Reconstruction of an Exact Newtonian-Form Scalar from the Kretschmann Curvature Invariant in Schwarzschild Spacetime

We demonstrate that a Newtonian-form specific-energy scalar can be exactly reconstructed from the Kretschmann curvature invariant in Schwarzschild spacetime through a sequence of four algebraic operations. The resulting identity, Φ(r) = −c²√(K(r)r⁴/48) = −GM/r, is expressed entirely in terms of curvature invariants and the invariant areal radius, requires no metric coefficients, and remains analytically well defined for all r > 0, including at the Schwarzschild radius where g_tt = 0 and g_rr is undefined. The construction introduces no new dynamics and does not modify the Einstein field equations. A comparison with the curvature-invariant methods of Abdelqader and Lake (Phys. Rev. D 91, 084017) is included. Extension to non-vacuum black hole spacetimes will be addressed in a subsequent work.