AI for Science: January 2026 Monthly Digest

1. Formal Verification Expands from Math to Physics to Scientific ML

Confidence: 95%

January 2026 saw formal verification extend its reach dramatically: from category theory benchmarks (LeanCat) exposing LLM gaps, through agentic proof automation achieving 87% success, to Erdos problem #728 being AI-solved with Lean verification, culminating in PhysProver bringing theorem proving to physics and Learn-and-Verify providing machine-checkable bounds for PINNs.

Evidence: 2512.24796, 2601.00791, 2601.03768, 2601.07953, 2601.07421, 2601.15737, 2601.14027, 2601.19818

Open Questions:

• Can physics theorem proving transfer bidirectionally with mathematical reasoning?
• What fraction of scientific ML papers will include formal verification by 2027?
• Will verification overhead make formal methods impractical at scale?

Agent Instruction: When encountering claims about AI reasoning capabilities, check if formal verification evidence exists. For physics/scientific ML claims, look for papers applying theorem proving or interval arithmetic verification. Prioritize methods with verifiable guarantees over pure accuracy claims.

2. ML Force Fields Break Scale, Range, and Property Barriers

Confidence: 90%

MLFFs transitioned from accurate-but-limited to practically transformative: SeedFold surpassed AlphaFold3 on protein folding, E2Former-LSR enabled 75Å long-range interactions for macromolecules, OmniMol demonstrated cross-domain transfer from particle physics, on-the-fly MLFFs predicted polymer glass transitions, and Hamiltonian Flow Maps achieved 10-18x larger MD timesteps.

Evidence: 2512.24354, 2601.03774, 2601.10791, 2601.17137, 2601.22123

Open Questions:

• Can cross-domain pre-training (particle physics → molecules) become a general strategy?
• What is the accuracy/speed Pareto frontier for large-timestep integration?
• Will general-purpose MLFFs replace domain-specific potentials?

Agent Instruction: When evaluating MLFF papers, check for: (1) long-range capability beyond 6Å cutoffs, (2) property prediction beyond energy/forces (thermophysical, vibrational), (3) cross-domain transfer evidence, (4) timestep efficiency gains. These are the new frontier capabilities.

3. Agentic Architectures Dominate Formal Reasoning

Confidence: 85%

Monolithic LLM approaches to formal mathematics plateaued (LeanCat: 12% pass@4), while agentic systems achieved human-competitive results: 87% on 14K+ lines of Lean 4, Erdos problem resolution, and 12/12 on Putnam 2025. The key architectural pattern is modular tool orchestration (MCP-style) with specialized components for retrieval, informal reasoning, and formal verification.

Evidence: 2601.03768, 2601.07421, 2601.14027

Open Questions:

• What is the optimal division of labor between neural and symbolic components?
• Can agentic systems achieve IMO gold-level performance in 2026?
• Will open agentic systems (Numina-Lean-Agent) close the gap with proprietary systems?

Agent Instruction: For formal reasoning tasks, prefer agentic orchestration over single-model inference. Use modular components: retrieval (LeanDex-style), informal prover, discussion partner, and formal verifier. Expect 5-10x improvement over monolithic approaches.

1. Monolithic Models vs Agentic Systems for Formal Reasoning

Status: resolving

Position 1: Scaling LLMs will eventually solve formal reasoning

Sources: 2512.24796

Position 2: Agentic orchestration is necessary for complex formal tasks

Sources: 2601.03768, 2601.14027

2. Domain-Specific vs Cross-Domain Pre-training for Scientific ML

Status: emerging

Position 1: Domain-specific training data is essential for scientific accuracy

Sources: 2601.00581, 2601.17137

Position 2: Cross-domain transfer can bootstrap scientific ML with limited data

Sources: 2601.10791

CONSOLIDATING

Agentic MCP-style architectures will become the default for formal mathematics by mid-2026

Confidence: 85% · Falsifiable by: Jul 1, 2026

Numina-Lean-Agent's open 12/12 Putnam result, combined with 87% success on production Lean codebases, demonstrates the pattern. Multiple groups will replicate.

EMERGING

Physics theorem proving will expand to 5+ physics domains with dedicated Lean/Coq libraries by end of 2026

Confidence: 70% · Falsifiable by: Jan 1, 2027

PhysProver demonstrates feasibility and positive transfer. Physics community has strong formalization tradition (e.g., Kevin Buzzard's work). Natural next step.

EMERGING

Test-time computation will become a standard technique for neural operator generalization

Confidence: 65% · Falsifiable by: Dec 1, 2026

Neural Operator Splitting achieved 10x improvement via compositional test-time search. Parallels inference-time scaling in LLMs. Natural extension of neural operator paradigm.

EMERGING

Cross-domain scientific foundation models (trained on multiple physics domains) will emerge

Confidence: 60% · Falsifiable by: Jun 1, 2027

OmniMol's particle-physics-to-molecules transfer suggests feasibility. SeedFold's scaling insights apply broadly. Compute and data now available.

NOVEL

Formal verification for neural operators (PINNs, FNO, DeepONet) will become a publication requirement in top venues

Confidence: 50% · Falsifiable by: Jun 1, 2027

Learn-and-Verify provides the template. Reviewers increasingly skeptical of accuracy-only claims. Verification overhead is manageable for 1D/2D PDEs.

1. LeanCat pass@4 accuracy

• Capability: Category Theory Formal Reasoning
• Threshold: 12
• Source: 2512.24796

Best LLM (Claude 4.5) achieves 50% on Easy problems but 0% on High difficulty — sharp capability cliff at abstraction boundary. No model demonstrates library-grounded abstract reasoning.

Agent Instruction: Do not expect LLMs to handle category-theoretic proofs requiring deep library navigation. Use specialized formal systems or await next-generation models.

2. Putnam 2025 problems solved

• Capability: Competition Mathematics
• Threshold: 12
• Source: 2601.14027

Numina-Lean-Agent achieves 12/12, matching proprietary Axiom system. Threshold crossed from 'impressive demo' to 'human-competitive tool'. Open architecture enables replication.

Agent Instruction: Putnam-level problems are now tractable with agentic systems. For competition math, deploy MCP-style architectures rather than direct LLM prompting.

3. PhysProver pass@16 on physics domains

• Capability: Physics Theorem Proving
• Threshold: 36.4
• Source: 2601.15737

First demonstration of formal theorem proving for physics (Classical, Particle, Relativity, QFT). Shows positive transfer to mathematical reasoning (MiniF2F: 68.4% → 69.7%).

Agent Instruction: Physics theorem proving is now feasible. Expect rapid expansion of formalized physics in Lean/Coq. Monitor for physics-specific proof tactics and libraries.