Phase 3 — Grounding the sub-Riemannian null detector in real data

Program charter. Phases 1–2 established that a magnetic environment carries genuine sub-Riemannian (SR) geometry, that the growth vector jumps 5 → 6 at a magnetic null (a detector), and that on one real SDO/HMI coronal field the detector fires. That is one data point and a detector that only reports presence + order. This phase asks the two questions that decide whether the SR framing is a tool or merely a language:

Grounding. Does the detector reproduce the standard coronal-null catalogue across many fields — synthetic-with-ground-truth and real? (a gallery, not an anecdote.)
Classification. Can SR invariants classify a null (radial vs spiral, sign) — and is that classification more robust to noise than the standard eigenvalue scheme? This is the only way the framing adds something, because the standard classifier is the linearization and the SR tangent cone is the linearization.

1. Literature — the standard scheme we must match or beat

A 3D magnetic null is a point where B = 0. Its structure is read from the Jacobian M = ∇B in the first-order Taylor expansion B ≈ M·r, with tr M = ∇·B = 0.

Classification by eigenvalues (Parnell, Smith, Neukirch & Priest 1996). The eigenvalues (λ₁, λ₂, λ₃), Σλ = 0, split into a spine (the odd-sign-out eigenvector) and a fan (the plane of the two like-sign eigenvectors). If all three eigenvalues are real the null is radial (an X-type in the fan 2D projection); if two are a complex-conjugate pair it is a spiral null (an O-type). The sign (positive / negative, historically A / B) is the sign of the spine eigenvalue: field lines run in along the spine and out along the fan, or vice-versa.
Radial → spiral is a current threshold. Writing M = S + J (symmetric + antisymmetric), the antisymmetric part is the current ∇×B. The component of current parallel to the spine rotates the fan; when it exceeds a threshold J_thr the two fan eigenvalues collide and go complex — the null turns from radial to spiral. This gives a continuous, controllable ground-truth knob.
Detection. On a Cartesian grid the field standards are the Poincaré-index method and the trilinear method (Haynes & Parnell 2007), with FOTE (first-order Taylor expansion) a fast alternative; a 2020 A&A inter-comparison finds FOTE and trilinear the most reliable on gridded fields. These are applied to potential / NLFFF extrapolations of SDO/HMI magnetograms; real catalogues exist (e.g. nulls associated with X-class flares in cycle 24).

Reading, and the caveats it hands us. The entire standard classification is a function of M — the linearization. The SR tangent cone (nilpotent approximation) at the null is also built from M. So an SR classifier that reads the tangent cone cannot, in principle, know more than the eigenvalues do at zero noise. The only room for a genuine contribution is conditioning: the standard route estimates M by finite-differencing a noisy field (an ill-conditioned derivative) and then tests a discriminant (real-vs-complex) that is fragile exactly at the J_thr boundary; the SR route can instead read the type from integrated geodesic quantities, and integration is far better conditioned than differentiation. Whether that theoretical advantage survives in practice is an empirical question — the decisive one below.

Sources:

Parnell et al. 1996, Phys. Plasmas 3, 759 — null classification (spine/fan, radial/spiral).
Haynes & Parnell 2007 — trilinear null-finding.
Olshevsky et al. 2020, A&A 644, A150 — comparison of null-finding methods.
Edwards & Parnell et al. 2024 (arXiv:2410.16778) — real HMI null-point catalogue (cycle 24).

2. Candidate edge and failure modes

Candidate edge. An SR null classifier that (a) reproduces the Parnell radial/spiral/sign labels on ground-truth fields, and (b) degrades more gracefully than direct eigenvalue classification as field noise rises, because it reads the type from integrated geodesic flow rather than a finite-difference Jacobian.

Failure modes to guard against (pre-registered).

Circularity. If the SR classifier internally finite-differences M, it is the standard classifier in disguise — no contribution. The SR classifier must use only integrated flow quantities (geodesic caustics / return maps), never a numerical ∇B.
Resolution limit (known). On a raw gridded field the reach estimator under-resolves the null (Phase-2 caveat: returns Q = 5); detection reads the measured tangent cone. The gallery must state, per detection, whether Q = 6 came from the resolved grid or the local Jacobian.
Threshold fragility is the point, not a bug. Near J_thr both classifiers should struggle; the test is which struggles less, measured, not asserted.
Selection / survivorship. Report every seeded null, including ones the detector misses; no quiet dropping of hard cases.

