Spatial Displacement Theory · Paper 04 · Prepared for Submission

Atomic Emission Without a Wavefunction: The Rotation Field v(r) = cα√(a₀/r), the Closure z·k² = 1, and the Movement Law λ = 2L(c/v)

J. C. Harvey

Melbourne, Australia · July 2026 · Domain 04 — Atomic Physics & Spectroscopy

Keywords: rotation field · closed circulation · Rydberg closure · z·k² = 1 · movement law λ = 2L(c/v) · drag factor

1. The rotation field and its closed states (APS01)

The substrate circulates about the nucleus; the electron — a hard localised vortex, never a cloud — occupies radii where the circulation closes on itself. The n-ladder is the set of closed-circulation radii; no quantised-action postulate is invoked. The apparent "cloud" of the textbook picture is a strobe artefact of a hard point sampled by emission.

2. The forced constant

Equation (2) is fixed by the derivation basis before any spectrum is consulted. The reduced-mass factor in (3) is classical two-body kinematics, not an imported field correction. Verification: hydrogen < 12.5 ppm; one-electron ions through Z = 30, < 263 ppm (NIST). The residual at few ppm is the next movement-budget term z = (Zα)² and is logged as a remainder, not fitted away.

3. The closure z·k² = 1 across the periodic table

Measured first-ionisation energies from hydrogen through lead — 126 levels — satisfy the closure identically. Nothing is fitted; the identity is counted on data. The hydrogen seat sits at k = 1/α = 137.036: the 137 rung is a k-value of the hierarchy, not a mystery number. The hydrogen c-boundary is ϟ_H = α²a₀ = r_e — the same koppa object that carries planetary fields in Paper 06, twenty orders of magnitude up.

4. The movement law of emission

Every emission wavelength is a geometric path 2L multiplied by the velocity gear c/v of the emitting seat. Across seven independently anchored floors the ratio of measured to constructed wavelength is 1.0000 (APS07 dossier). The grammar of the emitting atom sets the valence floor; core-breach transitions scale the path by small integers (×3–×9), and the pairing discipline of a shell is read as triton-integrity in the nuclear ledger — one bookkeeping across atomic and nuclear floors. King-plot non-linearities are, in this reading, the mass-defect ledger's shape imprinted on isotope shifts.

5. Multi-electron structure: the drag factor (class D)

D tracks the outer-shell electron count: lone s¹ opener D ≈ 1.76, paired s² D ≈ 1.40, full p⁶ D ≈ 1.02 — co-shell wakes draft constructively in the shared seat, as riders in a pace-line (APS02). Certification: class D, a computed/observed correlation; the (8/3) slot weight is not yet derived from geometry and is flagged, not smoothed over.

6. The Lamb interval: two observed centers

The hydrogen 2S–2P interval is accepted as measured: Lamb and Retherford’s microwave quench of a metastable 2S beam, 1057.845 MHz, 2S above 2P_1/2. In this framework that datum is a driven change of occupancy form (pair-form to plane-form) on the proton center, not a vacuum correction and not a number extracted from the trefoil. A prior engine constant asserting 1051.8 MHz was a bare literal with no derivation — the evaluating solver had never compiled — and was deleted from the canon with authorization on 2026-07-03. That retraction stands.

Helium-plus at n=2 is a second measured interval, 14041.13 MHz, on a different nuclear center: the α (two protons). Nuclear construction names that object. It does not compute 14041 from 1057. Kepler n⁻³ occupancy of the hydrogen sticker is a test of that sticker, not a derivation of Lamb. The APS04 nuclear-geometry addend remains +0.761 MHz — a small nuclear-structure term, not the bulk interval.

7. Certification

References