Hydrogen from One Cost: The Lattice Coulomb Field and the Bohr Spectrum

Is hydrogen fully derived? Charge the recognition cost J(x) = ½(x + 1/x) − 1 on each link's flux f of the cubic lattice ℤ³, so that a link costs cosh f − 1; the alternative charges it on log-ratios of levels. A point charge c has least-cost flux arsinh ∇μ, whose potential obeys Σ arsinh(μ(n) − μ(n′)) = c δ(n, 0), the sum running over the neighbours n′ of n. At every finite charge μ(n)|n| → c/(4π), with an inverse-square Coulomb field; a class of odd flux laws containing the alternative gives the same far field. With the cost on the flux, charges Q₁ and Q₂ carry the energy Q₁Q₂/(4πr)(1 + o(1)) as r → ∞, without a weak-field limit. This law, or on either law the one-charge field with a test-charge reading, gives the atom's potential −g/r. For −κ∇² − g/r on square-integrable states, bounded at the nucleus and twice continuously differentiable elsewhere, the negative energies are exactly −g²/(4κn²), each with n² independent states and no other bound states there; no square-integrable state has positive energy. Hydrogen is not fully derived: its spectrum's form, with frequency ratios such as Lyman-α : Balmer-α = 27 : 5, rests on J, named readings and one open continuum step, and its scale needs the fine-structure constant and the electron mass. Spin, fine and hyperfine structure, the Lamb shift, nuclear recoil, the zero-energy state and scattering states lie outside it.