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)
Melbourne, Australia · July 2026 · Domain 04 — Atomic Physics & Spectroscopy
Abstract. The atomic emission spectrum is derived from one rotation field of the substrate about the nucleus, with no wavefunction and no fitted parameter (APS01, RESOLVED). The circulation v(r) = cα√(a₀/r) admits closed-circulation radii as the states; the constant governing all series is forced by {me, c, α, h} as R∞ = mecα²/2h before any spectrum is consulted, identical to the measured value. Hydrogen lines verify to < 12.5 ppm and one-electron ions (Z ≤ 30) to < 263 ppm against the NIST tabulation; the closure z·k² = 1 holds exactly. Every emission satisfies the movement law λ = 2L·(c/v): a path length times a velocity gear at c, verified across seven independent floors with ratio 1.0000. The multi-electron drag factor D and the Lamb interval remain OPEN, the latter carrying a documented retraction.
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 whitelist 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 ϟH = α²a₀ = re — 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: retraction on the record
The 2S–2P interval is OPEN. A prior engine constant asserting a native value of 1051.8 MHz was found to be a bare literal with no derivation behind it — the evaluating solver had never compiled — and was deleted from the canon with authorization on 2026-07-03. The measured 1057.845 MHz stands in the engine labelled OBSERVED-INPUT only. The native candidate (the trefoil wake's C₃ structure and its r⁻³ component acting at the seat) is a mechanism candidate, not a number; its amplitude computation is PENDING a lattice wake solver. We consider the documented retraction itself a result: the ledger holds.
7. Certification
| Result | Value | Status |
|---|---|---|
| Rotation field v(r) = cα√(a₀/r) | eq. (1) | DERIVED — class B, passes delete-test |
| R∞ from whitelist | eq. (2), ~0 ppm | DERIVED — class B |
| H spectrum | < 12.5 ppm | VERIFIED vs NIST |
| H-like Z ≤ 30 | < 263 ppm | VERIFIED vs NIST |
| z·k² = 1, 126 levels | exact | IDENTITY counted on data — zero fits |
| λ = 2L(c/v), 7 floors | ratio 1.0000 | VERIFIED (APS07) |
| Drag factor D | 1.76 / 1.40 / 1.02 | class D — (8/3) weight underived, flagged |
| Lamb 2S–2P | 1057.845 MHz | OBSERVED input only; native amplitude OPEN; fabricated 1051.8 RETRACTED |
Methodological declaration. No wavefunction, no operator algebra, no quantised-action postulate, and no perturbative field expansion appears anywhere above. The NIST line tabulations enter as measured verification targets only. Both open items (the (8/3) slot weight; the Lamb amplitude) are flagged OPEN rather than fitted, and the one historical fabrication in this sector is disclosed as retracted, with the deletion logged in the canon.
References
- Engine/include/sdt/laws.hpp — namespace sdt::laws::atomic (rotation_field_velocity, R_inf_derived, emission_wavelength_m, outer_velocity_from_IE, drag_factor); law_VI::angular (retraction note in situ).
- Investigations/04_Atomic_Physics_and_Spectroscopy — APS01 (emissions, RESOLVED), APS02 (drag factor), APS04 (wake angular structure), APS07 (movement-law dossier; King-plot campaign).
- NIST Atomic Spectra Database (measured verification targets).
- CODATA 2018 (me, α, h, a₀, IE tabulations — measured inputs).
- Paper 06 of this series (the same koppa invariant at planetary scale).