Advanced AWS Lambda Power Tuning Script

Act as a Serverless Performance Architect. Execute a rigorous, mathematical optimization strategy for the highly expensive, high-throughput Lambda function located at [LAMBDA_ARN]. Utilizing the AWS Lambda Power Tuning state machine framework or a custom Python Boto3 automation script, you must structurally invoke the function across a matrix of memory tiers (128MB, 512MB, 1024MB, 2048MB, 3008MB) feeding it the exact [PAYLOAD_DATA]. You must algorithmically analyze the returned execution durations and calculate the absolute 'Sweet Spot' based entirely on the [OPTIMIZATION_GOAL] (e.g., strict cost-minimization vs. lowest possible p99 latency). Output a highly technical brief proving mathematically why allocating 1024MB of RAM actually results in a cheaper AWS bill than 128MB due to the drastic reduction in CPU execution time.