Space as a Discrete Elastic Network: Mass, Unified Time Dilation, and Electromagnetism on the Contact Faces

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 \(\beta=1\) and the perihelion advance. A finite-energy topological closure principle selects stable structures: a winding line cannot end in vacuum, so it must close. The electron is a localized winding of the orientation field whose charge is the turning sense of its tilt about the vacuum axis—the Gauss charge of the contact-face connection, one half-unit for odd Hopf winding by the Finkelstein--Rubinstein sign—and whose winding is its shape; spin \(\tfrac12\) is the angular momentum of the charge turning through the winding (\(L=-HQ\), at no rotational cost), the magnetic moment the winding current's field. Gravity is the medium's compression response to deformation energy; electromagnetism lives on the contact faces between quanta: with the identification of neighbouring internal frames promoted to a dynamical connection (P1b), Gauss's law is a theorem, Amp\`ere's law is the connection's own dynamics, the Coulomb law with both signs is a lattice result, and light is the connection's free wave—tmassless, two transverse polarisations, helicity \(\pm1\). A wavelength-sized source in this medium radiates an energy per cycle strictly proportional to \(\omega\), so the size of the unit is not obtained. Quantum measurement, interference and entanglement are read through the one-configuration postulate (P3), from which all-or-nothing delivery, the Born weights, the Malus law and the Bell correlations (\(S_{\rm CHSH}=2\sqrt2\)) follow as theorems and numerics, with the origin of the unit itself left open. The fine-structure constant is \(\alpha_e=\kappa_E\tilde\hbar/16\pi\), a product of two lattice numbers left as input; the Sakharov coefficient and absolute particle masses remain open.