1. Monotonicity in Strike Mathematical Form: C(K1,T)≥C(K2,T), for K1<K2 As the strike price increases, the call option price must not increase. A higher strike gives you a less favorable right (to buy at a higher price), so the call should be cheaper. Prevents: Vertical

2. Convexity in Strike Option prices as a function of strike must be convex. The price at an intermediate strike K2​ should lie below or on the straight line connecting C(K1) and C(K3). Prevents: Butterfly Arbitrage (constructing a butterfly spread for guaranteed profit).

3. Monotonicity in Maturity Mathematical Form: C(K,T1)≤C(K,T2), for T1<T2 Meaning: A longer-dated option (with the same strike) should always cost more or equal, since it offers more time for favorable movements. Prevents: Calendar Spread Arbitrage (profiting from mispricing

4. Cross (Four Prices) Mathematical Form: C(K2​,T2​)−C(K2​,T1​)≥C(K1​,T2​)−C(K1​,T1​) Ensures consistency between strike and maturity dimensions together. The increase in option value due to longer maturity should itself respect monotonicity in strike. Prevents:

A new RL method stabilizes LLM training for SOTA performance Introducing BAPO: Balanced Policy Optimization with Adaptive Clipping. This novel framework addresses key challenges in off-policy reinforcement learning for LLMs by dynamically adjusting clipping bounds to boost

Another interesting paper about how to scale weight decay with muP for AdamW, from a different perspective. They argue for «independent weight decay scaling» as a way to maintain relative update sizes and ensure hyperparameter transfer. https://arxiv.org/abs/2510.190...

I saw this while reading Anthropic’s “Toy models of superposition” (https://transformer-circuits.p... I was surprised to see J-L mentioned, and their paper shows an analogue of it can be true for NNs! The proof is really a one-liner (“apply J-L from big space to small”). The details:

Can LLM verbalize a new concept? Yes (we plug it back in to evaluate) Does LLM synonyms of such concept? Yes (and apparently *some* of them are common across models: machine synonym ) Can we use these new concepts to control and understand the model better? We think so, but you