R2 — classifying nulls under noise: integrate or differentiate?

Reproduce: ../cosmic-web/.venv/bin/python scripts/run_r2_classifier.py (≈45 min; --fast for a smoke run). Results → artifacts/r2_results.json, figure → public/img/posts/forbidden-directions-r2-noise.png. Protocol pre-registered in PROGRAM-P3-real-null-grounding.md.

Question

The standard null classification (Parnell radial/spiral ± sign) is a function of the Jacobian M = ∇B, estimated in practice by differentiating a noisy gridded field. The sub-Riemannian read-out can instead integrate: trajectories of the curvature-kernel flow (field lines), their winding, and their escape asymmetry — no derivative ever taken. Does integration classify more robustly under noise? (H-R2; H-R1 = does it classify at all.)

Protocol (fairness)

Same noisy grid instance per draw to all methods; true null location given to all; same information ball ρ = 0.5. Calibration battery (p,q) = (1.0, 0.4) fixes the winding threshold W_crit = 0.376 at σ = 0, then frozen; disjoint test battery (p,q) ∈ {(0.8,0.3), (1.2,0.5)} × j/j_thr ∈ {0, .5, .9, 1.1, 1.5, 3} × both signs, each in its own random frame (24 configs, N = 25 draws, 6 noise levels, two legs).

Methods: flow (integrated: winding + escape asymmetry + chirality) · fd-plain (central differences at the null — standard practice) · fd-lsq (least-squares linear fit over the ball — the strong baseline, ≈ the maximum-likelihood M̂ under white noise) · fd-lsq-small (same at ρ = 0.25).

Legs: clean (exactly linear field) and curved (a divergence- and current-free quadratic contaminant at 30% RMS of the linear part — a physical field the linear model cannot represent).

Results (4-class accuracy; n = 600 per point)

σ	flow	fd-plain	fd-lsq	fd-lsq-small
0.00	0.958	1.000	1.000	1.000
0.05	0.930	0.838	1.000	0.998
0.10	0.820	0.695	1.000	0.978
0.20	0.528	0.505	1.000	0.930
0.35	0.462	0.342	0.998	0.878
0.50	0.420	0.332	0.980	0.833

(clean leg; the curved leg is the same picture within error bars — see the JSON and figure.)

Verdicts

• H-R1 — CONFIRMED. At σ = 0 the integrated flow recovers the Parnell type at 0.958 (its only misses are the j/j_thr = 0.9, 1.1 cases, where the types genuinely merge). An SR-side classifier exists; the framework is not type-blind in principle.
• H-R2 — PARTIAL, and the interesting half is negative.
  • Against standard practice (pointwise finite differences): confirmed at every noise level, both legs. The integrate-don’t-differentiate mechanism is real.
  • Against the strong baseline (least-squares linear fit): refuted everywhere. fd-lsq stays ≥ 0.97 up to σ = 0.5 — even in the curved leg.

Why the strong baseline is unbeatable here (the honest anatomy)

1. Least squares is also an integral. The LSQ fit is an integral operator over the ball (≈ 4 000 nodes); white noise averages down by √4000 ≈ 63. “Integration beats differentiation” is true — and regression is integration, done with statistically optimal weights. The trajectory integral uses the same data budget less efficiently.
2. The curved leg was parity-protected (post-hoc insight, flagged as such): the quadratic contaminant is even under r → −r while the linear basis is odd, so on a symmetric fit ball the contamination is exactly orthogonal to the fit — zero bias. The first contaminant that aliases into a centred linear fit is cubic. The curved leg therefore under-tested model misspecification.
3. The winding statistic is noise-biased upward. The confusion matrix at σ = 0.2 shows the failure mode precisely: sign (escape-time asymmetry) stays near-perfect under noise, but noise-induced angular wander inflates winding, so radial nulls read as spiral. Under noise the flow classifier degrades into a good 2-class (sign) classifier — which is exactly the ~0.45 plateau observed.

The morale, stated plainly

For classifying a null the community’s local linear fit is already the right tool — statistically near-optimal — and the SR trajectory read-out cannot beat it there, only the naive pointwise practice. The SR framework’s genuine, non-redundant contribution in this domain remains what R1/P2 established: detection and order — the scale-covariant jump Q: 5 → 6 that exists before any Jacobian is estimable — plus, from P1, the gradient read-out δ = −ε². Classification is a language the framework speaks, not a tool it sharpens.

Open questions / next tests (derived, not run — no forking paths)

1. Off-centre reality. Real pipelines estimate the null position; an off-centre fit ball breaks the parity protection and biases the linear fit. Does the ranking tighten under realistic localisation error? (R3’s real-field classification is the first probe.)
2. Debiased winding. Subtracting the noise-only winding expectation (estimable from shuffled fields) should repair the radial→spiral bias; worth doing only if (1) shows a regime where the linear fit actually struggles.
3. Cubic contamination (odd parity) as the honest misspecification stress test.