S4 — The null collider, Earth arm: an audit, a cascade, and where folds live

Question. Can we catch two magnetic nulls annihilating at Earth — drive a model-magnetosphere pair through its fold with a physical dial and verify every signature the fold normal form predicts?

Answer (short). No — and the two reasons are findings. (1) The T96 empirical magnetosphere is null-free inside its own magnetopause; its advertised null population is a boundary-shell artifact. (2) The classical Dungey (IGRF + uniform IMF) field owns real nulls, but its topology change is a broken ring degeneracy, not clean isolated folds. The certified fold of the program lives on the Sun (see S5-solar-fold-hunt.md); Earth’s arm contributes the stability phase portrait and the walls.

Pre-registered hypotheses (as run, three designs)

• H-S4a (existence). A 1-parameter storm path of IGRF+T96 exhibits at least one interior pair creation/annihilation.
• H-S4b (fold signature). Opposite degrees; separation $C\sqrt{|\lambda-\lambda_c|}$; $\det\nabla B \to 0$.
• H-S4c (SR read). The measured 2-jet at the collision carries the S1 crossover structure (pre-registered as plateau-4/$Q=7$; refined by the solar catch — see S5c).

What actually happened (chronology kept, per program rules)

1. v1/v2 (Dst sweep, −5 → −150 nT and validity-capped −5 → −100 nT). The census EXPLODES with storm depth (143 → 1830 nulls over the full path; artifacts/s4_sweep_extended.json). All 47 tracked “creations” failed certification identically: separation ladders CONSTANT in Dst (slope ≈ 0.000 vs the fold’s 0.5) — they were finder-discovery artifacts, coexisting neighbours found late by the seed battery. The certificate did its job on its own pipeline.
2. Boundary audit (run_s4b_drive.py). Probing why driven pairs were rigid: B at the null positions is IDENTICAL for Dst −5/−50/−100 — the nulls do not feel the ring current. Testing every interior null against T96’s own magnetopause (geopack.t96_mgnp): all 70 interior census nulls sit < 1 R_E OUTSIDE the model boundary (dist 0.7–0.8, id = −1), and inside the valid domain the hunt finds zero nulls at every IMF Bz in [−9, +3] nT. The smooth average magnetosphere is null-free; the “149 nulls, 79 spiral” census of P4-S3 is structure of the model’s continuation past its own edge. (P4-S3’s blog section now carries this correction; the spiral/radial anatomy remains valid as anatomy of a realistically shaped non-force-free field.)
3. Dungey collider (run_s4c_dungey.py). IGRF(2012-03-07) + 5 nT uniform IMF, dial = IMF angle θ from northward. Census: the classical cusp-pair topology (2 nulls, degree sum 0, r ≈ 18–23 R_E) is structurally stable over 125° of IMF rotation, then cascades 2 → 3 → 4 → 6 → 8 → 12 → 17 approaching anti-parallel: the pure-dipole null RING (the μ<0 face of the S1 fold family) broken up by the real multipoles. Up close the cascade nulls live in a 0.02-nT-flat azimuthal valley (Jacobian eigenvalues ~ (0.83, −0.82, −0.009), condition ~ 10²): null identity dissolves; walked “deaths” are continuation-conditioning events, not certifiable isolated folds. Several backward-walked newborns died with det̂ → 0.01 (genuinely approaching degeneracy), but no partner could be resolved against the valley conditioning; the pre-registered certificate was therefore NOT issued at Earth.

Verdicts

• H-S4a: REFUTED as posed for T96 (no interior nulls to collide inside validity); for the Dungey family the count changes but through ring-cascade degeneracy, not certifiable isolated folds.
• H-S4b/c: not issued at Earth; issued on the Sun (S5c) where the four-signature certificate passed at machine precision.
• Byproducts kept: the stability phase portrait (det̂ vs disĉ; 135 Dungey states + the solar pair’s dive into the fold wall) and the cascade census — figure forbidden-directions-stability-walls.png.

Honesty box

The Dungey field is the classical vacuum superposition (no magnetopause currents); its neutral points at 5 nT sit at r ≈ 18–23 R_E, beyond the real magnetopause — it is the textbook outer-field model, not a calibrated magnetosphere. The T96 audit statement is about the model’s own boundary function; real magnetospheric nulls are dynamical current-sheet structures (Cluster/MMS territory), absent from smooth empirical averages by construction.

Reproduce

cd research/preferred-directions
../cosmic-web/.venv/bin/python scripts/run_s4_collider.py   # Dst sweep (context)
../cosmic-web/.venv/bin/python scripts/run_s4b_drive.py     # boundary audit
../cosmic-web/.venv/bin/python scripts/run_s4c_dungey.py    # Dungey cascade
../cosmic-web/.venv/bin/python scripts/render_s45_figures.py

Artifacts: s4_collider.json (Dungey census + portraits + T96 audit), s4_sweep_extended.json (full Dst-path census, exploratory beyond −100 nT).