Where we are

Part 5 settled the division of labour: detection, order, and gradient belong to the sub-Riemannian invariants; type classification belongs to the linear fit. This part goes where a pointwise fit cannot follow at all — the unstable configurations through which magnetic topology changes — and then takes the detector to a world with genuinely non-force-free currents.

A null is a place; reconnection is an event

Everything so far treated nulls as static. But coronal nulls appear and disappear as flux emerges and cancels, and topology dictates how: in pairs of opposite sign, through a fold (saddle–node) bifurcation. Run the film of a null birth backwards: two generic nulls — one positive, one negative — drift together, merge, and vanish. At the instant of merging the field vanishes not linearly but quadratically: a degenerate null, the transition state of the field’s topology. It is structurally unstable, which is exactly why it matters — it is the configuration the field passes through when its topology changes. (Whether any reconnection — non-ideal evolution with a parallel electric field — accompanies that change is a separate, dynamical question; see the section below. The definitive statement of that separation — reconnection needs a localised non-ideal region and can proceed with no null at all — is Pontin & Priest 2022, Living Rev. Solar Phys. 19, 1.)

The standard eigenvalue toolkit sees this moment only as $\det\nabla\mathbf B \to 0$, a numerical zero test with no scale attached. The growth vector has a law for it.

The stage, rendered from real data. The actual Sun of 2012-03-07 (the X5.4-flare day): the full-disk SDO/HMI magnetogram textured on the solar sphere — the dark and bright specks are real sunspot groups — with the AR11429 potential-field extrapolation embedded at its true disk position and height scale. Blue: the arcade over the strong flux. Orange: field lines threading the detected coronal null (★); the inset card is the close-up — its spine rising, its fan sweeping away. Every curve is a field line of the real extrapolation; nothing is drawn by hand. Pipeline: research/preferred-directions/scripts/render_real_assets.py.

The corona computed, over the corona seen. The real EUV corona of AR11429 at the same hour — SDO/AIA 171 Å, million-kelvin plasma lighting up the true field lines — with our potential-field extrapolation drawn on top (blue: the arcade; orange: the null's spine and fan, ★ the null). The circled +/− mark the bipole's two magnetic poles (flux-weighted centroids of the measured surface field); arrowheads give the direction of $\mathbf B$, running + → −; the inset shows where on the full solar disk this frame sits (N up). The two datasets are independent: AIA sees plasma, HMI measured the surface field the extrapolation grew from — and cross-registering them required correcting HMI's 180° camera rotation (CROTA2) against AIA's upright frame, with the disk geometry taken from the FITS headers. The arcade is traced in the flux-centred extrapolation window (which contains the full loop system) and the null's fan in the null-centred window (which gives its lines room) — two potential-field volumes of the same magnetogram. Departures are expected and honest: a potential field carries no currents, and this region was anything but current-free (it flared X5.4 the same day).

The skeleton evolving through the detonation. Thirty-two real AIA 171 Å frames spanning 00:00–03:06 UT (6-minute cadence), and the skeleton evolves with the Sun: each frame's overlay is re-extrapolated from its own HMI magnetogram (measured at the matching minute), the arcade re-traced from the same physical footpoints across the full field of view (every loop system in frame gets its lines, both poles included), and the null re-detected and identity-tracked in a stable co-rotating domain. Watch the measured track: the null descends from 3.9 px toward the surface exactly through the X5.4 flare (h = 1.5 px at the 00:24 peak), flickers at its detection floor through the X1.3 impulsive phase (lost 01:12–01:30), and is re-found from 01:36 onwards as the region settles, riding at h ≈ 2–3.5 px to the end — 24 of 32 frames in all, within this windowed potential model at survey resolution. And the deeper honest point stands: a potential field holds no free energy, so the skeleton can only breathe with the boundary data while the EUV corona reorganises violently — that difference is the flare. The null's fan lines are traced to their full extent: born at the null in its survey volume and continued past its walls in a wider extrapolation of the same magnetogram, ending where they reach the photosphere (crisp) or the edge of the modelled region (fading). Frames exposure-normalised, per-frame FITS registration. Pipeline: fetch_aia_seq.py + fetch_hmi_seq.py + render_aia_gif; null track in artifacts/aia_gif_null_track.json. Footnote: the full cinematic sequence of this region is on YouTube — AR11429 2012-03-07, SDO/AIA 171 Å (NASA SDO footage).

