K–R Dual-Constant Framework for Adaptive Optimization Across Computational, Energy, and Intelligent Systems

This work presents a universal dual-constant adaptive optimization framework governed by two dynamic parameters: K (scaling constant) and R (reduction or residual constant). The framework establishes a mathematically unified structure for regulating performance, efficiency, and stability across computational, engineering, and intelligent systems by maintaining a predictive balance between enhancement and stabilization.

The core operational behavior of the framework can be expressed through the generalized optimization relation:

where:

• P0P_0P0 represents baseline system performance, energy, or output,

• PPP represents the regulated or optimized system output,

• KKK is the adaptive scaling constant controlling amplification, acceleration, or efficiency improvement,

• RRR is the residual or corrective constant representing stabilization, damping, or loss compensation.

This formulation establishes a dynamic equilibrium between scaling and corrective regulation. As the scaling constant KKK increases, system efficiency and optimization improve through controlled amplification, while the residual constant RRR ensures stability, loss compensation, and predictive correction. Together, these constants enable real-time adaptive optimization across diverse operational environments.

The proposed framework is domain-independent and applicable across a wide range of technological fields including semiconductor and VLSI systems, artificial intelligence and machine learning optimization, Internet of Things (IoT) networks, robotics and automation, microprocessors, cloud computing, cryptographic systems, battery charging technologies, radar and sonar processing, power grid infrastructure, and advanced energy systems. By dynamically adjusting the constants using feedback-driven control, the framework enables improved convergence speed, reduced energy consumption, enhanced computational stability, and increased operational reliability.

Unlike traditional static or single-parameter optimization approaches, the dual-constant formulation introduces a scalable predictive equilibrium model capable of cross-domain implementation and hardware–software co-optimization. The framework can be embedded within analytical models, embedded firmware, digital hardware architectures, or intelligent software systems, providing a unified mathematical foundation for next-generation adaptive, energy-efficient, and self-regulating technological infrastructures.