For a century the Rydberg formula was taught as an empirical fit — a pattern Balmer spotted in 1885 and Bohr later justified with quantised orbits. Spatial Displacement Theory says it was never empirical. The entire emission spectrum — hydrogen, hydrogen-like ions, the first lines of every element — falls out of a single rotation field v(r) = cα√(a₀/r), with no quantum mechanics and no fitted numbers.
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SDT does not put the electron in an orbit. It puts it in a rotation field — a circulation of the spation medium around the proton whose speed falls off with radius like the wake of anything spinning in a fluid:
At the Bohr radius r = a₀ the field moves at exactly v = cα — the electron's ground-state speed is the fine-structure constant times light, full stop. There is nothing to quantise: the states are the standing radii where the circulation closes on itself, and at those radii the velocity steps down as v_n = αc/n. From a velocity, a displacement; from a displacement, an energy:
Feed that energy ladder into λ = hc/|ΔE| and the inverse wavelength collapses to the Rydberg form. But here is the move that matters: the constant out front is not a number anyone measured spectroscopically and copied in. It is built entirely from m_e, c, α, h:
R∞ derived ≡ NIST · 0.0 ppm The engine builds R∞ from first principles and compares it to the spectroscopic value. They agree to the last printed digit — and that agreement is itself a falsifier: change α or m_e and the whole spectrum walks off the sky.
This recomputes every classic series from R∞ = m_e·c·α²/(2h), rebuilt live out of the measured constants. Pick a series and a final level; the predicted wavelength and its deviation from the known NIST line appear in real time. Across all of Lyman, Balmer and Paschen the worst miss is 12.5 ppm — a higher-order remainder the leading-order law leaves on the table, not a failure of it. The next term in the SDT movement budget accounts for most of it (see the scoreboard).
What you should see: the famous H-α line at 656.47 nm (Balmer, n₂=3), Lyman-α at 121.568 nm (n₁=1, n₂=2) — each within ~12 ppm of NIST, from a constant that touched no spectrometer.
One honest refinement, also derived: the nucleus is not infinitely heavy, so the engine uses the reduced mass μ/m_e = m_p/(m_e+m_p) ≈ 0.999456 — built from the whitelisted m_e and m_p, no fit. Without it the law over-predicts by ~545 ppm; with it the worst H line lands at 12.5 ppm, the residual floor.
If the rotation field is real, it should not care which nucleus sits at the centre — only how much charge it carries. Strip an atom down to a single electron and the velocity law simply scales with the nuclear charge Z:
Helium-plus pulls its single electron at twice the speed, so its Lyman-α-equivalent line sits at one-quarter the wavelength — deep in the ultraviolet. The engine ran this for Z = 1…30 and checked against NIST He⁺, Li²⁺ and the full Z²-scaling table. The worst deviation across all thirty elements is 263 ppm — the residual climbing with Z exactly as the (Zα)² relativistic correction predicts.
Watch: the line marches into the extreme ultraviolet as 1/Z². At Z=2 (He⁺) it lands near 30.4 nm, matching NIST's 30.3785 nm to within a few hundred ppm — the gap is relativity, not error.
Underneath all of it sits the single conservation law of SDT — Law V, the movement budget. Anything that moves through space spends part of its lightspeed allowance on linear motion and the rest on circulation; the two always sum to c²:
Define the displacement z = v²/c² and the k-factor k = c/v. Then for every state, at every scale, the two multiply to one exactly:
The cleanest demonstration: take an element's first ionisation energy IE₁, turn it into the velocity it implies via v₁ = √(2·IE₁/m_e), and build z and k from that single number. The product is one — for hydrogen, for uranium, for everything in between.
Every element returns exactly 1. There is no tolerance being abused here — z·k² is algebraically identical to one, which is precisely why it can stitch the atomic, stellar and galactic regimes onto a single budget.
Here is what the engine printed, with nothing rounded away. Four phases, no fitted parameters, no quantum wavefunction anywhere in the chain:
| Phase | What it tests | Result | Worst error |
|---|---|---|---|
| R∞ | m_e c α²/(2h) ≡ NIST R∞ | PASS | 0.0 ppm |
| 1 | H spectra — Lyman/Balmer/Paschen | PASS | 12.5 ppm |
| 2 | H-like ions Z=2…30, λ∝1/Z² | PASS | 263 ppm |
| 3 | Neutral first resonance (21 elements) | 21/21 | < 5% |
| 4 | zk²=1 closure, all 118 elements | EXACT | 0 (identity) |
The Rydberg constant is built from {m_e, c, α, h} — all on the permitted-inputs whitelist — and the spectroscopic R∞ never entered the calculation. Its agreement is a correspondence, checked after the fact. Delete the comparison and the predicted wavelengths do not move. That is the signature of a real derivation, not a calibration.
The honest caveat sits in the ppm column. The leading-order law lands within ~12.5 ppm for hydrogen and ~263 ppm for the heaviest H-like ions, and the residual grows with Z² exactly as a relativistic correction must. SDT already carries that correction in the movement budget: the next term is the displacement z = (v/c)² = (Zα)² itself, and folding it in shrinks the residual on cue — the climb with Z is the budget's own relativistic term, nothing imported. The last sub-ppm sliver is a genuine open item for SDT to account for mechanically; it is logged as a remainder, never patched with an outside result.
Balmer's 1885 formula, Rydberg's constant, the whole grammar of atomic light — in SDT these are not empirical patterns awaiting a quantum justification. They are the bookkeeping of one rotation field v(r)=cα√(a₀/r) and one conservation law v²+v_circ²=c². The constant out front was forced by {m_e,c,α,h} before any spectrum was consulted.