Energetic Feedback Theory (Energy I): Foundations of a Common Quadratic Energy Geometry

Physical systems may contain many distinct energetic mechanisms, yet their behaviour can often be organized through the interraction between energy that is stored or structurally confined and energ that is active, transported, exchanged or dissipated. This study introduces Energetic Feedback Theory (EFT) as a general representational framework that organizes these contributions into complementary closed/storage-like and open/active/transport-like energetic sectors and treats their coupling as a primary geometric object. The present work develops the minimal quadratic representation of this organization. The two sectors constitute a reduced energetic partition rather than a restriction on the number of physical energy mechanisms within a system. The framework is examined through four case studies. First, a classical additive energy is lifted into a one-parameter family of homogenous quadratic representations that preserve the energy value and homogeneity-corrected first order behaviour at a selected working point. Second, independently sampled Kepler hydrogen-like systems recover closely related normalized signatures across macroscopic and microscopic scales. Third, mechanical and and plasma harmonic oscillators recover a common harmonic signature despite their different physical carriers and energy scales. Finally, a Duffing oscillator shows that the recovered geometry deforms continuously as nonlinearity increases while remaining definite over the investigated range. The results support the broader EFT hypothesis that physically distinct systems can be represented within a common quadratic energy space geometry. The geometry is common in form, whereas its coefficients remain system-and regime dependent. This establishes representational foundation for subsequent analysis of stability, energy exchange, and nonlinear energetic dynamics. Keywords: energetic feedback theory; quadratic energy geomerty; energetic sectors; Hessian stability; cross-scale analogy; nonlinear systems