Where we are

Part 6 ended with three candidate pieces of new science. A claim is not science until it has survived the attacks a referee would mount — so this part mounts them ourselves: a literature audit, a pre-registered race (in the series' self-registered sense — hypotheses and losing conditions committed to the repo before the run; no external registry) against the strongest statistical rival, a precision derivation, and a real-data hunt. Everything below, including the losses, was decided by rules written down before the experiments ran.

The audit, first

Candidate	Literature check	Outcome
No force-free field hosts a spiral null	the MHD-relaxation literature (Fuentes-Fernández & Parnell) already knew this dynamically	downgraded: our 3-line pointwise proof is a clarification; what remains ours is the audit corollary, correctly scoped — a spiral null in any smooth force-free (bounded-α) extrapolation catalogue is a numerical force-free violation (non-force-free and data-driven MHD extrapolations are outside the theorem)
The gyro-caustic series $\delta(\varepsilon)$	guiding-centre theory has the machinery (Littlejohn, Brizard), not the statistic	survives → hardened below
The scale-crossover pair read-out	null detection is pointwise; multifractal “local dimension” is a different object	survives the library — so we raced it against the strongest rival we could build

The claim that hardened: the refocusing average is an elliptic integral

Part 3 measured how a magnetic-field gradient delays the refocusing of Larmor orbits on the exponential profile (general one-dimensional profiles carry the $1-\tfrac34\beta$ calibration, and past $\beta = \tfrac43$ the leading-order effect flips sign): $\delta = -\varepsilon^2 - c_4\varepsilon^4 - \cdots$ with $\varepsilon = \lvert\nabla\ln B\rvert\,r_L$. Chasing $c_4$ to precision ($2.2497 \pm 0.0009 = 9/4$, refuting the $5/2$ a coarse fit once suggested) exposed a coefficient pattern — squared normalised central binomials — that is the fingerprint of one classical function. The whole series collapses into two symbols:

\[\boxed{\;\delta(\varepsilon) \;=\; 1 - \frac{2}{\pi}\,K(2\varepsilon)\;}\]

with $K$ the complete elliptic integral of the first kind (modulus convention). Nothing here is asymptotic hand-waving; every link is checked at referee grade:

• The mechanism is a one-line integral of motion. Along a geodesic, $\dot\theta = e^{\varepsilon x}$ and $\dot x = \cos\theta$ give $d\dot\theta/d\theta = \varepsilon\cos\theta$, hence $\dot\theta = 1 + \varepsilon(\sin\theta - \sin\theta_0)$ — the angular dynamics is integrable, a pendulum in disguise.
• Per launch angle, the refocusing time equals the $\theta$-period $2\pi\big[(1-\varepsilon\sin\theta_0)^2-\varepsilon^2\big]^{-1/2}$ — an identification verified against the Jacobian-zero conjugate times of the geodesic integrator to $10^{-8}$ relative, every angle, up to $\varepsilon = 0.45$; formally open (and it fails off the exponential profile), so every “refocusing” reading of the period formulas below is conditional on it.
• The launch-angle average equals $\tfrac2\pi K(2\varepsilon)$ — verified symbolically term-by-term through $O(\varepsilon^{12})$ (computer algebra) and numerically to $10^{-10}$ absolute through the full measurement pipeline.

The exact period-average law, against the caustic measurement. A: the mean refocusing delay $-\delta$ (log scale) measured by the geodesic code from conjugate times (dots) riding the exact mean-period curve $\tfrac2\pi K(2\varepsilon)-1$ (line) — the agreement across four decades is the numerical content of the period identification — up to the critical gradient $\varepsilon = 1/2$ where $K$ diverges. B: the per-angle measured conjugate time against the exact $\theta$-period formula at three gradients — the peak at $\theta_0 = \pi/2$ (launch along the gradient) grows toward divergence. Pipeline: research/preferred-directions/scripts/run_p5_series.py, render_closed_form.py.

The divergence is physics, not pathology. As $\varepsilon \to 1/2$ the mean $\theta$-period diverges — that is the theorem — and the measured conjugate times ride it: the slowest launch direction ($\sin\theta_0 = 1$) has period $T = 2\pi/\sqrt{1-2\varepsilon}$, and its measured $t_c$ matches at $\varepsilon = 0.45$ and $0.49$. Above the critical gradient the picture is sharper than a blanket no-refocusing claim (a referee-prompted probe measured it): for exactly the launch band $\sin\theta_0 \ge (1-\varepsilon)/\varepsilon$ no conjugate point is detected within the integration window of eight reference periods — 9 of 64 angles at $\varepsilon = 0.52$, 13 of 64 at $0.55$ — while every other angle keeps one, still equal to its period to $5\times10^{-5}$ (run_r6_supercritical.py; finite-horizon evidence, not a proof those orbits never refocus later). When the field changes by more than half across a Larmor radius: the mean period diverges — a theorem; the slowest orbits stop refocusing on every horizon we integrated — a finite-horizon measurement; and that the launch-averaged caustic opens with them is exactly as strong as the period identification. A critical gradient, delivered by a textbook special function. (The two steps a referee should still demand, stated plainly: a formal proof that the conjugate time equals the period for this integrable family — our evidence is $10^{-8}$ numerics — and one final literature pass on gyro-period integrals in exponential field profiles.)

The claim that lost: the crossover, demoted by its own race

Part 6’s scale crossover reads an unresolved null pair’s separation from one point — an observable with no analogue in the null-detection literature. No analogue does not mean no competitor. The obvious statistical rival: fit a divergence-free quadratic field to the same noisy grid and root-find it. We pre-registered the protocol and the demotion rule, then ran it: same gridded field to both sides, the crossover building its own potential from the data, a clean leg and a contaminated leg, three separations, three noise levels.