3. Glossary and metrics (single source of truth)

Symbol / term	Definition
Null	point where `B = 0`.
M = ∇B	field Jacobian at the null; `tr M = 0` (`∇·B = 0`).
(λ₁,λ₂,λ₃)	eigenvalues of `M`, `Σλ = 0`.
radial null	all `λ` real (fan projection is an X).
spiral null	one real `λ`, two complex-conjugate (fan projection is an O).
spine / fan	eigenvector of the odd-sign-out `λ` / plane of the other two.
sign (±, A/B)	sign of the spine eigenvalue.
J∥	spine-parallel current, `= ½(spine)·(∇×B)`; drives radial→spiral at `J_thr`.
Q (growth vector homog. dim.)	SR homogeneous dimension of the tangent cone; `Q=5` in bulk field, `Q=6` at a null.
δ (nilpotent deviation)	fractional departure of the geodesic refocusing pattern from the flat (nilpotent) model; a purely integrated quantity.
χ (SR chirality)	signed asymmetry of the geodesic caustic under fan rotation — the SR read-out of `J∥`’s sign; `0` for a radial null.

Metrics.

Detection recall = (# nulls with Q=6) / (# seeded nulls), per configuration.
Classification accuracy = (# type matches vs ground truth) / (# nulls); type ∈ {radial+, radial−, spiral+, spiral−}.
Confusion matrix over the four types.
Noise-robustness curve: accuracy vs relative field noise σ = ‖noise‖ / ‖B‖_rms (added as Gaussian per-component before either classifier runs), for the SR classifier and the finite-difference-eigenvalue baseline on the same noisy fields. Report the crossover σ* (if any) beyond which SR wins.
All curves from ≥ N independent noise draws with mean ± std; state N.

4. Hypotheses (falsifiable, ordered)

H-R1 (recovery). On noiseless synthetic nulls of every type, the SR classifier’s labels match the Parnell eigenvalue labels: confusion matrix within one off-diagonal count of diagonal. Prediction: yes (tangent cone = linearization). Judged by: classification accuracy.
H-R2 (robustness — decisive). As σ rises, SR classification accuracy stays above the finite-difference-eigenvalue baseline beyond some σ* > 0. Prediction (pre-registered): SR wins for σ ≳ 0.1 near the radial/spiral boundary; may be refuted if integration’s conditioning advantage is swamped by the null-localization error. Judged by: the two accuracy-vs-σ curves and σ*.
H-R3 (grounding). On real SDO/HMI fields the detector reproduces a standard null catalogue — a gallery of ≥ several detections whose types agree with the eigenvalue scheme where the field is resolved. Judged by: per-null agreement table.

5. Ordered experiments

R1 — baseline + synthetic ground truth + first gallery. nulltopo.py (Parnell classification, golden-tested); nullfields.py (linear_null(eigs, current) with known type; charge_field for realistic emergent nulls). Run the SR growth-vector detector on a battery covering all four types + the real HMI null; confirm Q=6 for every type (detection recall); assemble the first gallery with baseline labels. (H-R1 detection half.)
R2 — SR classifier + noise head-to-head. Define χ/δ-based SR type read-out from integrated geodesics only. Confusion matrix at σ=0 (H-R1). Accuracy-vs-σ for SR vs finite-difference baseline on identical noisy fields; report σ* (H-R2, decisive).
R3 — real gallery. Several real SDO/HMI magnetograms → extrapolate → detect + classify all nulls → a real detection gallery with a per-null agreement table (H-R3).

Each experiment ends with: confirmed / refuted / partial, magnitude, the honest read of what could still be wrong, and the next hypothesis. Results land in docs/R1..R3-*.md, the gallery figures under public/img/posts/, and a series write-up.

Reproduce: ../cosmic-web/.venv/bin/python scripts/run_r1_gallery.py (numpy/scipy/matplotlib; sunpy+astropy for R3).