The Exact Tension–Curvature Relation: Deriving General Relativity and Its Planck-Scale Corrections from the Projection Structure

Schmieke (2026n) established that the Einstein equation emerges from the projection π_Θ: M_s → M_Θ, with the Ricci curvature of the emergent spacetime related to the Hessian of the tension functional Φ and the stress-energy tensor related to the tension gradient. However, the derivation remained at the level of proportionality (∝) rather than equality (=), and the proof was given only in sketch form. This paper closes that gap. We provide the complete GaussCodazzi derivation of the embedding M_Θ ↪ M_s, determine the exact proportionality constant κ relating intrinsic curvature to the second fundamental form of the embedding, derive Newton’s gravitational constant G as a function of the pre-coherent parameters (D_s, Ω_s, β_max), and develop the systematic Planck-scale correction series to general relativity as an expansion in l_P/L. We classify every result as [Derived], [Constrained], or [Conjectured].