The race, lost. Median relative error of the estimated pair separation (log scale) versus noise, for the scale crossover (orange) and the divergence-free quadratic fit + root-finding (blue). The fit wins every cell by one to four orders of magnitude — even in the contaminated leg, because a smooth background is precisely what a polynomial fit absorbs. Per the pre-registered rule, the crossover is demoted from candidate tool to conceptual observable. Pipeline: research/preferred-directions/scripts/run_p5_race.py.

What survives the demotion, stated precisely: $Q = 7$ at the symmetric fold’s degenerate point ($\nabla\mathbf B \equiv 0$ normal form) is a statement about the field’s intrinsic geometry, not an estimate in competition — with Part 6’s correction attached: a generic rank-2 fold keeps a nonzero 1-jet and reads $Q = 6$; the 2/3/4 plateau dictionary remains the correct meaning of uniform/null/degenerate; and the dilation collapse remains the right mental model of what an unresolved pair is. But for measuring a separation on data, fit and root-find. This is the second time the series has measured this moral — classification fell to the linear fit in Part 5 the same way.

The hunt that came back empty: a fold on the real Sun

The transition state’s cleanest real-data signature would be a null pair being born in a flux-emergence region. We fetched eight real HMI frames tracking AR11158 — the textbook emergence region — from its birth through its X2.2 flare (2011-02-13 → 02-15), and ran the census on every frame.

A region builds its topology. AR11158 emerging (real SDO/HMI, 2011-02-13 → 02-15 through the X2.2 flare; red/blue = photospheric polarity; ★ = detected coronal nulls of the potential extrapolation). The census reads 0,0,0,(1),0,1,0,2: a young bipole is a simple arcade with no coronal nulls at all, and nulls appear as the region builds the multipolar structure that makes it flare-capable. Pipeline: research/preferred-directions/scripts/run_p5_sequence.py.

The honest verdict: no fold was caught. The final pair is same-signed, and fold-born pairs are necessarily opposite-signed (topological degree conservation) — so these two nulls arose independently as the region’s complexity grew. What the census does deliver is a real-data observation worth keeping as a candidate: the coronal null count as a topological complexity indicator of an emerging region, zero while it is a simple arcade. Candidate, not established — eight frames of a window-sensitive extrapolation are too sparse and too model-dependent to establish an index or any relation to flare capability; the G1 protocol (hourly cadence, tracked windows, census-grade counts, ensembles) is the test it must pass. Catching a birth needs hourly cadence, a co-moving window, and null identity tracking — specified for the follow-up, not claimed. (One methodological save worth confessing: the first tracking pass left the region in the window’s corner, and its quiet-Sun nulls briefly impersonated an “annihilation candidate” — caught by looking at the picture, fixed by recentring, discarded.)

What stands after review

Claim	Status after the litmus tests
$Q = d+k+2$, tested $k = 0,1,2$ in 3D; $Q=7$ at the symmetric fold, $Q=6$ at a generic rank-2 fold	stands as corrected by Part 6’s certified collision — intrinsic geometry, no estimation rival
$\delta = 1 - \tfrac2\pi K(2\varepsilon)$ (period average); critical gradient $\varepsilon = 1/2$	proven as a period-average identity; its caustic reading verified at $10^{-8}$–$10^{-10}$, formal step still open
Force-free ⇒ radial-only; magnetotail ⇒ 79 spirals	theorem stands; the census half is retracted — Part 6’s boundary audit puts all 149 T96 nulls outside the model’s own magnetopause, so no valid interior spiral example survives
Crossover as pair-metrology tool	demoted by pre-registered race; survives as concept
Type classification, pair metrology	belong to statistical fits — measured twice
Fold birth on the real Sun	not caught; requirements specified

The series’ deepest result is the shape of this table: the sub-Riemannian framing’s durable contributions are laws, exact structure, and dictionaries — things with no estimation competitor — while every head-to-head against a matched statistical estimator was lost and reported. That is what distinguishes a research program from an advertisement, and it is the standard the next phase (the moduli question, the hourly fold hunt, the in-situ cross-match) will be held to.

Glossary

• $K(k)$ — complete elliptic integral of the first kind, $\int_0^{\pi/2} d\phi/\sqrt{1-k^2\sin^2\phi}$ (modulus convention).
• Critical gradient — $\varepsilon = 1/2$: the value of $\lvert\nabla\ln B\rvert\,r_L$ at which the mean $\theta$-period diverges (theorem) and the slowest launch angle loses its finite period; its caustic reading — the mean refocusing time diverging — is conditional on the period identification (above it, measured: only the band $\sin\theta_0 \ge (1-\varepsilon)/\varepsilon$ shows no conjugate point within the finite integration window; the rest keep $t_c = T$).
• Pre-registered rule — the verdict criterion written into the phase charter before the experiment ran; the race’s demotion rule is one.
• Same-signed pair — two nulls whose spine eigenvalues share a sign; cannot be fold-born (degree conservation), hence the emergence verdict.

Reproduce

cd research/preferred-directions
../cosmic-web/.venv/bin/python scripts/run_p5_series.py       # the period-average law, 4 checks
../cosmic-web/.venv/bin/python scripts/render_closed_form.py  # its figure
../cosmic-web/.venv/bin/python scripts/run_r6_supercritical.py # above the critical gradient
../cosmic-web/.venv/bin/python scripts/run_p5_race.py         # the race (~75 min)
../cosmic-web/.venv/bin/python scripts/fetch_sequence.py      # 8 real HMI frames
../cosmic-web/.venv/bin/python scripts/run_p5_sequence.py     # the emergence census

Charter and reports: docs/PROGRAM-P5-litmus.md, T1-race.md, T2-closed-form.md, T3-emergence.md.