The anatomy of the two real nulls. Each row is one coronal null of the AR11429 volume, drawn in the null's own frame (the visual grammar of Pontin & Priest 2022: colour = topological role): a 3D skeleton with the translucent ideal fan disc, a view down the spine where the real traced fan lines (blue) radiate from the null, and a side view with the spine (orange) vertical and the fan edge-on. Gray lines are ambient field; the open circle at the null is coloured by topological degree (blue $+1$, red $-1$); a line that sweeps in along the fan and leaves along the spine is split at the null and carries both colours. The type schematic and the measured $\nabla\mathbf B$ eigenvalues are at right. Note $J_\parallel = 0.00$ on both: a potential field is current-free, exactly as the force-free theorem demands of its radial-only nulls. (These are nulls of the survey window's extrapolation; low potential-field nulls are window-sensitive — see the R3 report's window-sensitivity check.)

The fold, and the law’s third point

The divergence-free normal form of the collision is one family:

\[\mathbf B_\mu = \Big(xz,\;\; yz,\;\; \mu - z^2 + \tfrac{x^2+y^2}{2}\Big).\]

For $\mu > 0$ it has two nulls at $(0,0,\pm\sqrt\mu)$ — a radial pair of opposite sign, the canonical creation topology. At $\mu = 0$ they merge into one isolated null where $\lvert\mathbf B\rvert \sim r^2$: field vanishing of order $k = 2$. The law $Q = d + k + 2$ then demands $Q = 7$ — a value never measured before in this program.

Measured (with an exact vector potential, so the sub-Riemannian structure is genuine): weights $(1,1,1,4)$, $Q = 7$ at the fold point; $(1,1,1,3)$, $Q = 6$ at each split null. The law now stands tested at $k = 0, 1, 2$ in three dimensions. The transition state of magnetic topology has its own integer.

Reading a collision from one point

Here is the observable nothing pointwise can imitate. Stand at the midpoint of a split pair — not at either null; the field there is nonzero — and measure the local flux exponent $w_4(r)$: how the reachable holonomy scales with probe radius $r$.

The crossover, and its collapse. A: at the pair midpoint the exponent sits at 2 (locally uniform field) until the probe radius reaches the pair, then climbs to 4 (the degenerate parent); the knee marches with the half-separation $\sqrt\mu$ (dotted verticals). The $\mu=0$ control (orange) is flat at 4 across two decades. B: plotted against $r/\sqrt\mu$, all four separations lie on one universal curve — the dilation symmetry of the geometry. C: centred on a member null: plateau 3 (a single generic null), rising to 4 when the probe swallows the partner. Pipeline: research/preferred-directions/scripts/run_p4_fold.py.

Read the three plateaus as a dictionary: $w_4 = 2$ — you are in ordinary field; $3$ — a generic null is at hand; $4$ — a degenerate structure. And the knee of the curve reads the separation of a pair the probe never resolves individually. A pointwise method must distinguish two zeros to know there are two; this reads their distance from one point’s scaling. It is the first observable in this series with no standard analogue — and one honest catch came with it: the vector potential’s symmetric gradient at the probe point contributes curl-free junk at $r^2$ that masks the physics. It is the gradient of an exact form, so it subtracts off cleanly at the endpoint; the estimator now does this in general, and every earlier result was re-verified unchanged.

No analogue, however, does not mean no competitor — and we raced it, pre-registered, against the obvious statistical rival: fit a divergence-free quadratic field model to the same noisy grid and root-find it. The fit wins every cell, by one to four orders of magnitude, in a clean and a contaminated leg alike — so for quantitative pair metrology, fit and root-find; the crossover stands as the concept (and $Q=7$ at the fold as intrinsic geometry), not the estimator. That is the same division of labour Part 5 found for classification, now measured twice.

On the real Sun: a census, a partial, and a repair

Do real coronal fields carry such pairs? Scanning twelve active-region volumes across the three real days: four same-volume pairs, the closest at 23.4 px (2012-03-07). At its midpoint, on the real extrapolation’s own vector potential:

The crossover on the real Sun — honestly partial. The rising edge is there: $w_4$ starts at 2 and climbs through $\sim3$ as the probe reaches the pair scale. The degenerate plateau at 4 is not — correctly: a 23-px pair embedded in a busy active region is two independent nulls, not a fold in progress, and beyond the pair scale the background field takes over. The clean signature awaits a genuinely merging pair, i.e. a flux-emergence magnetogram sequence. Pipeline: research/preferred-directions/scripts/run_p4_solar_pairs.py.

