A Causal Set Framework for Geometric Phase Transitions and Emergent Physical Energy

This paper presents a discrete quantum gravitational framework formulated within Causal Set Theory to investigate the emergence of macroscopic geometry and physical energy from underlying causal structure. Rather than treating spacetime as a fundamental background or requiring the empty state |∅⟩ to lie outside the kinematical Hilbert space, we formulate |∅⟩ as a well-defined boundary condition within the Fock-like configuration space. By introducing a structural density operator N^ and evaluating the discrete four-dimensional Benincasa-Dowker action S_BD, we demonstrate the existence of a quantum critical point λ_c. At this transition boundary, the effective spectral dimension collapses (D_s → 0) alongside the vanishing of the effective physical energy (E_phys → 0). These results indicate that physical energy is an emergent bulk quantity dependent on causal ordering, rather than an essential prerequisite for structural existence.