S1 — the magnetic fold: Q = 7, and a null pair read as one scale crossover
CORRECTED (S5c certified collision, Part 6; article Cor. 5.1). This family’s degenerate point has $\nabla\mathbf B \equiv 0$ — the fully symmetric normal form. A generic rank-2 fold keeps a nonzero 1-jet and reads $Q = 6$, not 7; $Q = 7$ below is correct for this constructed symmetric family only. The scale-crossover tool was also later demoted by the pre-registered race (T1). Preserved as the original record.
Reproduce: ../cosmic-web/.venv/bin/python scripts/run_p4_fold.py (~6 min).
Results → artifacts/p4_fold.json, figure →
public/img/posts/forbidden-directions-fold-crossover.png.
Family and predictions: PROGRAM-P4-bifurcations-new-worlds.md.
H-P4a — the law’s first k = 2 point in 3D: confirmed
| Point | measured weights | Q | prediction |
|---|---|---|---|
| fold (degenerate) null, $\mu=0$ | $(1, 1, 0.99, 4.00)$ | 7 | $(1,1,1,4)$, $Q=7$ |
| split null, $\mu=0.09$ | $(1, 1, 0.99, 2.98)$ | 6 | $(1,1,1,3)$, $Q=6$ |
The split pair classifies radial− / radial+ — opposite signs, the canonical pair-creation topology. $Q = d + k + 2$ now stands measured at $k = 0, 1, 2$ in 3D.
H-P4b — the crossover and its dilation collapse: confirmed
The scale-resolved flux exponent $w_4(r)$ at the pair midpoint:
- $\mu = 0$ control: flat at $4$ over two decades (no spurious scale).
- Every $\mu > 0$: clean plateau at $2$ (the field is nonzero at the midpoint — locally uniform), then a knee to $4$ (the probe swallows the pair and sees the degenerate parent). The knee marches with $\sqrt\mu$.
- Collapse: plotted against $r/\sqrt\mu$, all four curves lie on one universal crossover (knee at $r/\sqrt\mu \approx 1!-!3$) — the dilation symmetry, verified.
- Centred on a member null: plateau at $3$ (single generic null), knee to $4$ when the partner enters the probe.
What this means, plainly. A pointwise method needs to resolve two nulls to know there are two; below its resolution a merging pair is indistinguishable from one null. The growth-vector read-out gets the same information from one point: the height of the plateau says what you are sitting on (uniform 2 / single null 3 / degenerate 4), and the knee location reads the pair separation — including for pairs it never resolves individually. This is the first observable in the program that the standard toolkit has no analogue of, and it is precisely the transition state of magnetic reconnection (null creation/annihilation) that it measures.
Post-hoc note (T1): “no analogue” does not mean “no competitor”. The pre-registered
race of P5 pitted this knee estimator against a divergence-free quadratic fit +
root-finding on shared noisy grids; the fit won every cell and the crossover is
demoted to a conceptual observable — see T1-race.md. The plateau
dictionary and $Q=7$ at the fold stand; the separation estimator does not.
The gauge lesson (caught and fixed)
The first run measured flux weight 2 at the split null — a gauge artefact: the
vector potential’s symmetric gradient at the probe point (curl-free junk,
$\partial_1A_2+\partial_2A_1 = \mu \neq 0$ there) contributes an $r^2$ term that masks
the physical $r^3$. Since that term is the gradient of
$\chi = A_0!\cdot!d + \tfrac12 d!\cdot!S\,d$ (an exact form), it is a pure endpoint
correction; mfield3d.weights_Q now subtracts it in general (gauge_terms). All
earlier results are unchanged (their potentials had $S = 0$ at the probe points —
re-verified: uniform 5, linear null 6, spiral 6); the fold’s split null is the first
configuration that exposed the trap the program’s own “gauge-adapt before measuring”
rule warned about.
Open
- The $\mu < 0$ face (the null ring) has its own SR signature — unmeasured.
- The eigenvalue-collision degeneracy (radial↔spiral boundary, improper node) keeps $k = 1$, so $Q$ stays 6 — the growth vector is blind to that transition (honest).
- S2 asks whether real solar extrapolations contain close pairs showing this crossover.