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J1 — Jupiter: the polar patch as a combination of separatrices

J1 — Jupiter: the polar patch as a combination of separatrices

Question (user-posed). Jupiter’s Juno-era field model shows a patch of reversed flux near the north pole. Is the patch a combination of separatrices — the solar parasitic-polarity skeleton (null + fan dome + spine) on a giant planet?

Answer. In the $l \le 18$ JRM33 potential continuation (the model carries coefficients to degree 30; 18 is the useful-information cut): yes — and the whole structure is buried in the molecular envelope, beneath the 1-bar surface. Robustness: the structure rides on the degree-14–18 coefficients (the least-determined tail); it is absent at degree-13 truncation and in JRM09. On the record as a falsifiable prediction for the next Juno model release.

Pre-registered hypotheses and verdicts

Machinery

Non-circular w4 reading (J1d, referee follow-up)

The referee’s M1 (tangent-cone Q is a consistency check, not detection) is answered for this world: jupfield gained the EXACT toroidal vector potential (A = -(1/l) r^{-(l+1)} rhat x grad_s S_lm per harmonic; golden-checked curl A = B), and w4(r) was measured on the FULL analytic field with no Jacobian input (run_j1d_raw_w4.py, artifacts/j1d_raw_w4.json): w4 = 2.8-3.1 across r = 0.003-0.06 R_J at both nulls, 1.9-2.1 at a generic control — the k = 1 flux weight read raw on the fourth world.

Two-resolution cell-prefiltered root census + coefficient ensemble (J2/J2b)

run_j2_census.py (artifacts/j2_census.json) + run_j2b_sensitivity.py (artifacts/j2b_sensitivity.json). Census: Haynes–Parnell-style corner-sign prefilter over every spherical cell of the shell, Newton from candidate-cell centres, two grid resolutions (24×72×144 and 36×108×216) — both converge on three nulls: the two known ones plus r = 0.857, lat +27.2°, lon 67.0°. NOT a completeness theorem (the prefilter is necessary-only; centre-started Newton can fail or leave its cell) — the CONVERGENCE evidence is: two-resolution agreement, floor-move invariance (floor lowered to 0.80 returns exactly the same three nulls in 0.855–1.0, while exposing a root-dense, truncation-sensitive and poorly constrained continuation layer below 0.85: 39 further roots in 0.80–0.855; noise vs continuation amplification vs unresolved real structure not diagnosed, no completeness claimed there), and a net-charge degree cross-check (deg B/|B| numerically consistent with zero — −0.000/−0.000/−0.005 — over spheres at r = 0.885/0.94/0.999 ⇒ no unexplained NET topological charge above 0.885; index-cancelling missed pairs NOT excluded — fold pairs are exactly such). The Haynes–Parnell trilinear cell census remains the stronger comparator (chartered in G1, not implemented). Anatomy (J2b): all three radial — polar (deg +1), low-lat (deg +1), census find (deg −1); 0 spirals out of 3, as the curl-free envelope demands. Vectorised evaluator golden-checked at 2.8e-16.

Stress ensemble (J2b, all three nulls, PAIRED design): JRM33 covariances are not distributed, so per-degree Gaussian perturbations are scaled by the JRM33−JRM09 per-degree difference (1.3% at l=2, 23% at l=10, log-linear extrapolation → ~100% at l≥14; an honest stress proxy, NOT a posterior — outputs are survival fractions under the chosen ensemble and sample position ranges, not probabilities; JRM33−JRM09 differences mix model evolution, secular variation and uncertainty). N = 40, capture radius 0.15, floor 0.856: polar patch survives 40/40 (at thresholds 0.25/0.5/1.0 G alike); polar null 32/40 with sample position range over those 32 survivors r 0.86–0.93, lat 64–73°, lon 265–301° (raw recaptures before the floor cut: 36/40, down to r = 0.75 — acceptance sets reported separately); the two low-lat nulls are less persistent (15/40, 19/40). Paired amplitude scaling (same 40 standardized draws ×0.5/×1/×2): polar 35/32/29 of 40, low-lat 17/15/14 and 28/19/15. Capture radius 0.10/0.25 → 29/40 / 32/40; floor 0.80 → 35/40 (members sink below the floor rather than vanish). Possible upgrade recorded: fold in the published JRM33 resolution-matrix diagonals as coefficient-specific constraint weights.

Bald-patch classification (J1c, referee follow-up)

A PIL can be a separatrix without a null (bald patches, (B·grad)Br > 0). Measured point-by-point (run_j1c_baldpatch.py, artifacts/j1c_baldpatch.json): polar patch at l18: BP fraction 0.000 (the dome carries ALL the separatrix character — the pure case); low-latitude patch l18: 0.234 (mixed); l13 surviving polar patch (null-free): 0.104 — mostly plain arcade, so the “combination of separatrices” answer genuinely degrades under truncation rather than surviving via another mechanism.

Honesty box

Potential-field treatment of the molecular envelope (standard, but neglects any currents in weakly conducting H2); downward continuation to r = 0.85 amplifies degree-l power by (1/0.85)^(l+2) (~13× at l = 14), which is both why the patch appears at depth and why the robustness caveat is load-bearing; magnetodisc/magnetopause fields are negligible below r = 1 (tens of nT vs 1e4–1e6 nT internal). Longitudes are System III as used by JRM33.

Reproduce

cd research/preferred-directions
../cosmic-web/.venv/bin/pip install planetmagfields
../cosmic-web/.venv/bin/python src/jupfield.py
../cosmic-web/.venv/bin/python scripts/run_j1_jupiter.py
../cosmic-web/.venv/bin/python scripts/run_j1b_dome.py
../cosmic-web/.venv/bin/python scripts/render_j1_figures.py

Artifacts: j1_jupiter.json, j1b_dome.json, j1_maps.npz, j1c_baldpatch.json, j1d_raw_w4.json, j2_census.json. Blog: Part 8 (_posts/2026-09-02-forbidden-directions-jupiter.md).