Scale Digit-Square Convergence Requests

Design a service that receives very high request volume for the digit-square convergence operation. For each request, the input is a positive integer n. The service repeatedly replaces n with the sum of the squares of its decimal digits and returns whether the sequence eventually reaches 1. Requirements: - Handle billions of requests over time. - Keep request latency low. - Avoid expensive repeated computation for common states. - Use the observation that after a small number of transformations, fixed-width integers collapse into a small numeric range, such as values below 1,000 or another small bound chosen by the implementation.