The same measurement paid an unexpected dividend. Part 4 reported that the raw gridded field could not resolve the null jump and fell back to the measured Jacobian. The pair study shows the failure was the probe window, not the grid: probed at sub-pixel radii the interpolation has flattened everything, but in the window above the grid cell and below the surrounding structure (1.5–10 px here), the raw real field returns the null flux weight $w_4 \approx 3$ directly. On gridded data the detector is not resolution-limited; it is scale-windowed — choose the window consciously and it works on the data as it comes.

The hunt, part one: does the Sun fold on camera?

The pair census above is static — one extrapolation per day. The flare sequence hands us the other axis: thirty-two re-extrapolations of the same co-rotating volume, six minutes apart, through two X-class flares. If a null pair collapsed or was born anywhere in those three hours, a census should catch it — and topology says exactly what to demand. Away from the volume’s boundaries the total topological degree (the sum of $\mathrm{sign}\det\nabla\mathbf B$ over all nulls) cannot change smoothly: the only interior event a divergence-free family allows is a fold — two nulls of opposite degree approaching, merging through one degenerate point, and vanishing (or the mirror, a pair born). So the hunt is a bookkeeping protocol: track every null, attribute every birth and death (photospheric floor, lateral wall, or interior), and demand of any interior candidate all four fold signatures at once — opposite degrees, separation $\propto\sqrt{\lvert\lambda-\lambda_c\rvert}$, $\det\nabla\mathbf B \to 0$, and the $w_4 \to 4$ plateau at the collision.

The solar hunt, and the impostor it caught. A: the per-frame census of the AR11429 wide volume (SDO/HMI, 2012-03-07 00:00–03:06 UT, 6-min cadence; shaded bands = the X5.4 and X1.3 flares): 12–21 nulls per frame, and a degree sum that swings between $-5$ and $-17$ — the churn of marginal, low-lying nulls of a windowed potential extrapolation flickering against the finder, not interior topology change. B: the one candidate that survived the first cut (an opposite-degree pair approaching $14.5 \to 5.3$ px and co-dying between the 01:07 and 01:19 magnetograms) interrogated by boundary continuation: blend the two measured magnetograms, $\mathrm{cut}(s) = (1-s)\,\mathrm{cut}_A + s\,\mathrm{cut}_B$, and track the pair in $s$. A genuine fold must close like $\sqrt{s_c - s}$ (gray dashed); the candidate stays flat at $\approx 6.2$ px (fitted slope $-0.015$), the "collision" point reads $w_4 \approx 2$ and growth vector $Q = 5$ — a generic point, not even a null — and neither $\pm16$-px shifted window reproduces it. The pair was never merging; the finder was losing a marginal null. (Counts here are Newton-finder censuses — dense seeding, dedup — not exhaustive cell censuses; the Haynes–Parnell-grade upgrade is specified in the G1 charter.) Pipeline: scripts/run_s5_solar_fold.py + scripts/run_s5b_pair.py; census and blend in artifacts/s5_solar_fold.json, s5b_pair.json.

The verdict is honestly negative — no certified fold at 6-minute cadence — and the refutation is the valuable part. The protocol worked: a candidate that naive track bookkeeping would have published was executed by its own physics — flat separation where a fold demands a square root, $w_4 \approx 2$ where it demands the plateau, and no window robustness. But time is not the only dial the measured Sun offers.

The hunt, part two: the collider — a fold in the boundary blend, certified

The census at 00:01 UT holds one null population; the census at 03:07 UT holds a different one. Between those two measured boundaries stretches a continuous family: blend the first and last magnetograms, $\mathrm{cut}(s) = (1-s)\,\mathrm{cut}{00:01} + s\,\mathrm{cut}{03:07}$, and extrapolate each blend. The Sun itself moved from one endpoint to the other in three hours — along some path in boundary-data space — and since the interior null count differs between the endpoints, every continuous path between them either crosses fold walls or passes a null through the window boundary (in a windowed model the count can change both ways, so the fold is not forced — it must be found). The straight path is the one we can walk with a microscope, and on it the change happens by an interior collision: the certified fold below.

