Anthropic mathematician Levent Alpöge posted on July 19-20 that he and a colleague used Claude Fable 5 to find a counterexample to the Jacobian Conjecture — a problem open since 1939 and listed in Smale's 1998 Mathematical Problems for the Next Century. The counterexample is a concrete C³→C³ polynomial map with constant Jacobian determinant of -2 that is demonstrably non-injective; three distinct points all map to the same output, verifiable via Wolfram Alpha. The 2D case remains open. Wolfram MathWorld already updated its entry. HN thread (746 points) noted "Claude Fable was able to find a counterexample in degree 7" and called it a new era for AI in mathematics.
Proposes a competition for distinguishing robust from spurious reasoning in frontier mathematical LLMs using internal model mechanisms rather than final-answer accuracy. Motivated by the gap between strong benchmark scores and unreliable internal reasoning: a model can get the right answer for the wrong reasons. Focuses on math olympiad-level problems (AI Mathematical Olympiad context), providing a structured interpretability challenge tied to an existing competition with real stakes.
📑 arXiv 3w ago
⭐ Editor's Pick
As RLVR increasingly uses an LLM judge as the reward model, this paper asks: how capable must the judge be, and how biased is it? Studies this calibration question specifically for citation quality in deep-research systems — a practically important but understudied reward signal. Finds that frontier models are not always necessary as citation judges, and characterizes the bias profile of smaller rubric LLMs. Directly informs design of reward models for training web-grounded research agents.
Addresses the depth-vs-breadth tradeoff that limits single-agent ReAct-style web search: one long trajectory cannot simultaneously go deep on sub-questions and cover broad coverage. WebSwarm uses recursive multi-agent orchestration where a coordinator dynamically spawns specialized sub-agents for sub-queries and synthesizes their outputs, enabling parallelism across breadth while each sub-agent can depth-search independently. Targets complex research-oriented queries where current single-agent and naive multi-agent approaches fall short.
🔶 Anthropic Jun 30
★ High Signal
Claude Sonnet 5 (released June 30) ships a 1M-token context window and 128k max output tokens as the new Sonnet default, with adaptive thinking on by default and manual extended thinking removed. Performance sits close to Opus 4.8 on coding, tool use, and agent benchmarks at lower price. Introductory API pricing: $2/$10 per Mtok through August 31, reverting to $3/$15 after. Now the default model for Free and Pro claude.ai plans and Claude Code's default model. Includes a new tokenizer producing ~30% more tokens for the same text than Sonnet 4.6 — migration requires recounting prompt sizes.
Z.ai released GLM-5.2, a 753B open-weight model (MIT license, no regional restrictions) with a production-grade 1M-token context window, outperforming GLM-5.1 substantially on long-horizon coding benchmarks: 81.0 vs 63.5 on Terminal-Bench 2.1, 62.1 vs 58.4 on SWE-bench Pro. On FrontierSWE it trails Claude Opus 4.8 by 1% while edging GPT-5.5 by 1%, and it's the highest-ranked open-source model across all three long-horizon coding benchmarks tested. Architecturally introduces IndexShare (shared sparse attention indexer across every 4 layers, reducing per-token FLOPs by 2.9× at 1M context) and improved MTP speculative decoding (+20% acceptance length). Post-training uses a critic-based PPO formulation for long-horizon RL with an anti-hack module to detect reward hacking in coding agents.
OpenAI and Molecule.one's Maria platform demonstrated GPT-5.4 operating as a near-autonomous chemist: the model generated research proposals for improving Chan-Lam coupling of primary sulfonamides, and one proposal (OAI-M1-03) identified TEMPO as a useful oxidant additive. Across 10,080 high-throughput reactions run by Maria Lab, mean yields rose from 16.6% to 25.2%, and the fraction of reactions exceeding 30% yield tripled (15.6% → 37.5%). The finding transferred to bench-scale validation: yields improved for 11/14 substrate pairs with >2× gain in most cases. Total timeline was 3 months; humans retained oversight at key decision points, and a cheaper analog (4-hydroxy-TEMPO) was found to match TEMPO performance.
MEM1 trains agents end-to-end via RL to compress and update an internal memory state at each step, maintaining constant context size across arbitrarily long multi-turn tasks. Unlike RAG or full-context retention, the memory management policy itself is learned. Demonstrated on multi-turn web and tool-use tasks; from MIT, accepted ICLR 2026.
Intuitor (ICLR 2026) trains LLMs to improve reasoning using only self-certainty as a reward signal—no labeled data, no external verifier, no human-crafted reward. The companion code release (RLIF framework) enables direct reproduction of the result that models can self-improve on reasoning benchmarks from internal feedback alone. Practically significant because it removes the dependency on curated verifiable datasets.
GPT-Rosalind is OpenAI's first domain-specific vertical model, tuned for biology, drug discovery, and genomics. It scores 0.751 pass@1 on BixBench vs. GPT-5.4's 0.732 and leads on 6 of 11 LABBench2 task families. Available only to US enterprise partners (Amgen, Moderna, Allen Institute, UCSF, Thermo Fisher) and ships with a Codex plugin connecting to 50+ scientific tools.