Phase 4 — Degenerate nulls, bifurcations, and new worlds
Charter. P3 grounded the SR null detector on the generic nulls of the real Sun and delivered the honest division of labour: detection/order/gradient are sub-Riemannian, type classification belongs to the linear fit. This phase attacks the two places that division leaves the SR side genuinely ahead, and expands the data beyond the Sun.
- Degenerate nulls and their bifurcations. Nulls are created and destroyed in
pairs (opposite topological sign) through a fold (saddle–node) bifurcation: two
generic nulls approach, merge into a degenerate null where the field vanishes to
second order, and annihilate (Priest & Titov-style topology; null creation/
annihilation is observed in data-driven coronal simulations). The degenerate
configuration is structurally unstable — which is exactly why it matters: it is the
transition state of magnetic topology change, i.e. of reconnection events.
- The eigenvalue method sees a degenerate null only as
det ∇B → 0— a fragile numerical zero test with no scale information. - The growth vector has a law for it: field vanishing to order $k$ gives
$Q = 3 + (k+2)$. A fold point ($k=2$) must read $\mathbf{Q = 7}$ — a new, untested
value of the law — and a split pair must read as a scale crossover: probe
radii below the pair half-separation see the local structure (uniform: flux weight
2; at a member null: 3), radii above it see the parent degenerate structure (weight
4). The SR detector, uniquely, reads the separation of an unresolved pair off
one scaling curve. Nothing pointwise can do that. (Post-hoc, P5-T1: a polynomial
fit + root-finding CAN, and does it better — the crossover was later demoted to a
conceptual observable by the pre-registered race; see
T1-race.md.)
- The eigenvalue method sees a degenerate null only as
- New worlds. The R5 theorem says spiral nulls need genuinely non-force-free currents — which solar extrapolations cannot supply but planetary magnetospheres can (magnetopause and tail current systems are real and strong). Empirical magnetosphere models (IGRF internal + Tsyganenko external, fitted to decades of spacecraft data) are the natural next real dataset: cusp and tail nulls, with spiral types finally possible. Vacuum multipole fields of the non-dipolar planets (Uranus/Neptune, real mission-fitted Gauss coefficients) extend the gallery; force-free analytic models of extreme objects (twisted magnetar dipole) are included only as clearly-labeled analytic demonstrations.
Pre-registered predictions
| # | Prediction | Judged by |
|---|---|---|
| H-P4a | At the fold’s degenerate null, $Q = 7$ (first $k=2$ point of the law in 3D); at each split null $Q = 6$ with opposite signs | measured weights $(1,1,1,4)$ vs $(1,1,1,3)$ |
| H-P4b | The local flux-reach exponent $w_4(r)$ at the pair midpoint crosses over $2 \to 4$ around $r \sim \sqrt{\mu}$ (half-separation), and the curves for different $\mu$ collapse when plotted against $r/\sqrt\mu$ (the dilation symmetry) | exponent-vs-scale curves, collapse quality |
| H-P4c | Real solar extrapolations contain close null pairs, and the real pair shows the same crossover | pair census + one real crossover curve |
| H-P4d | Empirical magnetosphere fields carry detectable nulls; where real (non-force-free) currents flow, spiral nulls exist — completing the R5 story on real-physics data | null census + Parnell types + SR $Q$ |
Failure modes pre-registered: (a) the crossover may be smeared over a decade in $r$ (exponents are asymptotic, not local) — report the width honestly; (b) the degenerate point’s basin is tiny, so the $Q=7$ measurement must verify insensitivity to the probe radius window; (c) magnetosphere models are models fitted to data, not field measurements — say so; (d) close pairs on the Sun may all be spurious (extrapolation ringing) — check pair persistence across window/resolution choices.
The family (S1)
\[\mathbf B_\mu = \big(xz,\; yz,\; \mu - z^2 + \tfrac{x^2+y^2}{2}\big),\qquad \nabla\cdot\mathbf B_\mu = z + z - 2z = 0,\]with exact vector potential (verified in code) $A = \big(\tfrac{yz^2}{2},\; -\tfrac{xz^2}{2} + \mu x + \tfrac{x^3}{6} + \tfrac{xy^2}{2},\; 0\big)$.
- $\mu > 0$: two point nulls at $(0,0,\pm\sqrt\mu)$, Jacobians $\mathrm{diag}(\pm\sqrt\mu, \pm\sqrt\mu, \mp2\sqrt\mu)$ — a radial pair of opposite sign, exactly the pair-creation topology of the literature.
- $\mu = 0$: one isolated null at the origin with $\nabla\mathbf B = 0$ and $|\mathbf B| \sim r^2$ — the fold point, $k = 2$.
- $\mu < 0$: the pair is gone; a null ring of radius $\sqrt{2|\mu|}$ appears in the $z=0$ plane — the divergence-free fold’s third face, noted but not this phase’s target.
Experiments
- S1 (heavy): $Q$ at the fold point and at split nulls; $w_4(r)$ crossover curves at
midpoint and at a member null across a $\mu$ grid; dilation collapse.
src/bifurcation.py,scripts/run_p4_fold.py. - S2: census of null pairs in the real R3 extrapolations (more windows, all three days); crossover measurement on the closest real pair.
- S3: Earth magnetosphere via IGRF+Tsyganenko (geopack) — null census, types, $Q$; spiral hunt. Vacuum multipoles of Uranus/Neptune from published coefficients. Analytic twisted-dipole demo (labeled analytic).
Reports land in docs/S1..S3-*.md; figures under public/img/posts/; write-up follows
the house series conventions once the results exist.