And the microscope here is unreasonably good. The potential extrapolation is a finite sum of Fourier modes — an analytic function of position of which the grid is merely a sampling — so Newton’s method on the mode sum locates nulls at machine precision, arbitrarily far below the pixel. A pair whose entire approach happens inside one grid cell (invisible to any gridded finder) is fully resolvable. With that: track the census along $s$, walk every newborn null backward to where it dies, recover its opposite-degree twin along the fold axis (the kernel of $\nabla\mathbf B$), and drive the pair into its collision with adaptive continuation.

Two coronal nulls collapse into one another — three of the four demanded signatures pass; the fourth (window robustness) fails and is reported below. The certified fold of the AR11429 boundary blend at $s_c = 0.14095804$ (window position $x \approx 11.8$, $y \approx 209$, height $z \approx 3.7$ px). A: the two nulls' paths (blue: degree $+1$, orange: degree $-1$; colour = blend $s$), walked with predictor–corrector continuation into the collision (★). B: the pair separation against $\delta = s - s_c$ on log–log axes: the fitted slope is 0.4999 against the fold's exact $\tfrac12$, holding over more than four decades down to a separation of $10^{-5}$ px — twenty-thousandth of a pixel, courtesy of the spectral microscope; both members' $\det\nabla\mathbf B \to 0$ alongside (orange triangles, scaled). C: the SR read: the $w_4(r)$ crossover knee marches to zero with the shrinking pair, and at the collision the curve settles on the generic-null plateau 3 ($Q = 6$) — not the plateau 4 ($Q = 7$) of this post's symmetric fold family. That deviation is a finding, not a failure: see below. Pipeline: scripts/run_s5c_blend_fold.py → artifacts/s5c_blend_fold.json.

The collision, in field lines. Field lines of the blended extrapolation traced in the fold plane (horizontal axis = the fold axis, the kernel direction of $\nabla\mathbf B$ at the collision; the same real AR11429 volume as the flare sequence). A: at $s_c + 0.05$ the two nulls (★, degrees labelled) sit $\approx 3$ px apart, each with its own X-type line structure. B: at $s_c + 0.004$ they nearly touch. C: at $s_c$ one degenerate point (gold ★) remains — below $s_c$ the field is null-free here: the pair has annihilated. This is the fold sequence of the normal form above, drawn by measured solar boundary data.

The catch also refined the law it was sent to test. The symmetric fold family $\mathbf B_\mu$ predicts plateau-4 and $Q = 7$ at the collision — but that family’s degenerate point has $\nabla\mathbf B = 0$ entirely, a symmetric normal form. A generic fold is milder: at the collision the Jacobian keeps rank 2 — exactly one eigenvalue crosses zero (that is what $\det \to 0$ with a $\sqrt{\delta}$ pair means) — so the field still vanishes linearly in two directions, and the growth vector correctly answers $k = 1$: $Q = 6$, flux exponent 3. (Stated with hypotheses and proved as Corollary 5.1 of the companion article: the homogeneous dimension is blind to any generic fold, because $k \ge 2$ would need the entire Jacobian to vanish — codimension 8 in the trace-free space $\nabla\cdot\mathbf B = 0$ leaves it, not the fold’s codimension 1.) The plateau-4 belongs to the fully degenerate normal form; what survives contact with real data is the collapsing knee — the $w_4(r)$ crossover whose elbow marches to zero like $\sqrt{s - s_c}$, visible in panel C above. The honesty boxes, filled: the two flanking window shifts do not reproduce this particular fold ($+16$ px pushes the event region out of frame; $-16$ px has different null content at the matching position — window-sensitivity operating at event level, on a low null at $z \approx 3.7$ px, exactly the fragile class Part 5 documented). So the precise claim is: a fold of the windowed family built from two measured magnetograms, certified to machine precision — the wall between the 00:01 and 03:07 topologies is real in that family, and its location is pinned; whether the physical corona crossed this particular wall needs the global-extrapolation upgrade the temporal hunt already ordered.

New worlds: where the spiral nulls actually live

Part 5 proved a door shut: in any force-free field the current $\mathbf J = \alpha\mathbf B$ vanishes wherever $\mathbf B$ does, so $\nabla\mathbf B$ is symmetric at every null — no smooth force-free extrapolation with bounded $\alpha$ can host a spiral null (non-force-free and data-driven MHD extrapolations are not covered by the theorem). The contrapositive says where to look: a place with genuinely non-force-free currents. The nearest one is over our heads.

