelicans-riding-bicycles SVGs with newly accessed GPT-6 Astra at low through max reasoning levels and rendered them in a comparison grid against GPT-5.6 Sol, Terra, and Luna. Astra's outputs were markedly more coherent, while even the best GPT-5.6-Sol images remained largely abstract shapes. Astra does not support a reasoning=none setting, so all comparisons involved reasoning-enabled runs.

Simon Willison·Sep 4, 23:59 UTC · 11d agoAI research

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Model-Aware Schedules Improve Generation via Fiberwise Optimal Transport

Researchers introduce model-aware diffusion schedules via fiberwise optimal transport, cutting flow-matching FID on CIFAR-10 by 38.6% at 16 function evaluations.

The paper proposes constructing diffusion and flow-matching sampling schedules from a fiberwise prediction risk defined via optimal transport, combined with coefficient-path kinetic action, yielding a closed-form time allocation. Across DDPM and flow-matching experiments spanning targets, datasets, and architectures, the schedules beat model-agnostic baselines, including a 38.6% relative FID reduction for flow matching on CIFAR-10 at 16 function evaluations. Normalized fiberwise-risk profiles from independently trained models align closely, suggesting empirical universality, and a frozen analytic allocation template retains most of the gains.

arXiv cs.AI / cs.LG / cs.CL·Sep 10, 17:30 UTC · 6d agoAI research

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NCP-ArchPreview Technical Report: Moving towards Latent Space Language Models through Next Concept Prediction

An 8.9B-parameter latent-space language model using next-concept prediction matches OLMo-3-7B pretraining loss with only 51.3% of the training tokens.

NCP-ArchPreview augments next-token prediction with Next Concept Prediction over a product-quantized concept vocabulary built from hidden states, trained jointly end-to-end. The 8.9B model was trained on 5.73T tokens from the Dolma-3 dataset, the largest latent-space language model demonstration to date. It consumes 51.3% of the tokens to reach OLMo-3-7B's final pretraining loss and outperforms it by 2.45 points on the downstream macro-average, including a 5.99-point GSM8K gain. The learned latent space also enables lightweight domain adaptation via a 17M-parameter VQ module and improves speculative drafting accepted length by 4.17%.

Hugging Face daily papers·Sep 8, 20:00 UTC · 8d agoAI research1

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Learning Length-Extrapolatable Recurrent Models

Researchers propose Credit Stabilization through Time, a training method letting recurrent models extrapolate up to 128x their training length.

The paper argues that length extrapolation failure in BPTT-trained recurrent models is better explained through state credit, the signal through which future losses reach earlier recurrent states. It introduces Credit Stabilization through Time (CST), which locally rescales the state-credit signal during backpropagation without rotating the corrected component or changing forward computation. Controlled experiments show improved performance beyond the training horizon, with gains at up to 128x the training length.

arXiv cs.AI / cs.LG / cs.CL·Sep 8, 17:59 UTC · 8d agoAI research1

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Local gradient neural operator

Researchers propose LGNO, a lightweight interpretable neural operator using learnable local stencils, matching global-operator accuracy on PDE benchmarks with fewer parameters.

LGNO builds on nonlinear gradient discretization priors and uses multilayer perceptron convolutional layers to learn translation-invariant local kernels resembling discrete stencils. A zero consistent stencil factorization separates coefficient learning from field reconstruction, and network folding shares equivalent components to cut parameter counts for symmetric problems. Evaluations on linear and nonlinear, static and dynamic, and low- and high-dimensional PDE benchmarks show maintained accuracy, parameter efficiency, and rollout stability, with applicability to diffusion, flow, and quantum problems.

arXiv cs.AI / cs.LG / cs.CL·Sep 7, 16:53 UTC · 9d agoAI research1

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Recursive Code World Models: Building Complex Worlds through Recursive Scene Programs

RCWM reconstructs complex 3D worlds as executable code from a single image using recursive scene programs with global-local-global solver recursion.

The paper introduces Recursive Code World Models, coupling a Recursive Scene Program representation with a recursive construction solver for image-to-3D-world reconstruction. Each solver call establishes the whole scene, recursively reconstructs unresolved parts, and revisits the whole to refine composition, while a vision-language coding agent compares reference images with scene renders to guide refinement. RCWM outperforms prior code-based image-to-scene reconstruction methods, and ablations show deeper recursive calls improve fine-scale reconstruction.

