CG V4.0 - The Law of the Universe by Boudehri et al.

The Law of the Universe: The Mechanical Pulse by Boudehri Z. et al. Abstract This research presents the formalization of the Law of the Universe, a deterministic framework that unifies galactic dynamics and subatomic particle saturation through a single mechanical modulus. By identifying the vacuum as a non-linear elastic substrate—the One Material—we derive a universal rigidity constant, the 0. 048 Lock. This work marks the transition from the developmental Coherence Gravity model to a unified mechanical law governing the structural integrity of the substrate. Scientific Discovery The core discovery is the Universal Scaling Equation, which demonstrates that the transition from Newtonian gravity to asymptotically flat rotation curves in 175 disk galaxies (SPARC) is governed by the same dimensionless rigidity ratio (0. 048) found in the transverse momentum (pT) saturation pedestals of high-energy proton collisions at the CERN Large Hadron Collider. We establish that: The Mechanical Pulse: The Universe operates as a vibrating mesh where the 0. 048 dex represents the yield strength of the vacuum. The Bi-role Operator: The Pion acts as the localized relief mechanism (the "Kill Switch") when the substrate hits the 0. 048 tension threshold. The One Material: Gravity is not a force of attraction but the surface tension of the substrate, while particles are localized "pinches" or nodes within the same medium. A single structural law governing curvature, matter, and cosmological evolution. It completes the Coherence Gravity program by identifying the invariant mechanical principle underlying gravitational, subatomic, and large-scale phenomena. The vacuum is not empty and not passive. It possesses finite elastic rigidity and thermodynamic relaxation. Spacetime curvature arises from local decoherence stress, and matter corresponds to stabilized nodes within this mechanically coherent structure. High-energy interactions represent phase-restoration events of the same medium. A universal logarithmic modulus (0. 048 dex) emerges as the invariant structural tolerance of the system. This constant governs: • The flattening of rotation curves across 175 SPARC galaxies• Lensing behavior in high-Mach cluster collisions• Saturation of transverse momentum in collider experiments (CERN) • Finite relaxation dynamics of metric restoration• Compatibility with gravitational-wave propagation constraints The framework is covariant, ghost-free, hyperbolic, and parameter-minimal. It reduces to General Relativity in low-stress regimes while eliminating the need for particle Dark Matter or additional stochastic sectors. The Law imposes explicit physical exclusions: infinite curvature singularities, unconstrained metric evolution without thermodynamic cost, and arbitrary force scaling beyond the invariant modulus are forbidden. This preprint consolidates the conceptual foundation (V1), structural clarifications (V2), and full Lagrangian formalism (V3) into its completed axiomatically unified form (V4): a deterministic structural principle governing coherence, elasticity, and curvature across all scales.