Φ-Identity: A Unified Measurable Invariant Underlying Energy, Force, and Curvature

AbstractPrimitive Energy Φ is established as a directly measurable invariant identity obtainedfrom observable residuals of classical physical closure relations.It is operationally defined through the measurable deviations:ΦE = E − mc2, ΦF = F − ma, ΦC = robs − rs (1)These residuals satisfy the unified invariant identity:ΦE = ΦF = ΦC = Φ (2)which certifies Φ as a single measurable invariant underlying multiple physical observables.Classical laws emerge as the exact closure condition Φ = 0, while Φ̸ = 0 represents measurablestructural deviation. Numerical certification confirms invariant consistency across distinctobservational faces. Thus, Φ is established as a unified measurable invariant