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The old school knowledge tells us that the matter can couple to Einstein gravity in either minimal or Brans-Dicke manner. Recently we discovered a novel type of matter coupling in Einstein gravity. This type of matter coupling is theoretically self-consistent in the sense that all constraints in the Hamiltonian are preserved to be first class. At low energy scale it recovers the classic standard predictions of Einstein gravity + minimal coupling, including the gravitational potential, equivalence principle and so on, while at the high energy scale much richer phenomenology is granted, for instance, the theory is free from cosmological singularity. This type of matter coupling also predicts a universal lower bound on the cross section between dark matter particle and Standard Model particles. Ref: for Hamiltonian analysis, see P. A. M. Dirac, "Lectures on Quantum Mechanics" (Yeshiva University, New York 1964).