Hugging Face daily papers·Sep 9, 20:00 UTC · 7d agoAI research1

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LexFlip: A Dissociation Diagnostic for Legal Meaning Preservation Metrics

LexFlip releases 373 minimal perturbations of Quebec statutory French that reverse legal force while preserving tokens, exposing weaknesses in embedding-based meaning preservation metrics.

LexFlip provides 373 minimal perturbations of Quebec statutory French that reverse legal force while preserving 0.93 of tokens, creating dissociation items that break monotone token-overlap metric validation. The seven embedding and BERTScore metrics tested register only 0.022-0.039 of their identical-to-unrelated range on these edits, versus 0.670 for bidirectional NLI. Against FrJudge, with a measured human ceiling of r=0.597, a bare length feature outscores every semantic metric tested.

arXiv cs.AI / cs.LG / cs.CL·Sep 4, 15:49 UTC · 12d agoAI research

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27.5KB language-agnostic WebGPU syntax highlighter

A developer released gpu-lexer, a 27.5KB language-agnostic syntax highlighter that uses a tiny WebGPU model to label code tokens in the browser.

gpu-lexer splits source into words, whitespace, and symbols, then a small WebGPU model uses local and whole-file context to assign nine token classes, working on languages never seen in training. On held-out files, 12.57% of token labels differ from Shiki, though this measures agreement with Shiki rather than objective correctness. In benchmarks against Shiki 4.4.3, Prism.js, Highlight.js, Sugar High, and Starry Night, it highlighted 10 concatenated copies of three.min.js (5.56M characters) about 10x faster on an Apple M4 Pro in Chrome 152. The author frames it as an experiment, not a grammar-equivalent highlighter.

Hacker News · security·Sep 9, 01:14 UTC · 7d agoAI tools & infraHN 21↑ · 5 comments1

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Searching for New Physics with Reinforcement Learning

Researchers apply reinforcement learning to identify SMEFT operators explaining particle physics anomalies, reproducing and improving known CDF W-mass results.

The paper introduces a reinforcement learning method to search the large Standard Model Effective Field Theory (SMEFT) operator space for explanations of measurement anomalies. It was validated on the CDF W-mass anomaly, reproducing and improving known results, then applied to a harder multi-anomaly scenario. RL efficiently navigates complex loop-level operator correlations that bias human-driven phenomenological analysis.

arXiv cs.AI / cs.LG / cs.CL·Sep 9, 16:07 UTC · 7d agoAI research

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Mercator ↔ Equal Earth

Simon Willison used GPT-6 Astra (medium) in ChatGPT Work to build an animated D3 transition between Mercator and Equal Earth map projections.

Willison built an animated transition between the Mercator and Equal Earth map projections using D3. The tool was generated by GPT-6 Astra (medium) in ChatGPT Work. Equal Earth is a projection recently voted on at the UN. The post is a vibe-coding demonstration rather than a security or major model event.

Simon Willison·Sep 7, 16:24 UTC · 9d agoAI tools & infra1

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Semigroup-JEPA: Latent Dynamics Consistency for Zero-Shot Physics Generalization

SG-JEPA world model conditions latent prediction on physical parameters, halving open-loop prediction error versus DINO-WM and boosting robotic control success.

SG-JEPA extends the LeWorldModel JEPA framework by supplying the governing physics parameter to the temporal model via action-conditioning and jointly training an encoder and predictor through autoregressive latent rollout. On out-of-distribution gravitational-field tasks it reduces open-loop prediction error by up to 2x versus DINO-WM on 2D datasets and increases 3D robotic control success rate up to 2.5x using independently trained diffusion policies. A linear feature analysis attributes most of the gain to the encoder learning features that the predictor can carry forward through rollout.

arXiv cs.AI / cs.LG / cs.CL·Sep 9, 17:08 UTC · 7d agoAI research

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AI models' written reasoning steps correspond to distinct internal patterns, a new study finds

KAIST and Naver AI Lab researchers show LLM reasoning steps like extraction and computation map to distinct activation patterns, strongest in middle layers.