The Earth’s magnetosphere carries real current systems — magnetopause, tail, ring — and empirical field models (internal IGRF plus the Tsyganenko T96 external field) encode them, fitted to decades of spacecraft data. On our solar storm day (2012-03-07, southward IMF, $D_{st}=-50$ nT), Newton descent from cusp and tail seeds finds 149 magnetic nulls — and 79 of them are spiral, sitting exactly where the theorem permits them. The growth vector returns $Q = 6$ at every one, radial and spiral alike.

The bifurcation hunt (below) forced a sharper audit of this census, and the result is worth stating precisely. Testing every null against the T96 model’s own magnetopause (the boundary function shipped with the model): the entire population sits in a thin shell, under one Earth radius outside the model boundary — and inside its valid domain the model has no nulls at all, at any IMF $B_z$ we probed. The smooth average magnetosphere of an empirical model is null-free; the shell nulls are mathematical structure of the field’s continuation past its own edge (which is also why they barely respond to the ring-current index: $\lvert\Delta\mathbf B\rvert \le 0.03$ nT for a 30-nT $D_{st}$ swing). That is physically sensible — real magnetospheric nulls are dynamical creatures of the reconnecting current sheet, which is exactly why the in-situ missions hunt them during events rather than in quiet averages — and it upgrades the honest reading of the two figures below: they are anatomy of spiral and radial null structure in a realistically shaped, genuinely non-force-free field — not spacecraft-confirmed standing nulls of the magnetosphere. The MMS cross-match stays on the books.

The fourth world. Noon–midnight cut of $\lvert\mathbf B\rvert$ (log scale) in the IGRF + Tsyganenko T96 field for 2012-03-07 storm conditions: the compressed dayside, the magnetopause, and the dark ribbon of the tail current sheet — the genuinely non-force-free structure the Sun's *force-free* extrapolations cannot have (non-force-free and data-driven MHD extrapolations are outside that theorem). The null population sits off this plane, at the flank/lobe boundary (next figure). Pipeline: research/preferred-directions/scripts/run_p4_magnetosphere.py.

The null constellation — continuation-only structure, outside T96's own validity domain (the boundary audit above places every one of these roots outside the model's magnetopause; they are anatomy of the model's continuation, not standing magnetospheric nulls). The census in two projections (ecliptic and noon–midnight; 24 far-tail nulls beyond $\lvert x\rvert = 40\,R_E$ omitted). Spiral nulls (orange) trace the nightside flank current regions and the high-latitude lobe boundary; radial nulls (blue) line the plasma-sheet flanks. Honest caveats: these are nulls of an empirical model fitted to data — not directly measured zeros — the population sits where the model's current closure is least constrained, and no dayside cusp nulls converge in T96's soft magnetopause. Cross-matching against in-situ Cluster/MMS null catalogues is the recorded next step.

The two species, dissected. The same null-frame anatomy (3D skeleton, down-the-spine, side view; colour = topological role as in the coronal sheet) for two core-census magnetospheric nulls. Top: a radial null — real eigenvalues, $J_\parallel \approx 0$; note how much of the local field is honest gray ambient: a T96 null is weak and buried in its surroundings. Bottom: a spiral null — complex fan pair and $J_\parallel = -7$: the field-aligned current the force-free theorem requires. Its winding is so rapid ($\lvert\mathrm{Im}/\mathrm{Re}\rvert \approx 20$ — many turns per e-fold of radius) that the fan renders as a tight in-plane bundle rather than a visible corkscrew — the dense tube is the visual signature of a fast spiral.

And the worlds beyond, honestly scoped: planetary internal fields (Jupiter’s included — Part 8 hunts its envelope and finds the polar patch dome-covered) are vacuum fields — curl-free, so the theorem allows only radial nulls there; magnetars have no observed field maps to detect anything in; and black-hole magnetospheres are general-relativistic objects whose flux lift must first be rebuilt on curved spacetime. That last one is a genuine frontier, not an afternoon.

