One Kernel Behind Gravity and Electromagnetism

We propose a conditional micro-to-macro construction in which gravity and electromagnetism arise as different infrared readouts of one complex Relator edge. The microscopic carriers are Gaussian-collared phase states on a one-dimensional complex generator space \(\mathbb C\), while physical fields and forces are read in the orthogonal space \(\mathbb R^3\). The elementary edge factorizes into a radial modulus, an electromagnetic phase, a spin transporter, and an internal channel metric,\[\mathcal C_{ij}=J_y(\ell_{ij})\,U^{\mathrm{em}}_{ij}\,U^{\mathrm{spin}}_{ij}\,\eta_{\mathcal R}(s_i,s_j).\]The common radial factor is the Gaussian--Coulomb collar kernel\[J_y(\ell)=\int_0^\infty \exp(-y^2t^2-\ell t)\,dt,\qquad y>0,\qquad \ell\ge0.\]It is positive, strictly decreasing in \(\ell\), finite at \(\ell=0\), and has the infrared expansion\[J_y(R)=\frac{1}{R}-\frac{2y^2}{R^3}+O(R^{-5}),\qquad R\gg y.\] The modulus reconstructs edge lengths and supplies the metric sector under the stated Regge-refinement assumptions. The compact phase factor \(U^{\mathrm{em}}_{ij}\) gives gauge-covariant electromagnetic links and gauge-invariant cycle holonomies, whose continuum normalization yields the Maxwell action. The spin transporter \(U^{\mathrm{spin}}_{ij}\) supplies the Dirac coupling. The internal channel metric \(\eta_{\mathcal R}\) keeps the mass, charge, and transverse-current sectors distinct. In particular, the Coulomb monopole is a scalar charge block, while magnetostatic and radiative content belongs to the transverse current block. In the static weak-field limit the same kernel gives the regularized two-body potential\[U_{ab}^{(y)}(R)=\hbar c\,J_y(R)\left[\left(\sqrt{\alpha_{\mathrm{EM}}}\frac{q_a}{e}\right)\left(\sqrt{\alpha_{\mathrm{EM}}}\frac{q_b}{e}\right)-\left(m_a\sqrt{\frac{G}{\hbar c}}\right)\left(m_b\sqrt{\frac{G}{\hbar c}}\right)\right].\]Equivalently,\[U_{ab}^{(y)}(R)=\hbar c\,J_y(R)\left[\alpha_{\mathrm{EM}}\frac{q_aq_b}{e^2}-\frac{Gm_am_b}{\hbar c}\right].\]Using\[\alpha_{\mathrm{EM}}=\frac{e^2}{4\pi\epsilon_0\hbar c},\]the infrared limit is\[U_{ab}^{(y)}(R)=\frac{q_aq_b}{4\pi\epsilon_0R}-\frac{Gm_am_b}{R}+O\!\left(\frac{\hbar c\,y^2}{R^3}\right).\]Thus the Newton and Coulomb inverse-distance potentials arise from the same regularized radial kernel, while their signs and source content are fixed by the internal channel metric. Coarse-grained source bilinears are mapped by the Hubbard--Stratonovich transformation to effective fields on \(\mathbb R^3\). In this reading, the gravitational and electromagnetic fields are continuum dual variables of the same Gaussian-collared \(\mathbb C\)-space carrier data, rather than independent microscopic primitives. The construction gives a controlled common kernel-level origin for the Newton--Coulomb limit and for the Einstein--Maxwell--Dirac infrared reading, subject to the refinement, locality, positivity, and normalization assumptions stated in the manuscript.