For seventy years, Bohmian trajectories have served as an interpretation of quantum mechanics, rarely as a computational medium — and the spin-dependent guidance law, in which the Pauli current carries an additional circulation about the spin direction that removes the resting ground-state electron of the 1952 theory, has so far been used to draw trajectories from previously solved eigenstates, not as a computational medium in its own right. This work shows that it is one. Structure comes from the density family and circulation from the law; the laboratory test reported here is a test of the family. What is propagated is that law applied to per-electron beable spin vectors: for one electron in a spin eigenstate this is the Pauli current with its spin term exactly, so hydrogen calibrates it; for a closed-shell singlet the Pauli current with its spin term vanishes identically, so what is used there is a declared extension and is marked as one. A structure-preserving integrator is derived from the two exact invariants of the law; it is reversible to about 10⁻¹¹ Bohr over a thousand steps forward and back — the step alone; with the level projector, whose projection cannot be undone, about 10⁻⁸ — and of second order along the trajectory (convergence factors 4.00, 4.14, 4.09), and a level projector — now a safeguard rather than a repair — holds the density-level theorem with a logged correction that falls at the order of the method. A spin relaxation term derived from the density family itself makes the triplet fixed point linearly unstable, which, with the beable spin sector identified with the spectroscopic one, the observed metastability of ortho-helium 2³S excludes; the term is withdrawn.
On a variationally determined density family for hydrogen, helium, and lithium — whose inputs are the Coulomb interaction, spin, and the asymptotic decay rates fixed by the ionization energies, and in which no constant is fitted to data, though which channels the helium family carries was prompted by an earlier comparison (Section 4): every constant either source-pinned, internally stationary, or declared as structure — ensembles of spin trajectories are propagated in configuration space. Hydrogen calibrates the momentum pipeline to machine precision. For helium, the family has the dip below the Hartree–Fock level at small momentum and lies 4.9 to 5.2 per cent above a full configuration interaction (aug-cc-pV6Z) at the lowest measured momentum, in either reading of the data convention — a comparison between two calculations that depends on no measurement, stated without the uncertainty of the family's stationary constants, which is not computed. Against two (e,2e) measurements of the ionization ratio into excited ion states, family and full configuration interaction move together: the 1984 work, whose ratios are formed at equal momentum, quotes a lowest value that the family exceeds by 28 and the full configuration interaction by 23 per cent; for the 1976 data the source does not say whether the ratio was formed at equal momentum or at equal angle, and depending on that the remaining tension sits at the lowest point or in the rise, for both calculations alike.
A 1024-trajectory lithium ensemble keeps the 1s² 2s shell structure of the underlying density intact, held by two exact invariants of the motion rather than produced by it; the dynamics preserves the statics, the medians of the core radii showing no trend over 600 atomic time units (linear trends of 0.3 and 0.1 per cent, within about one standard error); and its core pair circulates in opposite senses — in all 256 walkers of a run recorded at every step — one statement in this work that the stationary density itself cannot make, since a stationary density knows no direction of rotation. The momentum profile of the lithium 2s carries a node minimum at p_node = 0.8940 ± 0.003 a.u.: a registered value, kept in four documented generations, whose uncertainty is statistical at fixed structure; structural revisions moved it by up to 0.04 a.u. No momentum profile of atomic lithium has been measured — (e,2e) experiments on the free atom exist only at low impact energies, where the cross section does not map the bound momentum distribution — and the value does not separate the family from our own Hartree–Fock and s-limit CI values. A documented correction protocol accompanies all claims.
This is the manuscript as prepared for journal submission. Version 0.3 corrects the attribution of the guidance law: it is the spin term of the Pauli current as the non-relativistic limit of the Dirac current (Bohm and Hiley 1993), not the law of Bohm, Schiller and Tiomno (1955), whose velocity is the ordinary convective current (Correction Protocol, entry 24); no computed value changes. It also repairs a sign error in a kernel branch no reported run uses, with every reported run recomputed bit-identically (entry 25), and states for every ± what it is (entry 26). Versions 0.1 and 0.2 stay unchanged under their own DOIs, 10.5281/zenodo.23001950 and 10.5281/zenodo.23146678; the registered value v4 = 0.8940 ± 0.003 a.u. is unchanged and dates from version 0.2. Main text and supplement belong together; cross-pointers in the main text refer to the supplement. No third-party figures are reproduced; our readings of the (e,2e) measurements used for comparison are listed in Table S2 of the supplement and included in the reproducibility package. Everything in the package is in English: folder and file names, names in the code, comments, messages and data headers.