S5 — The solar fold hunt: nulls colliding on the real Sun

Question. Does the real Sun show a null pair collapsing or splitting on camera — in the 32-frame, 6-min-cadence AR11429 sequence (2012-03-07 00:00–03:06 UT, through the X5.4 and X1.3 flares)?

Pre-registered hypotheses

• H-S5 (temporal fold). Somewhere in the sequence an opposite-degree null pair of the co-rotating wide-window potential extrapolation annihilates or is born: a tracked pair whose separation decreases over ≥ 3 frames below the gate, both members then vanishing away from the volume walls (or the time-mirror). Interior events must survive the window-shift test (±16 px) — the R3 window-sensitivity caveat applied to events.
• H-S5b-1 (the inhabited transition state). The fine-dedup census of the null-rich subvolume near window (33, 194) tracks ONE persistent opposite-degree pair at ~1.6 px separation and healthy height (z ≈ 8–12 px) through both flares — a stable nearly-degenerate configuration: the fold’s neighbourhood is inhabited but not crossed.
• H-S5b-2 (the fold, pinned by boundary continuation). For the candidate annihilation bracketed by the measured frames 01:07 and 01:19 UT, the straight-line blend of the two measured boundary magnetograms, cut(s) = (1−s)·cut_A + s·cut_B, exhibits a genuine fold at some s_c ∈ (0,1): separation C·√(s_c − s), opposite degrees, det ∇B → 0, and the SR 2-jet plateau at the collision. The real Sun moved from one side of the fold to the other between two measurements 12 minutes apart; the blend pins where the wall stands between them.

Protocol

1. Census: per-frame potential re-extrapolation of the co-rotating WINL = 360 px window (CUTL = 225, NZL = 64), ALL nulls per frame (Newton from a 14³ grid + warm starts at the previous frame’s nulls + a dense patch on the null-rich cluster), height filter z > 1.4 px, lateral margins 6 px.
2. Identity tracking (6 px gate, 1-frame miss allowed); every birth/death attributed: floor (submergence through the height filter), wall (window-domain artifact), interior (fold candidate).
3. Interior candidates: opposite-degree co-death/co-birth with monotone approach; window-shift robustness required.
4. S5b: fine-dedup (0.6 px) subvolume tracking of the persistent pair; boundary-blend continuation of the bracketed candidate with bisection, √ fit, degree/det checks and the SR 2-jet read (grid Poincaré gauge).

Results

H-S5: REFUTED on this sequence. The warm-started census finds 12–21 nulls per frame, but the per-frame degree sum swings between −5 and −17: births/deaths are dominated by detection-boundary effects (height-floor submergence, lateral walls, marginal low nulls flickering against the finder), and no interior opposite-degree co-death passes the pairing criteria. One relaxed candidate (approach 14.5 → 5.3 px, co-death within 2 frames at 00:55–01:13) motivated S5b-2.

H-S5b-1: REFUTED. The fine-dedup (0.6 px) subvolume census does not sustain one persistent pair: the closest opposite-degree separation churns (0.7 → 12 px frame to frame, some frames pairless), and the window-shift test reproduces the separation in 0/6 shifted windows. The 1.6-px encounters at 01:49 and 02:49 are transient configurations of a churning low cluster, not a stable transition state.

H-S5b-2: REFUTED — and the refutation is the result. The blend continuation 01:07 → 01:19 “loses” the pair at s_c = 0.426, but with every fold signature ABSENT: the separation ladder is flat at ≈ 6.2 px (log–log slope −0.015 vs the predicted 0.500), w₄(r) at the “collision” reads ≈ 2 (generic point, no plateau), the growth vector reads Q = 5 (not even a null), and neither ±16-px shifted window reproduces any s_c. The candidate was a detection artifact — the pair stops being found (one member slips below the height floor), it does not merge. The certification protocol (√ law + degree bookkeeping + SR plateau + window shifts) killed a candidate that simple track-bookkeeping would have published.

Verdict (temporal arm). On this data — windowed potential extrapolations of 6-min line-of-sight magnetograms — no fold is certified in TIME across the 3-hour sequence.

S5c — the certified fold (boundary continuation, spectral stage)

H-S5c-a/b (CONFIRMED, machine precision). Blending the first and last measured magnetograms (00:01 → 03:07 UT; endpoint censuses 9 vs 11 interior nulls) and walking newborn nulls backward with predictor–corrector continuation on the analytic mode sum of the extrapolation (the FFT extrapolation is a finite Fourier series — the grid is only its sampling; Newton on the mode sum resolves nulls to ~1e-5 px):

• fold at s_c = 0.14095804, collision at window (11.84, 209.05, z = 3.67 px);
• opposite degrees (+1, −1); separation walked from 3.4 px down to 1.2e-5 px; log–log slope 0.4999 (fold predicts 0.500) over more than four decades of δ = s − s_c; det ∇B → 0 for both members.

H-S5c-c (REFINED — the deviation is the finding). Pre-registered as “plateau-4 / Q = 7” from the S1 fold family; the catch corrected the expectation: a GENERIC fold’s collision keeps a rank-2 Jacobian (exactly one eigenvalue crosses zero — that is what det → 0 with a √ pair means), so the field still vanishes linearly in two directions and the growth vector correctly reads k = 1: Q = 6, flux exponent flat-3. The plateau-4 belongs to the fully degenerate symmetric normal form (∇B ≡ 0), which is what S1 built. What survives on real data is the collapsing crossover knee: w4(r) elbows at r ~ separation, marching to zero like √δ (measured at δ = 2e-2 → 1e-4 → 0).

H-S5c-d (REFUTED — reported plainly). The fold is NOT window-invariant: +16 px shifts push the event region out of frame (geometric); the −16 px family has different null content at the matching position (checked by sub-box census across s). This is R3’s window-sensitivity operating at event level, on a low null (z ≈ 3.7 px). Precise claim: a fold of the windowed family built from two measured magnetograms, certified to machine precision; whether the physical corona crossed THIS wall needs a global or NLFFF extrapolation.

Three impostor candidates were executed along the way by the certificate (flat ladders, w4 ≈ 2, Q = 5, no robustness) — the protocol kills what it should kill.

Reproduce

cd research/preferred-directions
../cosmic-web/.venv/bin/python scripts/run_s5_solar_fold.py   # temporal census
../cosmic-web/.venv/bin/python scripts/run_s5b_pair.py        # the impostor, executed
../cosmic-web/.venv/bin/python scripts/run_s5c_blend_fold.py  # the certified fold
../cosmic-web/.venv/bin/python scripts/render_s45_figures.py

Artifacts: s5_solar_fold.json, s5b_pair.json, s5c_blend_fold.json.

Honesty box

These are nulls of a windowed potential extrapolation of measured line-of-sight magnetograms — the same model class whose window-sensitivity R3 documented; that is exactly why the event protocol carries the window-shift test and why deaths at the height floor or lateral walls are excluded. The blend path is linear interpolation between two measurements 12 minutes apart, not the Sun’s actual trajectory through boundary-data space; what is pinned is that the two measured states lie on opposite sides of a fold wall, and where the straight path crosses it.