The hunt at Earth: stability, and the wall the collider maps

Could the collision be staged at Earth too? The audit above answers for T96: its valid interior owns no nulls, so there is nothing to collide. But the classical outer magnetosphere — the Chapman–Ferraro/Dungey vacuum superposition of the planet’s real internal field (IGRF) and a uniform interplanetary field, here 5 nT — genuinely owns the textbook pair: two polar neutral points at $r \approx (2B_0/B_{sw})^{1/3} \approx 18\text{–}23\,R_E$. Rotating the IMF direction $\theta$ from northward to southward is the natural dial, and the census tells a two-act story. For $125°$ of rotation the two nulls just migrate — the configuration is structurally stable, the census flat (sampled every $5°$ with a fixed seed battery; “stable” here means no event at that resolution). Then, approaching the anti-parallel orientation, the count cascades: $2 \to 3 \to 6 \to 17$. The pure-dipole textbook says why — and we ran the control rather than citing it: replace IGRF by the same epoch’s pure dipole and set the IMF exactly anti-parallel, and $\lvert\mathbf B\rvert$ stays below $0.007$ nT around the entire circle of azimuths at $r^\ast = 18.15\,R_E$ — a degenerate null ring to numerical precision, the third face of the fold family from the top of this post ($\mu < 0$). Restore the real multipoles and the same circle is modulated eleven-fold (up to $0.078$ nT), its zeros surviving only at isolated azimuths: the ring breaks into the swarm (scripts/run_s4d_dipole_control.py). Measured up close, the swarm’s nulls live in a $0.02$-nT-flat azimuthal valley with Jacobian condition numbers near $10^2$: null identity itself dissolves there, folds merging into the broken ring rather than standing cleanly apart. The certified fold belongs to the Sun; Earth’s contribution is the phase portrait — where nulls are stable, and which walls they die on.

The stability map. A: every null state as a point in the $(\widehat{\det\nabla\mathbf B},\, \widehat{\mathrm{disc}})$ plane — normalised Jacobian determinant (its sign is the topological degree) against the normalised fan-eigenvalue discriminant (positive = radial, negative = spiral). Pale blue: the 135 states of the Dungey $\theta$-census (a vacuum field, so the type wall is untouchable — all radial, by Part 5's theorem). Dark blue/orange: the certified solar pair of the collider figure, whose two branches dive into the fold wall $\det = 0$ from opposite sides and annihilate — the only way a null count can change in the interior of a smooth divergence-free family. The type wall $\mathrm{disc} = 0$ is the other codimension-1 crossing (radial ↔ spiral conversion); crossing it needs the field-aligned current the force-free theorem demands. Stability = distance from the walls. B: the Dungey census versus IMF angle $\theta$: the classical cusp pair is structurally stable over $125°$ of IMF rotation (at $5°$ sampling), then the anti-parallel ring degeneracy (the fold family's $\mu<0$ face) breaks up into a cascade — real multipoles shattering a textbook symmetric bifurcation into a swarm. The ring is measured, not cited: the pure-dipole control holds $\lvert\mathbf B\rvert < 0.007$ nT around the whole $r^\ast = 18.15\,R_E$ circle; IGRF modulates it $11\times$. Degree-sum wobbles inside the shaded band are census resolution in the near-degenerate valley, not boundary flux. Pipeline: scripts/run_s4c_dungey.py, run_s4b_drive.py (the T96 boundary audit) → artifacts/s4_collider.json.

Where the series now stands

Question about a null	Right tool	Status
Is one here?	$Q\colon 5\to6$	grounded on 3 worlds: the synthetic battery, the Sun’s extrapolations, and the analytic Dungey/vacuum magnetosphere model (the empirical T96 census is retracted — continuation-only)
What order? Is it a transition state?	$Q = k+5$: generic 6; symmetric fold 7; generic (rank-2) fold reads 6 + collapsing knee	law tested at $k = 0,1,2$; refined by the certified collision
An unresolved pair’s separation?	concept: the $w_4(r)$ knee · tool: a polynomial fit + roots	knee confirmed synthetically; the fit won the pre-registered race (Part 7)
When do nulls appear/disappear?	the fold certificate: opposite degrees + $C\sqrt{\lvert\lambda-\lambda_c\rvert}$ + $\det\to 0$ + knee collapse	one collision certified (boundary blend, slope 0.4999); protocol also kills impostors
How is the field changing nearby?	period average: $\delta = 1-\tfrac2\pi K(2\varepsilon)$ exactly (exponential profile); the caustic agrees with it numerically ($10^{-8}$) there, identification formally open	exact period-average law; caustic status separate (Part 7)
Radial or spiral, which sign?	the local linear fit	stays with the standard toolkit (Part 5)

