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Microbiologists have been growing microbes on agar plates for nearly 150 years, but agar’s discovery dates back to a happy accident in a mid-17th-century kitchen. Legend has it that on a cold winter day, a Japanese innkeeper cooked tokoroten soup, a Chinese agar seaweed recipe known in Japan for centuries. After the meal, the innkeeper discarded the leftovers outside and noticed the next morning that the sun had turned the defrosting jelly into a porous mass. Intrigued, the innkeeper was said to have boiled the substance again, reconstituting the jelly. Since this discovery, agar has become a staple in many Japanese desserts, from yokan to anmitsu.
The model must be autoregressive. It receives a token sequence as input and predicts the next token. Output digits are generated one at a time, with each new token fed back as input for predicting the next. The carry propagation must emerge from this autoregressive process — not from explicit state variables passed between steps in Python.