Top-Down Memoization: Mental Model

Start with the idea

Begin with the concrete question: Return Fibonacci number F(12) with F(0)=0 and F(1)=1. For the visible input n=12, the stated result is 144. Before reading code, name the smallest piece of information that must survive from one step to the next. A one-dimensional DP state summarizes a prefix or a single parameter. Define its meaning before writing a recurrence. That is the mental model for Top-Down Memoization; it gives you a reason for each update instead of a line of code to memorise.

The core mechanism is this: Cache a recursive state the first time it is solved so overlapping subproblems run once. Treat it as a claim to test after every operation. Write down the state before the first operation, update it once, and ask which part of the claim is still true. If an item is skipped, inserted, or removed, explain why that action cannot destroy an answer that should be retained. A useful trace has columns for the current input position, the state before the step, the action, and the state afterward. Use the visible sample until the expected result becomes inevitable rather than merely plausible.

Now inspect the Rust sketch, not just its final expression. The supplied Rust sketch is a starting point. Identify what it stores, how one iteration changes it, and whether it already solves the exact task or only demonstrates a related operation.