Researchers at KAIST and Naver AI Lab defined eight recurring reasoning operations, including extraction, decomposition, formula recall, deduction, and computation, and showed they correspond to separable activation patterns in Qwen2.5-7B, Qwen3-8B, and Gemma4-31B on math tasks, with GPT-5 labeling solution segments. The separation peaks in middle layers, holds even when a computation step produces a wrong answer, and goes beyond surface-level token choice. Findings replicated on Llama-3-8B, and classifiers trained on Qwen3-8B transferred to GPQA-Diamond and MATH-500. The authors note that using internal states for error detection or mid-generation steering remains future work.

The Decoder·Sep 12, 13:39 UTC · 4d agoAI research2

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AI Doesn't Mean the End of Mathematics—at Least Not Yet

Schneier and Rafi argue frontier AI models produce notable mathematical results but cannot yet build genuinely new conceptual frameworks.

Bruce Schneier and Kasra Rafi, writing in The Guardian, argue current AI models are not yet as capable as experienced academic mathematicians despite striking results. They cite OpenAI's disproof of the unit distance conjecture, Anthropic's published cryptanalysis results, and Claude's attempt at the Riemann hypothesis as achievements in counterexample search and recombining known techniques. They contend AI has not yet developed substantial new conceptual frameworks, though they expect that capability sooner rather than later.

Schneier on Security·Aug 28, 11:03 UTC · 19d agoAI research1

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Stellar Colosseum: A Many-Agent Harness for Long-Horizon Research in Mathematics and Theoretical Computer Science

Stellar Colosseum, a many-agent harness for long-horizon math and TCS research, solves open problems and reaches 71% on TCS-Bench with Gemini models.

Stellar Colosseum is a model-agnostic harness that allocates inference across long-horizon research in mathematics and theoretical computer science, using strategy exploration, a readiness gate, section-level decomposition, and verifier feedback routing. Integrated into Google Antigravity's Teamwork framework as the Long Proof pattern, it obtains new results on open problems from FOCS and JMLR papers using Gemini 3.1 Pro. On TCS-Bench it achieves 71.0% accuracy with Gemini 3.1 Pro and Gemini 3.7 Flash, and a Codeforces evaluation with Gemini 3.1 Pro solves 218 of 222 problems.

arXiv cs.AI / cs.LG / cs.CL·Sep 14, 17:58 UTC · 2d agoAI research

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3D Point Splatting for mmWave Radar Novel View Synthesis

Researchers propose 3DPS, a differentiable point renderer for mmWave radar novel view synthesis that outperforms optical-NVS baselines by 1.7x-5.2x.

The paper introduces 3D Point Splatting (3DPS), the first differentiable point renderer for radar, derived from the solid-angle form of the radar equation with ITU-R P.2040 material models and complex phasor splatting. On six outdoor ColoRadar scenes it reaches 0.587 mean Pearson correlation on held-out range-azimuth images, between 1.7x and 5.2x the RadarSplat, Radar Fields, and DART baselines. The same optimized scene produces ADC, complex range profile, and RA outputs via standard FFT pipelines, and training takes about 3 minutes per scene on an RTX 4090.

arXiv cs.AI / cs.LG / cs.CL·Sep 10, 17:52 UTC · 6d agoAI research

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Technical Manual for a Toolkit for Measuring Contextual Individuation in Transformer Language Models

An open methodology toolkit measures whether transformer language models contextualize fixed word forms across domains using bridge forms and layer-wise silhouette analysis.

The manual documents an open toolkit built around 'bridge forms' - identical written words recurring across two or more subject domains with a different sense in each - to test whether transformer language models individuate word occurrences by context beyond the embedding layer. It covers declarative specification of bridge forms, Wikipedia corpus acquisition, occurrence localization, layer-wise representation extraction, domain-pairwise silhouette measurement, and visualization, justifying each choice against failure modes such as sense contamination and subword-tokenization misalignment. It is a methodological and implementation reference and reports no empirical results.

arXiv cs.AI / cs.LG / cs.CL·Sep 4, 16:38 UTC · 12d agoAI research

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A Chosen Future Can Still Be Rewritten: Causal Writability in Video Models

Study shows video models often learn correct physics but fail to use it; low-dimensional 'causal writability' edits can restore correct motion.

