Hydrogen from the Foam: The Coulomb Law from Cell Adjacency, Charge as Scalar Dressing, and the Rydberg Scale with No New Parameters

We construct the hydrogen atom directly from cell dynamics on the BCC lattice of truncated octahedra, with no continuum potential written in by hand. Four results. (1) The lattice Green's function of the scalar (A1g) channel is Coulombic: G(r) fits A/r with tail strength A = 1/(4pi) exact, box-stable and isotropic to five digits. The 1/r law is a property of cell adjacency. (2) Electric charge is the A1g content of a defect. A bare matter-pattern (T1u) defect is foam-neutral, with no monopole tail; the A1g x T1u cross-coupling vanishes identically (parity selection rule, verified to 1e-17 in both inter-cell coupling conventions); a charged particle is a T1u defect dressed with a unit of the A1g mode. (3) A light dressed defect bound in the well of a heavy one reproduces the hydrogen ladder, including the near-degeneracy of 2s and 2p at the same rung (spread below 1 percent), the fingerprint of the hidden SO(4) symmetry of a true 1/r force. With the framework's derived fine-structure constant and electron mass, the Rydberg energy is alpha^2 m_e c^2 / 2 = 13.605693 eV against CODATA 13.605693 eV, and the reduced-mass ground state is 13.598287 eV against the measured 13.598434 eV; the 1.1e-5 residual is the relativistic and QED structure a Schrodinger-level calculation is supposed to leave behind. The bound-state problem introduces no new parameter. (4) The foam leaves a derived signature: the ground state deviates from ideal Coulomb by delta = -K/a_B^2 with both the exponent (contact mechanism from the quartic lattice term) and the coefficient K = 256(D0 - Dnn - 7/320) = 3.810 in closed form. At the physical scale a_B/l_P ~ 3.3e24 this gives |delta| ~ 3e-49: the medium computes its own invisibility. Three verification scripts accompany the paper.