Glossary

• Fold (saddle–node) bifurcation — the generic way nulls are created/destroyed: a pair of opposite-sign nulls merging through one degenerate null.
• Degenerate null — $\mathbf B = 0$ with $\det\nabla\mathbf B = 0$; here the fully quadratic point ($\lvert\mathbf B\rvert \sim r^2$, $k=2$), giving $Q = 7$.
• $w_4(r)$ — scale-resolved flux exponent: the local log–log slope of the reachable holonomy versus probe radius; plateaus 2 / 3 / 4 = uniform / generic null / degenerate.
• Scale window — on gridded data, the probe range (cell size $\lesssim r \lesssim$ structure scale) in which reach exponents are meaningful.
• GSM / $R_E$ — geocentric solar-magnetospheric coordinates ($x$ to the Sun); Earth radii.
• T96 — Tsyganenko 1996 empirical external-field model (magnetopause, tail, ring currents), parameterised by solar-wind pressure, $D_{st}$, and IMF; here $(3\,\text{nPa}, -50\,\text{nT}, B_z = -5\,\text{nT})$.
• Topological degree — $\mathrm{sign}\det\nabla\mathbf B$ at a null ($\pm1$). Its sum over a region is conserved under smooth deformation until a null crosses the boundary or a fold happens inside: folds create/destroy $(+1,-1)$ pairs only.
• Boundary blend — the straight path $\mathrm{cut}(s) = (1-s)\,\mathrm{cut}_A + s\,\mathrm{cut}_B$ between two measured magnetograms; extrapolating each blend gives a smooth 1-parameter family of coronal fields connecting two measured states.
• Fold certificate — the four signatures demanded of any claimed pair creation/annihilation: opposite degrees; separation $= C\sqrt{\lvert\lambda - \lambda_c\rvert}$; $\det\nabla\mathbf B \to 0$ for both members; the $w_4$ knee collapsing to zero. Candidates failing any one are impostors (three died that way in this post).
• Pre-registered (as used in this series) — the hypothesis and its falsification criteria are written into the experiment’s docstring and committed before the run, in the same repository; there is no external timestamped registry. The practice has teeth (several pre-registered claims lost and the losses are published), but the term is used in this weaker, self-registered sense.
• Spectral (mode-sum) Newton — locating nulls by Newton’s method on the analytic Fourier-mode sum of the extrapolation rather than its sampled grid; resolves pairs to $\sim 10^{-5}$ px, far below the grid cell.
• Ring face / cascade — at exactly anti-parallel dipole + uniform field the null set degenerates to a circle (the fold family’s $\mu<0$ face); real multipoles break it into a swarm of near-degenerate nulls with dissolving identities.
• Structural stability / walls — a null configuration is stable while its $(\det, \mathrm{disc})$ state stays off the two codimension-1 walls: $\det = 0$ (fold: the count changes) and $\mathrm{disc} = 0$ (type: radial ↔ spiral).

Reproduce

cd research/preferred-directions
../cosmic-web/.venv/bin/python scripts/run_p4_fold.py           # Q=7 + the crossover
../cosmic-web/.venv/bin/python scripts/run_p4_solar_pairs.py    # real pair census
../cosmic-web/.venv/bin/python scripts/run_p4_magnetosphere.py  # the 149-null census
../cosmic-web/.venv/bin/python scripts/render_real_assets.py    # the real-data renderings
# the bifurcation hunt
../cosmic-web/.venv/bin/python scripts/run_s5_solar_fold.py     # temporal census (32 frames)
../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/run_s4b_drive.py         # T96 boundary audit
../cosmic-web/.venv/bin/python scripts/run_s4c_dungey.py        # Dungey cascade census
../cosmic-web/.venv/bin/python scripts/run_s4d_dipole_control.py # the ring, measured
../cosmic-web/.venv/bin/python scripts/render_s45_figures.py    # the four hunt figures

Environment of record: sunpy 8.0.0, geopack 1.0.13, scipy 1.18.0, numpy 2.5.0, astropy 8.0.0 (the cosmic-web venv).

Charter and reports: docs/PROGRAM-P4-bifurcations-new-worlds.md, S1-fold-crossover.md, S2-solar-pairs.md, S3-new-worlds.md, S4-null-collider.md, S5-solar-fold-hunt.md.