The Hamiltonian–Lagrangian Formulation of Spacetime Rigidity (v2)

We present a Hamiltonian–Lagrangian formulation of spacetime rigidity, a structural principle rooted at the Planck scale. The guiding paradigm is reversed: physical reality is fundamentally discrete, while the continuum of real numbers arises only as an emergent approximation. Starting from discrete canonical patches, we derive a reduced Hamiltonian whose continuum limit yields a causal propagation speed naturally identified with the speed of light. The associated effective Lagrangian describes a free entropic field with constant potential, reproducing the observed dark energy and the cosmological constant without fine-tuning. Within this framework, singularities in black holes and the Big Bang are regularized, while cosmological acceleration and early inflation appear as natural phases of rigidity. A structural relation of the form S · ρrig ∼ MP^4 constrains entropy and energy across bulk and boundaries. At the quantum level, entanglement and uncertainty are embedded directly in the canonical backbone, showing that the theory has both relativistic and quantum declinations. Taken together, these results suggest that several key features of spacetime dynamics—normally treated as separate inputs—can instead be understood as coherent consequences of rigidity, pointing toward a unified and economical foundation for cosmology and black-hole physics. This version (v2) corrects a duplicated sentence and adds an explicit reference to dark energy.