Tetrahedral Emergent Gravity : The Quaternion Vacuum and the Projection Paradigm Dark Matter as the Gravitational Shadow of H → R3, the 5-Cell as Fundamental Polytope in S3, and an Algebraic Prediction of ΩDM

Tetrahedral Emergent Gravity (TEG H) extends and builds upon TEG v8 by proposing that the quantum vacuum is fundamentally a quaternion field in \(H\) (topologically \(S^3\)), whose natural projection \(π: H → R^3\) produces the observed tetrahedral structure and galactic dynamics in three dimensions. Dark matter is identified as the gravitational shadow of the lost scalar component in the projection. The 5-cell (4-simplex) is identified as the fundamental polytope in \(S^3\). **This work is a direct extension of TEG v8.** All numerical results and validations of TEG v8 (SPARC galaxies, black hole entropy, etc.) remain valid. TEG v2 resolves Open Problem 2 of TEG v8 (Newtonian limit) and provides a deeper ontological foundation for the tetrahedral axiom. This document is a **working draft** submitted for priority registration and idea preservation. Several parts remain under active development. Open problems are clearly labelled.