S3 — new worlds: the Earth’s magnetosphere, and the first real spiral nulls

SUPERSEDED (S4b boundary audit, Part 6). Every null in the census below sits outside the T96 model’s own magnetopause (geopack t96_mgnp id = −1, typically 0.7–0.8 $R_E$ beyond it), where the empirical model is extrapolated, not fitted; the valid interior is null-free across the tested driving range. The headline “the spiral class exists in real physics” is retracted — what stands is the force-free theorem, the methodology, and the lesson that census claims need a model-domain audit. Per the independent review, any future magnetospheric null claim must be tested against event-conditioned in-situ data (Cluster/MMS) or a validated reconstruction inside the model domain — that gate is recorded here; no such claim is currently made. This report is preserved unedited below as the original record.

Reproduce: ../cosmic-web/.venv/bin/python scripts/run_p4_magnetosphere.py (~3 min; needs geopack, pip). Results → artifacts/p4_magnetosphere.json, figure → public/img/posts/forbidden-directions-magnetosphere.png.

Why the magnetosphere

The R5 theorem closed a door: any force-free field ($\mathbf J = \alpha\mathbf B$, bounded $\alpha$) has only radial nulls, so solar extrapolations can never show the spiral class on real data. The Earth’s magnetosphere is the natural place the door is open: the magnetopause, tail, and ring current systems are genuinely non-force-free, and the empirical field models that encode them (internal IGRF + external Tsyganenko T96) are fitted to decades of real spacecraft measurements. Epoch used: 2012-03-07 00:00 UT — the program’s solar storm day — under moderately disturbed conditions (Pdyn = 3 nPa, Dst = −50 nT, IMF Bz = −5 nT).

Result (H-P4d): 149 nulls, 79 spiral — the spiral class exists in real physics

Newton descent from cusp/tail seeds finds 149 nulls (deduped, $|B| < 0.5$ nT, outside $2.5\,R_E$). Standard classification: 79 spiral, 70 radial — the first spiral nulls of the entire program on a field with observational pedigree, exactly where the theorem said they must live: in the non-force-free current regions. The SR growth-vector detector returns $\mathbf{Q = 6}$ at every null (tangent-cone measurement), radial and spiral alike — the fourth world (after the synthetic battery, the ABC-like dynamo field, and the solar corona) in which the detector is grounded.

Honest caveats

1. These are nulls of an empirical model, not directly measured zeros. T96 is fitted to data, but a null’s existence is a delicate topological property; the population we find sits off the noon–midnight plane at $|y| \approx 17!-!25\,R_E$, $x \approx -12\ldots-32\,R_E$ (nightside flank / lobe boundary), where the model’s current closure is its least-constrained part. In-situ confirmation is the business of multi-spacecraft missions (Cluster/MMS null catalogues) — matching against those is the natural next step.
2. No dayside cusp nulls converged in this model/state: the classical Chapman–Ferraro neutral points belong to closed-magnetosphere idealisations; T96’s soft magnetopause under southward IMF does not produce exact dayside zeros along the seeded meridian.
3. The census depends on storm state (parmod); this is one epoch, not a climatology.

The rest of the “new worlds” ledger (scoped honestly)

• Planetary internal fields (Uranus/Neptune vacuum multipoles from mission-fitted Gauss coefficients): curl-free ⇒ the R5 theorem again forbids spirals; their exterior nulls (if any within validity) would extend the radial gallery — left as the program’s next cheap data extension.
• Magnetars / black holes: no observational field maps exist — magnetar fields are inferred scalars, and black-hole magnetospheres are GR objects whose sub-Riemannian treatment needs the flux lift rebuilt on curved spacetime. Recorded as a genuine frontier, not attempted with toy stand-ins here.

Correction (2026-07-12, from the S4 boundary audit)

Testing every census null against T96’s own magnetopause (t96_mgnp): the entire null population sits in a shell < 1 R_E OUTSIDE the model boundary, and inside its valid domain the model is null-free at every IMF Bz probed ([-9, +3] nT). The shell nulls barely respond to Dst (|dB| <= 0.03 nT per 30 nT) because the ring/tail currents are interior. The census stands as anatomy of null structure in a realistically shaped non-force-free field — NOT as standing nulls of the valid model interior. Blog Part 6 and S4-null-collider.md carry the full statement.