Vacuum Impedance as the Organizing Principle of Atomic Spectra: A Universal Gap at IE/4 and Recovery of the Electron Mass

A direct empirical test of the hypothesis that the vacuum impedance Z₀ = √(μ₀/ε₀) organises atomic transition spectra at a universal boundary. The derivation proceeds without free parameters, fitted inputs, or ion-specific assumptions. When the orbital impedance of a bound electron, Zorb(n) = Z₀n/α, matches the vacuum impedance Z₀, the resulting field configuration is simultaneously required by the impedance-match condition and forbidden by the closure condition that defines a bound state. The consequence is a structural zero-observation interval in every ion's transition spectrum at the energy ΔE = IE/4, where IE is the first ionization energy and the nuclear charge Z cancels exactly. The boundary is a property of the electromagnetic vacuum, expressed in each atom's own energy units. A direct test against the NIST Atomic Spectra Database across 80 ions spanning Z = 1–82, all four subshell families (s, p, d, f), and multiple ionization stages confirms a zero-observation interval at IE/4 in 71 ions with zero falsifications. The remaining 9 ions have catalogs that do not bracket IE/4 from both sides. In 42 of the 71 confirmed ions the gap ranks in the top 10% by width among all consecutive-pair intervals, with a median rank at the 92.5th percentile. Gap width stratifies by subshell family in the predicted direction (d < f < s/p, Mann-Whitney p = 1.1×10⁻⁴ and p = 4.8×10⁻⁵). An independent precision analysis using all pairwise energy-level differences finds that five sub-0.1 meV ions (Fe II, Mo I, Mn II, Fe I, Cr II) return a corrected mean residual of −2.8 ± 2.7 ppm (SEM, n = 5), consistent with zero. The correction of +533.8 ppm is the proton reduced-mass shift, derivable from CODATA constants and not fitted to the data. Control evaluations at IE/3 and IE/5 return −250,133 ± 1.1 ppm and +250,129 ± 4.6 ppm respectively, in agreement with the algebraic predictions for wrong-fraction evaluations, confirming the uniqueness of IE/4. Because α = Z₀/2R_K, the gap centre simultaneously encodes the electron rest mass and Bohr radius through m_e c² = 8G/α² and a₀ = ℏcα/8G, where G is the geometric mean of the gap boundaries measured from NIST photon energies alone. This is the first determination of m_e c² and a₀ from the location of a structural absence in a transition spectrum rather than from the positions of spectral lines. The half-filled shell ions (3d⁵, 4d⁵, 4f⁷) achieve the highest precision and are independently confirmed as magnetically symmetric by EPR spectroscopy (g = 2.000, no orbital contribution, L = 0), establishing that the spectroscopic gap sharpness and the magnetic character are two measurements of the same electromagnetic boundary. Ho II returns zero hitpairs in the E-sector sweep despite abundant level data, the predicted outcome for an ion organised entirely by the inductive mode, demonstrating that the E and H modes of the vacuum are geometrically orthogonal. At the nuclear scale, all four doubly magic nuclei (¹⁶O, ⁴⁸Ca, ¹³²Sn, ²⁰⁸Pb) show complete depletion of observed levels below Sn/4; all 20 non-doubly-magic nuclides surveyed show no gap; the two populations are separated by Δκ = 0.100 with zero overlap. The gadolinium decoupling — sharp atomic IE/4 gap in Gd II, no nuclear Sn/4 gap in ¹⁵⁵⁻¹⁵⁸Gd, same element — confirms that the atomic and nuclear closure conditions are independent instantiations of the same vacuum boundary. Related works:- Electromagnetic Closure and the Fine-Structure Constant: A Geometric Derivation (Heaton, 2026): https://doi.org/10.5281/zenodo.19157339- Determinacy Under Quotient Representations (Heaton, 2026): https://doi.org/10.5281/zenodo.18868210- Hydrogenic Alignment as a Coordinate Principle for Atomic Spectra (Heaton, 2026): https://doi.org/10.5281/zenodo.18167643- Recursive Geometry, Coordinate Compression and Invariant Structure in Electromagnetism (forthcoming): https://doi.org/10.5281/zenodo.19194036