The paper demonstrates 'causal writability' in video generation models: physically correct motion remains available inside the model even when the model outputs incorrect motion. In a red/blue mass oscillation setup, a low-dimensional edit predicted from simple physical variables restores correct fast motion, with a sharp depth boundary marking commitment. Early causal writability predicts which training errors later get corrected, and both writability and closure reproduce in a pretrained 1.3B video model.

arXiv cs.AI / cs.LG / cs.CL·Sep 14, 17:58 UTC · 2d agoAI research

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🔬“We have foundation models for language, not for physics” — Anima Anandkumar, Bren Professor of Computing

Caltech professor Anima Anandkumar discusses Neural Operators and FourCastNet for physics modeling, arguing inductive biases beat pure token scaling.

Anima Anandkumar, Bren Professor at Caltech and co-founder of Accelerated Understanding, describes Fourier Neural Operators that learn in frequency and spherical-harmonic domains to model weather, fusion, and fluid or heat flow. Her team built FourCastNet 3, a global weather model competitive with physics-based simulations that runs on consumer-grade GPUs. She also introduced TorchLean, a framework for writing PyTorch-style networks inside the Lean proof assistant for formal verification, and was appointed to the United Nations Scientific Advisory Board. She argues physical domains resist scaling due to tiny datasets and context lengths in the hundreds of billions, so progress comes from built-in structure and physical priors.

Latent Space·Aug 26, 15:15 UTC · 21d agoAI research1

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AdamX: Cosine similarity meets gradient descent

Researchers propose AdamX, a cosine-similarity-based first-order optimizer with variance rectification that matches Adam-class convergence across benchmark datasets and architectures.

The paper introduces AdamX, a first-order optimizer that uses cosine similarity as an adaptive mechanism for controlling update magnitudes, plus a variance rectification scheme for smoother optimization early in training. The method is described as scalable, model-agnostic, and straightforward to integrate into existing pipelines. Empirically, AdamX shows competitive convergence rates measured by epochs to reach performance thresholds under a fixed hyperparameter budget, with code and experiments released on GitHub.

arXiv cs.AI / cs.LG / cs.CL·Sep 10, 17:42 UTC · 6d agoAI research

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Diffusion Models and Concept Formation

Paper argues diffusion models implicitly form Cobweb-like concept hierarchies, with a basic level emerging at intermediate noise levels.

The authors draw a formal correspondence between diffusion models and Cobweb, a classic incremental concept-hierarchy learner, noting both are hierarchical Bayesian density models with Gaussian prototypes. Modes of the diffusion model's noisy marginals form a hierarchy whose basic level sits at intermediate noise, where class identity commits. The correspondence is tested on MNIST and Fashion-MNIST via mode-finding. Diffusion is reframed as a cognitive model of concept formation.

arXiv cs.AI / cs.LG / cs.CL·Sep 11, 16:43 UTC · 5d agoAI research

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MathKernel: An evidence-aware multi-engine mathematics kernel and MCP server

MathKernel is an open-source, evidence-aware multi-engine mathematics kernel that exposes verification workflows to AI agents via an MCP server.

MathKernel, published on GitHub, is a mathematics kernel that combines multiple computation engines with evidence-aware outputs. It ships as an MCP server, enabling AI agents and coding assistants to perform and verify calculations. The project drew moderate attention on Hacker News.

Hacker News · security·Sep 7, 00:21 UTC · 9d agoAI tools & infraHN 36↑ · 5 comments

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Prior Labs Releases TabPFN-3.5: A Tabular Foundation Model That Beats the Winning Otto Kaggle Solution With Default Settings

Prior Labs releases TabPFN-3.5, a 220M-parameter open-weights tabular foundation model that beats the 2015 Otto Kaggle winning score with default settings.

Prior Labs released TabPFN-3.5, a tabular foundation model that predicts in a single forward pass without per-dataset training or tuning. The base model grew from 53M to 220M parameters with a single multitask checkpoint, learned Fourier features, and in-context ECDF rank encodings. It scores 0.375 on the 2015 Otto Kaggle private leaderboard versus the winning 0.382 and claims first place on seven tabular benchmarks including TabArena and BeyondArena. Open weights cover the base, Fast (84M), and Thinking variants, but production use requires the Prior Labs API or a commercial license.

MarkTechPost·06:52 UTC · 15h agoModel release

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