Duality between Complementary Energy and Torque Based on the F Research Framework: Algebraic Solution, Geometric Construction, and Cross-Disciplinary Applications

Abstract Within the F Research framework, this paper establishes a unified algebraic relationship between complementary energy (a scalar) and torque (the magnitude of a vector). The core formulas are: total energy E = W + U, and torque τ = sqrt(E^2 - W^2), where W is work and U is complementary energy. Forward and reverse solution formulas are derived. This paper further proposes an innovative geometric construction method—the generalized power triangle—in which the vertical leg represents work W, the horizontal leg represents torque τ, and the hypotenuse represents total energy E. The complementary energy U = E - W is naturally extended to the complementary torque q = E - τ, which is interpreted as the non-orthogonal occupied margin. The five quantities are found to be divisible into three groups: total energy E; work/complementary energy group W, U; and torque/complementary torque group τ, q. Knowing any one quantity from any two groups allows the remaining three quantities to be determined. In particular, the formula for solving E from U and q is E = U + q + sqrt(2Uq). This paper also explains the origin of "linear sum" and "sum of squares" from the perspective of algebraic order: the dot product produces a 0th-order scalar (linear superposition), while the cross product produces a 2nd-order bivector (sum of squares). Through cross-disciplinary examples in mechanics, electrical engineering, motor science, materials mechanics, and elastoplastic structural mechanics, the universality of the framework is verified. A teaching experiment design is proposed to validate the cognitive benefits of the framework. Finally, a digital mini-program called the "Five-Quantity Solver" is designed. The framework also provides a new geometric perspective for understanding the bias-variance tradeoff in machine learning regularization. Keywords: complementary energy; complementary torque; non-orthogonal occupied margin; torque; work; generalized power triangle; F Research; duality; algebraic order; cross-disciplinary teaching