Space as a Discrete Elastic Network: Mass, Unified Time Dilation, and Electromagnetism on the Contact Faces
Abstract
We propose a field-quantum framework in which discrete field quanta constitute space (A), attract their neighbours (P1), each carry a two-component complex amplitude whose normalised states carry the Hopf fibration $S^1\to S^3\to\mathrm{CP}^1$ (P2), and every coherent excitation is one configuration whose energy, orientation and integer counts—action $\hbar$ per cycle—are attributes of the whole (P3), from which one-quantum wholeness follows as a theorem. The continuum limit is a Lorentz-invariant nonlinear Klein–Gordon theory, and the numerical checks below use lattice simulations of this action. Rest mass is the vacuum-subtracted deformation energy, $mc^2=\int(T^{00}-T^{00}_{\rm vac})\,d^3x$, verified by 3D decomposition; a boosted soliton recovers $E=\gamma mc^2$. Velocity and gravitational time dilation issue from one internal light clock: a moving clock's internal signal slants at unchanged transverse size; a static clock's spacing and update rate are each scaled by $\lambda_g$ (shorter steps, slower steps). The two combine into $\omega=\omega_0\sqrt{1-v_{\rm eff}^2/c^2}$ ($v_{\rm eff}=v$ for a moving clock, $v_{\rm esc}$ for a static one; one rule applied level by level, the factors multiplying when both act), with the conserved product law $E_\infty=S\,m_{\rm loc}$ verified to 0.65%. The coordinate light speed $c\lambda_g^2$ gives the $1.751''$ light deflection and the Shapiro delay by Fermat's principle, and the composition law $\lambda_g=e^{\Phi/c^2}$ yields…
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