S2 — null pairs on the real Sun: a partial crossover, and a resolution repair

Reproduce: ../cosmic-web/.venv/bin/python scripts/run_p4_solar_pairs.py (~15 s; needs the R3 magnetograms in artifacts/hmi/). Results → artifacts/p4_solar_pairs.json, figure → public/img/posts/forbidden-directions-solar-pair.png.

Census (H-P4c, first half)

Twelve active-region volumes across the three real days hold 4 same-volume null pairs; the closest: separation 23.4 px (2012-03-07), then 30.9, 52.1, 58.7 px. Real coronal extrapolations do carry pairs — but none near-degenerate: the closest pair is well separated, two independent nulls embedded in a busy active region, not a fold-in-progress.

The crossover on the real field (H-P4c, second half): partial

The scale-resolved flux exponent $w_4(r)$ at the closest pair’s midpoint, measured on the real extrapolation’s own vector potential (trilinear-interpolated):

• $w_4 = 2.0$ at the smallest resolvable radius (locally uniform — correct);
• a clean rising edge through $\sim 3$ as the probe reaches the pair scale — the null-ness of the neighbourhood is read from a point that is not a null (the S1 mechanism, qualitatively on real data);
• no degenerate-parent plateau at 4: beyond the pair separation the probe leaves the pair’s zone of influence and the active region’s background field dominates, so $w_4$ relaxes ($\approx 2.7$ at $2.4\times$ half-separation).

Honest verdict: the fold picture’s rising edge is visible on the real Sun; the full $2\to4$ crossover is not, and should not be — a 23-px pair in a structured region is not the fold normal form. The clean signature needs a genuinely merging pair (separation of a few px, e.g. flux-emergence null creation watched in a magnetogram sequence) — recorded as the concrete next data need.

The resolution repair (upgrades the P2 caveat)

P2 reported that the raw-grid reach estimator fails at a real null (returns the bulk value) and fell back to the null’s measured Jacobian. S2 locates the actual cause: P2 probed at sub-pixel radii (0.02–0.2 px), inside the interpolation-flattened zone. Probing the same real field at super-pixel radii (1.5–10 px, above the grid cell, below the structure scale) returns

\[w_4 \approx 3\ \ (\text{mean } 2.93 \text{ in } 1.5\!-\!6\ \text{px})\]

at a real coronal null — the single-null flux weight, read directly off the raw gridded field. The detector’s scale window on gridded data is $\text{cell} \lesssim r \lesssim \text{structure}$; within it, no tangent-cone substitution is needed. (The gauge adaptation of S1 — subtracting the potential’s symmetric gradient — is required here too; without it the reading is masked at 2.)

Status

• Pair census: real, 4 pairs, closest 23.4 px.
• Real crossover: rising edge yes, full fold signature no (needs a merging pair).
• Raw-grid real-data detection: works in the correct scale window — P2’s caveat is now a scale-window rule, not a limitation.