The Divisor Convolution Prime Power Theorem explores the relationship between divisors of prime powers and their applications in mathematical optimization, network systems, and cryptography. For any positive integer n, the theorem defines a function f(n) based on the sum of the divisors of n, weighted by both the divisor and the number of divisors of n/d. For any prime number p and non-negative integer k, the theorem provides an explicit formula for f(����), which is expressed as f(����) = ��^��+2 − (�� + 2)�� + (�� + 1)/(�� − 1)^2 This formula plays a crucial role in optimizing load balancing, hierarchical network routing, and fault tolerance in decentralized systems. By understanding the divisibility properties of prime powers, the theorem allows for more efficient distribution of traffic or data across multiple nodes in a network, ensuring scalability and redundancy. The applications extend to areas such as cloud computing,distributed storage systems, and blockchain technologies, where optimized access routes or permissions are vital for system performance. This paper delves into the mathematical formulation of the Divisor Convolution Prime Power Theorem and its practical significance in modern computational and network-based systems.Keywords: Divisor Convolution, Prime Power Theorem, Network Load Balancing, HierarchicalStructures, Cryptography, Optimization, Divisors, Traffic Distribution, Distributed Systems, Fault Tolerance, Cloud Computing, Blockchain, Network Routing, Mathematical Optimization, Access Permissions.